Contents
Chapter 1
Whole Numbers
1.1
Introduction to Whole Numbers
1
1.2
Addition of Whole Numbers and Perimeter
3
1.3
Subtraction of Whole Numbers
9
1.4
Rounding and Estimating
14
1.5
Multiplication of Whole Numbers and Area
16
1.6
Division of Whole Numbers
22
Problem Recognition Exercises: Operations on Whole Numbers
29
1.7
Exponents, Square Roots, and the Order of Operations
31
1.8
Problem-Solving Strategies
34
Chapter 1 Review Exercises
41
Chapter 1 Test
47
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
2.1
Introduction to Fractions and Mixed Numbers
49
2.2
Prime Numbers and Factorization
53
2.3
Simplifying Fractions to Lowest Terms
56
2.4
Multiplication of Fractions and Applications
60
2.5
Division of Fractions and Applications
65
Problem Recognition Exercises: Multiplication and Division of Fractions
70
Multiplication and Division of Mixed Numbers
72
Chapter 2 Review Exercises
77
Chapter 2 Test
81
Chapters 1 – 2 Cumulative Review Exercises
84
2.6
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
3.1
Addition and Subtraction of Like Fractions
86
3.2
Least Common Multiple
90
3.3
Addition and Subtraction of Unlike Fractions
95
3.4
Addition and Subtraction of Mixed Numbers
101
Problem Recognition Exercises: Operations on Fractions and Mixed Numbers
107
Order of Operations and Applications of Fractions and Mixed Numbers
109
Chapter 3 Review Exercises
115
Chapter 3 Test
120
3.5
i
Chapters 1 – 3 Cumulative Review Exercises
Chapter 4
121
Decimals
4.1
Decimal Notation and Rounding
124
4.2
Addition and Subtraction of Decimals
126
4.3
Multiplication of Decimals
131
4.4
Division of Decimals
135
Problem Recognition Exercises: Operations on Decimals
143
4.5
Fractions as Decimals
145
4.6
Order of Operations and Applications of Decimals
151
Chapter 4 Review Exercises
161
Chapter 4 Test
166
Chapters 1 – 4 Cumulative Review Exercises
169
Chapter 5
Ratio and Proportion
5.1
Ratios
172
5.2
Rates and Unit Cost
175
5.3
Proportions
178
Problem Recognition Exercises: Operations on Fractions versus Solving Proportions
185
Applications of Proportions and Similar Figures
186
Chapter 5 Review Exercises
194
Chapter 5 Test
199
Chapters 1 – 5 Cumulative Review Exercises
201
5.4
Chapter 6
Percents
6.1
Percents and Their Fraction and Decimal Forms
204
6.2
Fractions and Decimals and Their Percent Forms
207
6.3
Percent Proportions and Applications
212
6.4
Percent Equations and Applications
220
Problem Recognition Exercises: Percents
225
6.5
Applications Involving Sales Tax, Commission, Discount, and Markup
227
6.6
Percent Increase and Decrease
231
6.7
Simple and Compound Interest
235
Chapter 6 Review Exercises
237
Chapter 6 Test
243
Chapters 1 – 6 Cumulative Review Exercises
245
ii
Chapter 7
Measurement
7.1
Converting U.S. Customary Units of Length
248
7.2
Converting U.S. Customary Units of Time, Weight, and Capacity
253
7.3
Metric Units of Length
257
7.4
Metric Units of Mass, Capacity, and Medical Applications
259
Problem Recognition Exercises: U.S. Customary and Metric Conversions
263
Converting Between U.S. Customary and Metric Units
263
Chapter 7 Review Exercises
267
Chapter 7 Test
270
Chapters 1 – 7 Cumulative Review Exercises
271
7.5
Chapter 8
Geometry
8.1
Lines and Angles
274
8.2
Triangles and the Pythagorean Theorem
277
8.3
Quadrilaterals, Perimeter, and Area
282
8.4
Circles, Circumference, and Area
285
Problem Recognition Exercises: Area, Perimeter, and Circumference
289
Volume
290
Chapter 8 Review Exercises
294
Chapter 8 Test
297
Chapters 1 – 8 Cumulative Review Exercises
299
8.5
Chapter 9
Introduction to Statistics
9.1
Tables, Bar Graphs, Pictographs, and Line Graphs
302
9.2
Frequency Distributions and Histograms
304
9.3
Circle Graphs
307
9.4
Mean, Median, and Mode
310
9.5
Introduction to Probability
315
Chapter 9 Review Exercises
318
Chapter 9 Test
321
Chapters 1 – 9 Cumulative Review Exercises
323
Chapter 10
Real Numbers
10.1
Real Numbers and the Real Number Line
326
10.2
Addition of Real Numbers
329
10.3
Subtraction of Real Numbers
332
iii
10.4
10.5
Problem Recognition Exercises: Addition and Subtraction of Real Numbers
335
Multiplication and Division of Real Numbers
337
Problem Recognition Exercises: Operations on Real Numbers
340
Order of Operations
341
Chapter 10 Review Exercises
345
Chapter 10 Test
348
Chapters 1 – 10 Cumulative Review Exercises
349
Chapter 11
Solving Equations
11.1
Properties of Real Numbers
352
11.2
Simplifying Expressions
356
11.3
Addition and Subtraction of Properties of Equality
360
11.4
Multiplication and Division Properties of Equality
365
11.5
Solving Equations with Multiple Steps
371
Problem Recognition Exercises: Equations versus Expressions
377
Applications and Problem Solving
379
Chapter 11 Review Exercises
385
Chapter 11 Test
390
Chapters 1 – 11 Cumulative Review Exercises
392
11.6
Appendix
A.1
Energy and Power
396
A.2
Scientific Notation
398
A.3
Rectangular Coordinate System
399
iv
Chapter 1
Whole Numbers
Chapter Opener Puzzle
Section 1.1
Introduction to Whole Numbers
Section 1.1 Practice Exercises
1. (a) periods
(b) hundreds
(c) thousands
5. 321 tens
2. 1: ones
9: tens
7: hundreds
6: thousands
3: ten-thousands
7. 214 ones
6. 689 tens
8. 738 ones
9. 8,710 hundreds
10. 2,293 hundreds
3. 8,213,457
7: ones
5: tens
4: hundreds
3: thousands
1: ten-thousands
2: hundred-thousands
8: millions
11. 1,430 thousands
12. 3,101 thousands
13. 452,723 hundred-thousands
14. 655,878 hundred thousands
15. 1,023,676,207 billions
4. 103,596
6: ones
9: tens
5: hundreds
3: thousands
0: ten-thousands
1: hundred-thousands
16. 3,111,901,211 billions
17. 22,422 ten-thousands
18. 58,106 ten-thousands
19. 51,033,201 millions
20. 93,971,224 millions
1
Chapter 1
21.
Whole Numbers
10,677,881 ten-millions
22. 31,820 m
2
49. One hundred thousand, two hundred
thirty-four
thousands
50. Four hundred thousand, one hundred
ninety-nine
23. 7,653,468,440 billions
24. 31,000 ten-thousands
51. Nine thousand, five hundred thirty-five
25. 5 tens + 8 ones
26. 7 tens + 1 one
52. Five hundred ninety thousand, seven
hundred twelve
27. 5 hundreds + 3 tens + 9 ones
53. Twenty thousand, three hundred twenty
28. 3 hundreds + 8 tens + 2 ones
54. One thousand, eight hundred
29. 5 hundreds + 3 ones
55. One thousand, three hundred seventyseven
30. 8 hundreds + 9 ones
56. Sixty million
31. 1 ten-thousand + 2 hundreds + 4 tens + 1
one
57. 6,005
32. 2 ten-thousands + 8 hundreds + 7 tens + 3
ones
58. 4,004
59. 672,000
33. 524
60. 248,000
34. 318
61. 1,484,250
35. 150
62. 2,647,520
36. 620
63.
37. 1,906
64.
38. 4,201
65. Counting on a number line, 10 is 4 units to
the right of 6.
39. 85,007
40. 26,002
66. Counting on a number line, 3 is 8 units to
the left of 11.
41. ones, thousands, millions, billions
42. ones, tens, hundreds, thousands
67. Counting on a number line, 4 is 3 units to
the left of 7.
43. Two hundred forty-one
68. Counting on a number line, 5 is 5 units to
the right of 0.
44. Three hundred twenty-seven
45. Six hundred three
69. 8 > 2
8 is greater than 2, or 2 is less than 8.
46. One hundred eight
70. 6 < 11
6 is less than 11, or 11 is greater than 6.
47. Thirty-one thousand, five hundred thirty
48. Fifty-two thousand, one hundred sixty
2
Section 1.1
71. 3 < 7
3 is less than 7, or 7 is greater than 3.
Introduction to Whole Numbers
83. 90 < 91
84. 48 > 47
72. 14 > 12
14 is greater than 12, or 12 is less than 14.
85. False; 12 is made up of the digits 1 and 2.
73. 6 < 11
86. False; 26 is made up of the digits 2 and 6.
74. 14 > 13
87. 99
75. 21 > 18
88. 999
76. 5 < 7
89. There is no greatest whole number.
77. 3 < 7
90. 0 is the least whole number.
78. 14 < 24
91. 10,000,000
79. 95 > 89
92. 100,000,000,000
80. 28 < 30
93. 964
81. 0 < 3
94. 840
7 zeros
11 zeros
82. 8 > 0
Section 1.2
Addition of Whole Numbers and Perimeter
Section 1.2 Practice Exercises
3. 3 hundreds + 5 tens + 1 one
1. (a) addends
(b) sum
(c) commutative
(d) 4; 4
(e) associative
(f) polygon
(g) perimeter
4. Three hundred fifty-one
5. 1 hundred + 7 ones
6. 2004
7. 4012
2. 5 thousands + 2 tens + 4 ones
8. 6206
3
Chapter 1
Whole Numbers
9. Fill in the table. Use the number line if necessary.
+
0
1
2
3
4
5
6
7
8
9
0
0
1
2
3
4
5
6
7
8
9
1
1
2
3
4
5
6
7
8
9
10
2
2
3
4
5
6
7
8
9
10
11
3
3
4
5
6
7
8
9
10
11
12
4
4
5
6
7
8
9
10
11
12
13
5
5
6
7
8
9
10
11
12
13
14
6
6
7
8
9
10
11
12
13
14
15
7
7
8
9
10
11
12
13
14
15
16
8
8
9
10
11
12
13
14
15
16
17
9
9
10
11
12
13
14
15
16
17
18
10. 5 + 9 = 14
Addends: 5, 9
Sum: 14
18.
39 = 3 tens + 9 ones
+ 20 = 2 tens + 0 ones
59 = 5 tens + 9 ones
11. 2 + 8 = 10
Addends: 2, 8
Sum: 10
19.
15 = 1 ten + 5 ones
+ 43 = 4 tens + 3 ones
58 = 5 tens + 8 ones
12. 12 + 5 = 17
Addends: 12, 15
Sum: 17
20.
12 = 1 ten + 2 ones
15 = 1 ten + 5 ones
+ 32 = 3 tens + 2 ones
59 = 5 tens + 9 ones
13. 11 + 10 = 21
Addends: 11, 10
Sum: 21
21. 10 = 1 ten + 0 ones
8 = 0 tens + 8 ones
30 = 3 tens + 0 ones
48 = 4 tens + 8 ones
14. 1 + 13 + 4 = 18
Addends: 1, 13, 4
Sum: 18
22.
7 = 0 tens + 7 ones
21 = 2 tens + 1 one
+ 10 = 1 ten + 0 ones
38 = 3 tens + 8 ones
23.
6 = 0 tens + 6 ones
11 = 1 ten + 1 one
+ 2 = 0 tens + 2 ones
19 = 1 ten + 9 ones
24.
341
+ 225
566
15. 5 + 8 + 2 = 15
Addends: 5, 8, 2
Sum: 15
16.
42 = 4 tens + 2 ones
+ 33 = 3 tens + 3 ones
75 = 7 tens + 5 ones
17.
21 = 2 tens + 1 one
+ 53 = 5 tens + 3 ones
74 = 7 tens + 4 ones
4
Section 1.2
25.
407
+ 181
588
26.
890
+ 107
997
27.
444
+ 354
798
28.
29.
30.
31.
32.
4
13
+ 102
119
11
221
+ 5
237
Addition of Whole Numbers and Perimeter
36.
658
+ 231
889
37.
642
+ 295
937
1
11
38.
152
+ 549
701
39.
462
+ 388
850
40.
15
5
+9
29
11
1
31
7
+ 430
468
1
24
14
+ 160
198
41.
2
31
+8
41
42.
14
9
+ 17
40
2
1
76
+ 45
121
1
1
33.
25
+ 59
84
34.
87
+ 24
111
35.
38
+ 77
115
43.
1
7
18
+4
29
11
44.
1
79
112
+ 12
203
11
45.
5
62
907
+ 34
1003
Chapter 1
Whole Numbers
61. The sum of any number and 0 is that
number.
(a) 423 + 0 = 423
(b) 0 + 25 = 25
(c) 67 + 0 = 67
1
46.
331
422
+ 76
829
11
47.
87
119
+ 630
836
62. 13 + 7
63. 100 + 42
11
48.
100
+ 42
142
4980
+ 10223
15, 203
64. 7 + 45
11
49.
1
13
+7
20
23112
892
24,004
1
7
+ 45
52
65. 23 + 81
23
+ 81
104
11 1
50.
10 223
25 782
4980
40,985
18
+5
23
67. 76 + 2
76
+2
78
11 1 1
51.
92 377
5 622
34 659
132,658
1
66. 18 + 5
68. 1523 + 90
52. 12 + 6 = 6 + 12
53. 30 + 21 = 21 + 30
54. 101 + 44 = 44 + 101
69. 1320 + 448
55. 8 + 13 = 13 + 8
56. (4 + 8) + 13 = 4 + (8 + 13)
1
1 523
+ 90
1,613
1 320
+ 448
1,768
1
70. 5 + 39 + 81
57. (23 + 9) + 10 = 23 + (9 + 10)
58. 7 + (12 + 8) = (7 + 12) + 8
59. 41 + (3 + 22) = (41 + 3) + 22
5
39
+ 81
125
71. For example: The sum of 54 and 24
60. The commutative property changes the
order of the addends, and the associative
property changes the grouping.
72. For example: The sum of 33 and 15
73. For example: 88 added to 12
6
Section 1.2
74. For example: 15 added to 70
Addition of Whole Numbers and Perimeter
1
75. For example: The total of 4, 23, and 77
85.
60
52
75
+ 58
245
The total for the checks written is $245.
86.
115
104
93
+ 111
423
423 desks were delivered.
87.
2 787
1 956
991
1 817
1 567
715
+ 3 705
13,538
There are 13,538 participants.
76. For example: The total of 11, 41, and 53
77. For example: 10 increased by 8
78. For example: 25 increased by 14
79.
11
103
112
+ 61
276
276 people attended the play.
3
533
38
80.
54
44
61
3 97
103
+ 124
521
521 deliveries were made.
1
81.
82.
2
11
21, 209,000
20,836,000
+ 16, 448,000
58, 493,000
The shows had a total of
58,493,000 viewers.
88. 1494
155
+ 42
1691
There are 1691 thousand teachers.
111 11
11
195 mi
+ 228 mi
423 mi
She will travel 423 mi.
83. $43,000
+ 2,500
$45,500
Nora earns $45,500.
89.
100,052
675,038
+ 45,934
821,024
There are 821,024 nonteachers.
90.
$7 329
9 560
1 248
+ 3 500
$21,637
The total cost is $21,637.
1 11
84. 1, 205,655
+ 1,000
1,206,655
1,206,655 athletes are participating.
7
Chapter 1
Whole Numbers
1
91.
35 cm
35 cm
+ 34 cm
104 cm
92.
27 in.
13 in.
+ 20 in.
60 in.
98.
1
99. 9,084,037 + 452,903 = 9,536,940
100. 899,382 + 9406 = 908,788
101. 7,201,529 + 962,411 = 8,163,940
2
21 m
93.
20 m
18 m
19 m
11 m
+ 21 m
110 m
102.
45, 418
81,990
9,063
+ 56,309
192,780
103.
9,300,050
7,803,513
3, 480,009
+ 907,822
21, 491,394
104.
3, 421,019
822,761
1,003,721
+
9,678
5, 257,179
105.
64,700,000
36,500,000
24,100,000
+ 23, 200,000
$148,500,000
2
94.
95.
15 m
7m
6m
+7m
35 m
2
6 yd
10 yd
11 yd
3 yd
5 yd
+ 7 yd
42 yd
96.
200 yd
136 yd
142 yd
98 yd
58 yd
+ 38 yd
672 yd
97.
94 ft
94 ft
50 ft
+ 50 ft
288 ft
90 ft
90 ft
90 ft
+ 90 ft
360 ft
106.
8
2 211 1
65,899,660
60,932,152
1, 275,804 votes
128,107,616
Section 1.3
Section 1.3
Subtraction of Whole Numbers
Subtraction of Whole Numbers
Section 1.3 Practice Exercises
1. minuend; subtrahend; difference
12. 32 − 2 = 30
minuend: 32
subtrahend: 2
difference: 30
2. 134
3.
330
+ 821
1151
13.
9
−6
3
minuend: 9
subtrahend: 6
difference: 3
14.
17
−3
14
minuend: 17
subtrahend: 3
difference: 14
1
4.
782
21
+ 1 046
1,849
1
5.
46
804
+ 49
899
6. 14 < 21
15. 27 − 9 = 18 because 18 + 9 = 27.
7. 0 < 10
16. 20 − 8 = 12 because 12 + 8 = 20.
8. Twenty-two is less than twenty-five.
17. 102 − 75 = 27 because 27 + 75 = 102.
9. 12 − 8 = 4
minuend: 12
subtrahend: 8
difference: 4
18. 211 − 45 = 166 because 166 + 45 = 211.
10. 6 − 1 = 5
minuend: 6
subtrahend: 1
difference: 5
11. 21 − 12 = 9
minuend: 21
subtrahend: 12
difference: 9
9
19. 8 − 3 = 5
Check: 5 + 3 = 8
20. 7 − 2 = 5
Check: 5 + 2 = 7
21. 4 − 1 = 3
Check: 3 + 1 = 4
22. 9 − 1 = 8
Check: 8 + 1 = 9
23. 6 − 0 = 6
Check: 6 + 0 = 6
24. 3 − 0 = 3
Check: 3 + 0 = 3
25.
68
− 23
45
Check:
45
+ 23
68 
26.
54
− 31
23
Check:
23
+ 31
54 
Chapter 1
Whole Numbers
27.
88
− 27
61
Check:
61
+ 27
88 
28.
75
− 50
25
Check:
25
+ 50
75 
29.
30.
1347
− 221
1126
Check: 1126
+ 221
1347 

4865
− 713
4152
Check: 4152
+ 713
4865 
31.
1525
− 1204
321
Check: 1204
+ 321
1525 

32.
8843
− 5612
3231
Check:
Check: 10 004
+ 2 802
12,806 

34. 12,771
− 1 240
11,531
Check: 11 531
+ 1 240
12,771 

14,356
− 13, 253
1,103
Check:
36.
34,550
− 31, 450
3,100
Check:
6 16
37.
76
− 59
17
64
− 48
16
94
− 75
19
41.
19
+ 75
94 
1
240
−136
104
Check:
104
+ 136
240 
1
5 10
42.
360
− 225
135
Check: 135
+ 225
360 
10
6 0 10
43.
71 0
−1 89
5 21
44.
85 0
− 30 3
54 7
45.
435 0
− 432 7
23
46.
729 3
− 725 5
38
11
Check:
521
+ 189
710 
1
4 10
Check:
547
+ 303
850 
1
4 10
Check:
23
+ 4327
 4350 
1
8 13
3 100
+ 31 450
34,550 
17
+ 59
 76 
Check:
38
+ 7255
7293 
9 9
5 10 1012
47.
1
Check:
49
+ 38
87 
1
Check:
1
Check:
5 14
38.
Check:
3 10
1 103
+ 13 253
14,356 
35.
87
− 38
49
8 14
40.
3231
+ 5612
8843 
12 806
− 2 802
10,004
33.
1
7 17
39.
16
+ 48
 64 
10
60 02
−1 2 3 8
47 64
1 11
Check:
4764
+ 1238
6002 
Section 1.3
9 9
210 1010
48.
30 0 0
−2 3 5 6
6 44
49.
10 ,425
− 9 022
1, 403
Check:
0 10
Check:
9
1 13 8 10 11
50. 2 3, 9 0 1
−8 0 6 4
15,8 3 7
1 403
+ 9 022
10,425 

Check:
32 , 1 1 2
− 28 3 3 4
3, 7 7 8
53.
47 0
−9 2
37 8
16
5 6 14
54.
67 4
−8 9
58 5
37 0 0
− 29 8 7
7 13
9
7 1010
59.
1 3 778
+ 28 334
 32,112 
Check: 378
+ 92
470 
Check:
62.
45
−1 7
2 8
63.
78
−6
72
64.
50
−12
38
65.
422
− 100
322
11
1 1
1
Check:
5 662 119
+ 2 345 115
8,007,234 
Check:
1 174 072
+ 1 871 495
 3,045,567 
4 16
78
− 23
55
4 10
1 713
+ 2987
 3700 
2 14
61.
11
11
1
17 212
Check: + 4 123
21,335 
3 0 45 5 67
− 1 8 71 4 95
1, 1 74,0 72
3 15
Check: 585
+ 89
674 

1 1
60.
11
4212
+ 3788
8000 
Check: 30 941
+ 1 498
32,439 

8, 0 07, 23 4
− 2, 3 45,11 5
5, 6 62,11 9
9
2 1014
1 11
16
2 6 10 10
55.
1 11
2 217
+ 59 871
 62,088 
Check:
32 , 4 39
− 1 4 98
30 ,9 41
21 335
58. − 4 123
17, 212
1
Check:
13
3 13
11
1
62 088
− 59 871
2, 217
16
3 6 10
57.
111
800 0
− 378 8
4 212
1
15 837
Check: + 8 064
23,901 
1110 10
2 1 0 012
52.
56.
+ 2356
 3000 
11
5 1 10
51.
9 9
7 10 10 10
11
1 644
1
Subtraction of Whole Numbers
1
1
Chapter 1
66.
Whole Numbers
4 10
89
− 42
47
79. $5 0
−17
$3 3
$33 change was received.
8 10
67. 109 0
−72
101 8
4 15
80.
0 11
68. 3111
− 60
3051
0 11
81. 1 18
− 63
55
Lennon and McCartney had 55 more hits.
4 10
69.
50
−1 3
37
70.
405
− 103
302
71.
10 3
−35
68
55
−39
16
16 DVDs are left.
4 10
82.
5 05
−2 00
30 5
305 ft more
83.
26
−1 8
8
Lily needs 8 more plants.
84.
$50
− 37
$13
$13 more is needed.
9 13
1 16
8 11
72.
91
−1 4
77
73. For example: 93 minus 27
74. For example: 80 decreased by 20
10 13
4 0 14
75. For example: Subtract 85 from 165.
85.
76. For example: 42 less than 171
77. The expression 7 − 4 means 7 minus 4,
yielding a difference of 3. The expression
4 − 7 means 4 minus 7 which results in a
difference of −3.
5 1 4 9
−2 6 7 0
2 4 79
The Lion King had been performed 2,479
more times.
12 13
1 2 3 14
86.
78. Subtraction is not associative. For
example, 10 − (6 − 2) = 10 − 4 = 6, and
(10 − 6) − 2 = 4 − 2 = 2. Therefore
10 − (6 − 2) does not equal (10 − 6) − 2.
12
3 2 3 44
−3 0 6 46
16 98
Brees needs 1698 more yd.
Section 1.3
87.
14 m
39 m
+ 12 m
− 26 m
26 m
13 m
The missing length is 13 m.
88.
139 cm
87 cm
547 cm
+ 201 cm
− 427 cm
427 cm
120 cm
The missing length is 120 cm.
1 10
94.
11
89.
4
96. 953, 400, 415
− 56,341,902
897,058,513
14
14
+ 10
46 yd
The missing side is 10 yd long.
90.
6
+5
97.
82,025,160
− 79,118,705
2,906, 455
98.
103,718 mi 2
− 54,310 mi 2
11
15ft
49, 408 mi 2
99.
−11ft
4 ft
100. 103, 718 mi 2
− 1, 045 mi 2
2279000
− 2249000
30,000
The difference is 30,000 marriages.
102, 673 mi 2
The difference in land area between
Colorado and Rhode Island is
102,673 mi2 .
1 14
92.
93.
41, 217 mi 2
− 24, 078 mi 2
17,139 mi 2
The missing side is 4 ft long.
91.
2, 2 0 5,000
− 2, 1 6 0,000
4 5,000
The greatest increase occurred between
Year 5 and Year 6; the increase was
45,000.
95. 4,905,620
− 458,318
4, 447,302
56 yd
− 46 yd
10 yd
14
Subtraction of Whole Numbers
2, 2 4 9,000
− 2, 1 6 0,000
89,000
The decrease is 89,000 marriages.
101.
54,310 mi 2
− 41, 217 mi 2
13,093 mi 2
2279000
− 2160000
119,000
The difference is 119,000 marriages.
Wisconsin has 13,093 mi 2 more than
Tennessee.
13
Chapter 1
Whole Numbers
Section 1.4
Rounding and Estimating
Section 1.4 Practice Exercises
1. rounding
16. 8363 ≈ 8400
2. 30 ft
17. 8539 ≈ 8500
3.
59
− 33
26
18. 9817 ≈ 9800
0 12 10
20. 76,831 ≈ 77,000
19. 34,992 ≈ 35,000
4. 1 3 0
−98
32
21. 2578 ≈ 3000
22. 3511 ≈ 4000
1 11
5. 4 009
+ 998
5, 007
6.
23. 9982 ≈ 10000
24. 7974 ≈ 8000
25. 109,337 ≈ 109,000
12,033
+ 23,441
35,474
26. 437,208 ≈ 437,000
7. Ten-thousands
27. 489,090 ≈ 490,000
8. Hundreds
28. 388,725 ≈ 390,000
9. If the digit in the tens place is 0, 1, 2, 3, or
4, then change the tens and ones digits to
0. If the digit in the tens place is 5, 6, 7, 8,
or 9, increase the digit in the hundreds
place by 1 and change the tens and ones
digits to 0.
29. $77,025,481 ≈ $77,000,000
30. $33,050 ≈ $33,000
31. 238,863 mi ≈ 239,000 mi
32. 492,000 m 2 ≈ 500,000 m 2
10. If the digit in the ones place is 0, 1, 2, 3, or
4, then change the ones digits to 0. If the
digit in the ones place is 5, 6, 7, 8, or 9,
increase the digit in the tens place by 1
and change the ones digit to 0.
11. 342 ≈ 340
33.
57
82
+ 21
→
→
→
60
80
+ 20
160
34.
33
78
+ 41
→
→
→
30
80
+ 40
150
35.
41
12
+ 129
12. 834 ≈ 830
13. 725 ≈ 730
14. 445 ≈ 450
15. 9384 ≈ 9400
14
→
→
→
40
10
+ 130
180
Section 1.4
29
73
+ 113
→
→
→
37.
898
− 422
→
→
900
− 400
500
38.
731
− 584
→
→
700
− 600
100
39.
3412
− 1252
→
→
3400
− 1300
2100
40.
9771
− 4544
→
→
9800
− 4500
5300
41.
97,404,576
+ 53,695,428
→
→
97, 000,000
+ 54, 000,000
151, 000,000
$151,000,000 was brought in by Mars.
42.
81, 296,784
54,391, 268
+ 38,168,580
→
→
→
36.
46.
1 30
70
+ 110
210
$3,470,295
3,173,050
+ 1,970,380
Rounding and Estimating
$3,500,000
3, 200,000
+ 2,000,000
$8,700,000
→
→
→
47. (a) Year 4; $3,470,295 → $3,500,000
(b) Year 6; $1,970,380 → $2,000,000
48.
$3,500,000
− 2,000,000
$1,500,000
49. Massachusetts; 78,815 → 79,000 students
50. Vermont; 8059 → 8000 students
51.
79,000
− 8,000
71,000
The difference is 71,000 students.
1
4
52. 45,879
9137
16,756
78,815
17,422
13,172
+ 8059
→
→
→
→
→
→
→
46,000
9,000
17,000
79,000
17,000
13,000
+ 8,000
189,000
The total is 189,000 students.
1
81,000,000
54,000,000
+ 38,000,000
173,000,000
$173,000,000 was brought in by Hershey.
53. Answers may vary.
43.
71,000,000
− 60,000,000
11,000,000
Neil Diamond earned $11,000,000 more.
54. Thousands place
4208 − 932 + 1294 ≈ 4000 − 1000 + 1000
≈ 3000 + 1000
≈ 4000
44.
63,640
− 43,130
55.
3045 mm
1892 mm
3045 mm
+ 1892 mm
56.
1851 cm
1782 cm
1851 cm
+ 1782 cm
71,339,710
− 59,684,076
→
→
→
→
64,000
− 43,000
21,000
A California teacher makes about $21,000
more.
1
45.
$3,316,897 →
3, 272,028 →
+ 3,360, 289 →
$3,300,000
3,300,000
+ 3, 400,000
$10,000,000
15
→
→
→
→
→
→
→
→
3000 mm
2000 mm
3000 mm
+ 2000 mm
10,000 mm
2000 cm
2000 cm
2000 cm
+ 2000 cm
8000 cm
Chapter 1
57.
Whole Numbers
105 in.
57 in.
57 in.
105 in.
57 in.
+ 57 in.
→
→
→
→
→
→
Section 1.5
58.
2
110 in.
60 in.
60 in.
110 in.
60 in.
+ 60 in.
460 in.
→
→
→
→
→
182 ft
121 ft
182 ft
169 ft
+ 169 ft
200 ft
100 ft
200 ft
200 ft
+ 200 ft
900 ft
Multiplication of Whole Numbers and Area
Section 1.5 Practice Exercises
1. (a) factors; product
2. 13,000
1
(b) commutative
3.
869, 240 →
870,000
34,921 →
30,000
+ 108,332 → + 110,000
1,010,000
4.
907,801 →
900,000
− 413,560 → − 400,000
500,000
5.
8821 →
8800
− 3401 → − 3400
5400
(c) associative
(d) 0; 0
(e) 7; 7
(f) distributive
(g) area
(h) l × w
6.
×
0
1
2
3
4
5
6
7
8
9
0
0
0
0
0
0
0
0
0
0
0
1
0
1
2
3
4
5
6
7
8
9
2
0
2
4
6
8
10
12
14
16
18
3
0
3
6
9
12
15
18
21
24
27
4
0
4
8
12
16
20
24
28
32
36
5
0
5
10
15
20
25
30
35
40
45
6
0
6
12
18
24
30
36
42
48
54
7
0
7
14
21
28
35
42
49
56
63
8
0
8
16
24
32
40
48
56
64
72
9
0
9
18
27
36
45
54
63
72
81
16
Section 1.5
7. 5 + 5 + 5 + 5 + 5 + 5 = 6 × 5 = 30
Multiplication of Whole Numbers and Area
30.
18
×5
40 Multiply 5 × 8.
+ 50 Multiply 5 × 10.
90 Add.
31.
26
×2
12 Multiply 2 × 6.
+ 40 Multiply 2 × 20.
52 Add.
32.
71
×3
3 Multiply 3 × 1.
+ 210 Multiply 3 × 70.
213 Add.
33.
131
× 5
5
150
+ 500
655
8. 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 + 2 = 9 × 2
= 18
9. 9 + 9 + 9 = 3 × 9 = 27
10. 7 + 7 + 7 + 7 = 4 × 7 = 28
11. 13 × 42 = 546
factors: 13, 42; product: 546
12. 26 × 9 = 234
factors: 26, 9; product: 234
13. 3 ⋅ 5 ⋅ 2 = 30
factors: 3, 5, 2; product: 30
14. 4 ⋅ 3 ⋅ 8 = 96
factors: 4, 3, 8; product: 96
15. For example: 5 × 12; 5 ⋅ 12; 5(12)
16. For example: 23 × 14; 23 ⋅ 14; 23(14)
Multiply 5 × 1.
Multiply 5 × 30.
Multiply 5 × 100.
Add.
17. d
34.
18. a
19. e
20. b
21. c
35.
22. a
23. 14 × 8 = 8 × 14
24. 3 × 9 = 9 × 3
25. 6 × (2 × 10) = (6 × 2) × 10
36.
26. (4 × 15) × 5 = 4 × (15 × 5)
27. 5(7 + 4) = (5 × 7) + (5 × 4)
28. 3(2 + 6) = (3 × 2) + (3 × 6)
29.
24
×6
24 Multiply 6 × 4.
+ 120 Multiply 6 × 20.
144 Add.
37.
17
725
× 3
15
60
+ 2100
2175
Multiply 3 × 0.
Multiply 3 × 20.
Multiply 3 × 700.
Add.
344
× 4
16
160
+ 1200
1376
Multiply 4 × 4.
Multiply 4 × 40.
Multiply 4 × 300.
Add.
105
× 9
45
00
+ 900
945
3
1410
×
8
11, 280
Multiply 9 × 5.
Multiply 9 × 0.
Multiply 9 × 100.
Add.
Chapter 1
38.
Whole Numbers
3
47.
2016
×
6
12,096
2 1
3312
39.
×
7
23,184
1
1
48.
13
× 46
78
+ 520
598
49.
143
× 17
1001
+ 1430
2431
4
40.
72
× 12
144
+ 720
864
4801
×
5
24,005
32
1
13
41.
42,014
×
9
378,126
42.
51,006
×
8
408,048
4
43.
44.
45.
11
50.
1 11
32
× 14
128
+ 320
448
5 776
+ 14 440
20,216
48
41
× 21
41
+ 820
861
1
3
51.
349
19
3141
+ 3490
6631
52.
512
31
512
+ 15 360
15,872
68
× 24
1
272
+ 1360
1632
53.
×
×
1
3
151
127
1 057
3 020
+ 15 100
19,177
×
2
46.
×
722
28
55
× 41
55
+ 2200
2255
18
Section 1.5
Multiplication of Whole Numbers and Area
1
1
54.
703
× 146
1 4 218
28 120
+ 70 300
102,638
59.
111
11
4122
982
8 244
329 760
+ 3 709 800
4,047,804
×
11
13
1
24
222
× 841
55.
1
60.
222
8 880
+ 177 600
186,702
11
56.
43
54
6 00
61. 600 →
× 40 → × 4 0
24 000 = 24,000
387
506
×
2 322
0 000
+ 193 500
195,822
62. 900 → 9 00
× 50 → × 5 0
45 000 = 45,000
3 11
21
57.
3 000
63. 3000 →
× 700 → × 7 00
21 00000 = 2,100,000
3532
6014
14 128
35 320
000 000
+ 21192 000
21, 241, 448
×
4 000
64. 4000 →
× 400 → × 4 00
16 00000 = 1,600,000
65.
8000 → 8 000
× 9000 → × 9 000
72 000000 = 72,000,000
66.
1000 →
1 000
× 2000 → × 2 000
2 000000 = 2,000,000
2
7
58.
7026
528
56 208
140 520
+ 3513 000
3,709,728
×
2810
1039
1 25 290
84 300
000 000
+ 2 810 000
2,919,590
×
9 0000
67. 90,000 →
× 400 → × 4 00
36 000000 = 36,000,000
19
Chapter 1
Whole Numbers
5 0000
68. 50,000 →
× 6,000 → × 6 000
30 0000000 =
300,000,000
69.
78.
11,784 →
12,000
× 5 201 → ×
5,000
60,000,000
1
3
4
79.
$45
37
315
+ 1 350
$1,665
80.
12
× 12
24
+ 120
144
A case contains 144 fl oz.
70. 45,046 →
45,000
× 7 812 → ×
8,000
360,000,000
71.
700
× 15
3500
+ 7000
10,500
15 CDs hold 10,500 MB of data
82,941 →
80,000
× 29,740 → ×
30,000
2, 400,000,000
2
72. 630, 229 →
630,000
× 71,907 → ×
70,000
44,100,000,000
×
2
81. 115
×5
575
73. $189 → $200
× 5
×
5
$1000
32
$130
74. $129 →
× 28 → × 30
$3,900
75.
76.
57 5
× 5 00
287,5 00
287,500 sheets of paper are delivered.
10, 256 →
1 0000
× $272 → × 272
272 0000 =
$2,720,000
48 → 5 0
× 12 → × 1 0
5 00
500
× 7
$3500 per week
77. 1000
× 4
4000
4000 minutes can be stored.
20
4
82.
14
28
×2
× 6
28
168
She gets 168 g of protein.
83.
31
× 12
62
+ 310
372
He can travel 372 miles.
84.
23
× 32
46
+ 690
736
Sherica schedules 736 hr.
Section 1.5
Multiplication of Whole Numbers and Area
90. A = l × w
A = (130 yd) × (150 yd)
85. A = l × w
A = (23 ft) × (12 ft)
23
× 12
46
+ 230
276
6
130
× 150
000
6500
+ 13000
19,500
2
The area is 276 ft .
The area is 19,500 yd 2 .
86. A = l × w
A = (31 m) × (2 m) = 62 m 2
91. (a) A = l × w
A = (40 in.) × (60 in.)
87. A = l × w
A = (73 cm) × (73 cm)
2
3
2
40
60
00
+ 2400
2
2400 in.
73
× 73
219
+ 5110
5329
×
2
The area is 5329 cm .
1
(b) 14
×3
42
There are 42 windows.
88. A = l × w
A = (41 yd) × (41 yd)
41
× 41
41
+ 1640
1681
The area is 1681 yd 2 .
1
(c)
89. A = l × w
A = (390 mi) × (270 mi)
2400
×
42
4 800
+ 96 000
100,800
The total area is 100,800 in.2
1
6
92. A = l × w
A = (50 ft.) × (30 ft.)
390
× 270
000
27300
+ 78000
105,300
8
50
× 30
000
+ 1500
1500
The area is 105,300 mi2 .
The area is 1500 ft 2 .
21
Chapter 1
Whole Numbers
93. A = l × w
A = (8 ft) × (16 ft)
94. A = l × w
A = (10 yd) × (15 yd) = 150 yd 2 .
4
16
× 8
128
2
The area is 128 ft .
Section 1.6
Division of Whole Numbers
Section 1.6 Practice Exercises
1. (a) dividend; divisor; quotient
(b) 1
(c) 5
(d) 0
(e) undefined
(f) remainder
2. (a)
(b)
(c)
(d)
3.
12
7.
11
36 610
104 600
+ 523 000
664, 210
5+2
5·2
(3 + 10) + 2
(3 · 10) · 2
5.
6.
11
44
8.
789
× 25
1
2
11
103
×
48
824
+ 4 120
4,944
3 945
+ 15 780
19,725
3 18 8 10
5 17
4.
×
5230
127
9.
4 89 0
− 3 98 8
90 2
10.
38 002
+ 3 902
41,904
678
− 83
595
1
1008
+ 245
1253
1
11. Dividend: 72
divisor: 8
quotient: 9
220
× 14
1 880
2 200
3, 080
12. Dividend: 32
divisor: 4
quotient: 8
13. Dividend: 64
divisor: 8
quotient: 8
22
Section 1.6
14. Dividend: 35
divisor: 5
quotient: 7
31. 6 ÷ 3 = 2 because 2 × 3 = 6.
3 ÷ 6 ≠ 2 because 2 × 6 ≠ 3.
32. (36 ÷ 12) ÷ 3 = 3 ÷ 3 = 1 but
36 ÷ (12 ÷ 3) = 36 ÷ 4 = 9.
15. Dividend: 45
divisor: 9
quotient: 5
33. To check a division problem without a
remainder you should multiply the
quotient and the divisor to get the
dividend.
16. Dividend: 20
divisor: 5
quotient: 4
34. To check 0 ÷ 5 = 0 we multiply 0 × 5 = 0
which is true. If we try to check 5 ÷ 0 = ?
we need to find a number to multiply by 0
to get 5. Since no such number exists, the
answer to 5 ÷ 0 is undefined.
17. You cannot divide a number by zero (the
quotient is undefined). If you divide zero
by a number (other than zero), the quotient
is always zero.
18. A number divided or multiplied by 1
remains unchanged.
35. 6
19. 15 ÷ 1 = 15 because 15 × 1 = 15.
20. 21 21 = 1 because 1 × 21 = 21.
21. 0 ÷ 10 = 0 because 0 × 10 = 0.
22.
Division of Whole Numbers
0
= 0 because 0 × 3 = 0.
3
23. 0 9 is undefined because division by zero
13
78
−6
18
−18
0
1
13
×6
78 
52
36. 7 364
−35
14
−14
0
52
×7
364 
41
205
−20
05
−5
0
41
×5
205 
1
is undefined.
24. 4 ÷ 0 is undefined because division by
zero is undefined.
25.
37. 5
20
= 1 because 1 × 20 = 20.
20
26. 1 9 = 9 because 9 × 1 = 9.
27.
28.
19
38. 8 152
−8
72
−72
0
16
is undefined because division by zero
0
is undefined.
5
= 5 because 5 × 1 = 5.
1
29. 8 0 = 0 because 0 × 8 = 0.
30. 13 ÷ 13 = 1 because 13 × 1 = 13.
23
19
× 8
152 
Chapter 1
39. 2
40. 6
Whole Numbers
486
972
−8
17
−16
12
−12
0
97
582
−54
42
−42
0
409
41. 3 1227
−12
02
−0
27
−27
0
42. 4
59
236
−20
36
−36
0
203
43. 5 1015
−10
01
−0
15
−15
0
407
44. 5 2035
−20
03
−0
35
−35
0
822
45. 6 4932
−48
13
−12
12
−12
0
11
486
× 2
972 
517
46. 7 3619
−35
11
−7
49
−49
0
4
97
× 6
582 
2
409
× 3
1227 
11
822
× 6
4932 
14
517
× 7
3619 
2
47.
56
× 4
224 correct
48.
82
×7
574 correct
1
3
59
× 4
236 
1
49. 253
×3
759
incorrect
1
203
× 5
1015 
1
50. 120
×5
600
3
407
× 5
2035 
24
incorrect
253 R2
3 761
−6
16
−15
11
−9
2
120 R4
5 604
−5
10
−10
04
−0
4
Section 1.6
51.
12
14 R4
5
74
58.
−5
24
−20
4
113
× 9
1
1017
+ 4 Add the remainder.
1021 Correct
27 R1
59. 2 55
−4
15
−14
1
14
52.
218
× 6
1308
+ 3 Add the remainder.
1311 Correct
53.
25
× 8
200
+ 6
206 incorrect
4
14
54. 117
×7
819
+ 5
824 incorrect
55. 8
7 R5
61
−56
5
29 R2
56. 3 89
−6
29
−27
2
10 R2
57. 9 92
−9
02
Division of Whole Numbers
16 R1
60. 3 49
−3
19
−18
1
25 R 3
8 203
−16
43
−40
3
117 R2
7 821
−7
12
−7
51
−49
2
61. 3
197 R2
593
−3
29
−27
23
−21
2
14 × 5 + 4 = 70 + 4
= 74 

27 × 2 + 1 = 54 + 1
= 55 

16 × 3 + 1 = 48 + 1
= 49 

197 × 3 + 2 = 591+ 2
= 593 

200 R1
62. 4 801
−8
001
200 × 4 + 1 = 800 + 1
= 801 
7 × 8 + 5 = 56 + 5
= 61 

29 × 3 + 2 = 87 + 2
= 89 

42 R4
63. 9 382
−36
22
−18
4
10 × 9 + 2 = 90 + 2
= 92 

53 R4
64. 8 428
−40
28
−24
4
25
42 × 9 + 4 = 378 + 4
= 382 

53 × 8 + 4 = 424 + 4
= 428 

Chapter 1
65. 2
Whole Numbers
1557 R1
3115
−2
11
−10
11
−10
15
−14
1
550 R1
70. 2 1101
−10
10
−10
01
00
1
111
1557
× 2
3114
+ 1
3115 
71. 19
785 R5
66. 6 4715
−42
51
−48
35
−30
5
785
× 6
4710
+ 5
4715 
751 R6
67. 8 6014
−56
41
−40
14
−8
6
751
× 8
6008
+ 6
6014 
1287 R4
68. 7 9013
−7
20
−14
61
−56
53
−49
4
69. 6
835 R2
5012
−48
21
−18
32
−30
2
53
479 R9
9110
−76
151
−133
180
−171
9
269 R8
72. 13 3505
−26
90
−78
125
−117
8
4
43 R19
73. 24 1051
−96
91
−72
19
264
1 2 87
× 7
9009
+ 4
9013 
197 R27
74. 41 8104
−41
400
−369
314
−287
27
23
835
× 6
5010
+ 2
5012 
26
1
550
× 2
1100
+ 1
1101 
Section 1.6
75. 26
308
8008
−78
20
−0
208
−208
0
Division of Whole Numbers
231 R56
80. 221 51107
−442
690
−663
277
−221
56
612
76. 15 9180
−90
18
−15
30
−30
0
302
81. 114 34428
−342
228
−228
0
209
82. 421 87989
−842
3789
−3789
0
1259 R26
77. 54 68012
−54
140
−108
321
−270
512
−486
26
83. 497 ÷ 71 = 7
7
71 497
−497
0
84. 890 ÷ 45 = 42
42
45 1890
−180
90
−90
0
2628 R33
78. 35 92,013
−70
220
−21 0
1 01
−70
313
−280
33
85. 877 ÷ 14 = 62 R9
62 R9
14 877
−84
37
−28
9
229 R96
79. 304 69712
−608
891
−608
2832
−2736
96
27
Chapter 1
Whole Numbers
86. 722 ÷ 53 = 13 R33
13 R33
53 722
−53
192
−159
33
48
94. 3 144
−12
24
−24
0
$48 per room
87. 42 ÷ 6 = 7
22 lb
95. 100 2200
−200
200
−200
0
88. 108 ÷ 9 = 12
12
9 108
−9
18
−18
0
28 acres
96. 260 7280
−520
2080
−2080
0
14 classrooms
89. 28 392
−28
112
−112
0
97. 1200 ÷ 20 = 60
60
20 1200
−120
00
−0
0
Approximately 60 words per minute
15 tables
90. 8 120
−8
40
−40
0
98. 2800 ÷ 400
7
400 2800
−2800
0
Approximately 7 tanks of gas
5 R8
91. 32 168
−160
8
5 cases; 8 cans left over
25
99. 18 450
−36
90
−90
0
Yes they can all attend if they sit in the
second balcony.
8 R9
92. 52 425
−416
9
Yes; $9 left over
52 mph
93. 6 312
−30
12
−12
0
28
Section 1.6
3 000
100. 12 36,000
−36
0
Teacher: $3000
10,000
12 120,000
−12
0
CEO: $10,000
Division of Whole Numbers
117 cars are waiting in line.
5 000
12 60,000
−60
0
Professor: $5,000
4 000
12 48,000
−48
0
Programmer: $4,000
12 R2
4 50
−4
10
−8
2
12 loads can be done.
(b) 2 ounces of detergent are left over.
101. (a)
103.
21,000,000
×
365
7,665,000,000 bbl
104.
52
5
×
260
× 50
13,000 min
105. 3552 ÷ 4 = 888
$888 billion
106.
34,080
− 9 600
24,480
24,480 ÷ 96 = 255
Each crate weighs 255 lb.
102. 26 ÷ 2 = 13
2
13
× 9
117
Problem Recognition Exercises: Operations on Whole Numbers
1. (a) 52
+ 13
65
(b) 52
× 13
156
+ 520
676
6
2. (a) 17 102
102
0
9 12
(b) 1 0 2
− 17
85
4 12
(c)
1
102
× 17
714
+ 1020
1734
(d) 102
+ 17
119
(c)
5 2
− 1 3
39
4
(d) 13 52
52
0
29
Chapter 1
Whole Numbers
5064
× 58
40512
+ 253200
293,712
(b) 5064
+ 58
5122
87 R18
(c) 58 5064
−464
424
− 406
18
5. (a) 156
+ 41
197
(b) 197
− 41
156
3. (a)
6. (a) 6004
+ 221
6225
(b) 6004
− 221
6004
7. 4,180
5 14
8. 41,800
(d) 5 0 6 4
− 58
5006
9. 418,000
10. 4,180,000
4. (a) 1226
−114
1112
11. 35,000
12. 3,500
10 R86
(b) 114 1226
−114
86
0
86
13. 350
14. 35
15. 246,000
1
16. 2,820,000
(c) 1226
+ 114
1340
(d)
17. 20,000
12
18. 540,000
1226
× 114
4904
12260
+ 122600
139,764
30
Section 1.7
Section 1.7
Exponents, Square Roots, and the Order of Operations
Exponents, Square Roots, and the Order of Operations
Section 1.7 Practice Exercises
1. (a) base; 4
(b) powers
(c) square root; 81
(d) order; operations
(e) variable; constants
(f) mean
3
21. 2 = 2 ⋅ 2 ⋅ 2 = 4 ⋅ 2 = 8
2. False
24. 52 = 5 ⋅ 5 = 25
2
22. 4 = 4 ⋅ 4 = 16
23. 32 = 3 ⋅ 3 = 9
3. True: 5 + 3 = 8 and 3 + 5 = 8
3
25. 3 = 3 ⋅ 3 ⋅ 3 = 9 ⋅ 3 = 27
4. False: 5 − 3 = 2, but 3 − 5 ≠ 2
26. 112 = 11 ⋅ 11 = 121
5. False: 6 × 0 = 0
3
27. 5 = 5 ⋅ 5 ⋅ 5 = 25 ⋅ 5 = 125
6. True: 0 ÷ 8 = 0
7. True: 0 × 8 = 0
3
28. 4 = 4 ⋅ 4 ⋅ 4 = 16 ⋅ 4 = 64
8. True: 5 ÷ 0 is undefined.
29. 25 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 4 ⋅ 4 ⋅ 2 = 16 ⋅ 2 = 32
9. 94
30. 63 = 6 ⋅ 6 ⋅ 6 = 36 ⋅ 6 = 216
8
10. 3
4
31. 3 = 3 ⋅ 3 ⋅ 3 ⋅ 3 = 9 ⋅ 9 = 81
11. 27
4
32. 5 = 5 ⋅ 5 ⋅ 5 ⋅ 5 = 25 ⋅ 25 = 625
5
12. 6
33. 12 = 1 ⋅ 1 = 1
6
13. 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 3 = 3
34. 13 = 1 ⋅ 1 ⋅ 1 = 1
14. 7 ⋅ 7 ⋅ 7 ⋅ 7 = 74
35. 14 = 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1
15. 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 2 ⋅ 2 ⋅ 2 = 4 4 ⋅ 23
36. 15 = 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1
3
3
16. 5 ⋅ 5 ⋅ 5 ⋅10 ⋅10 ⋅10 = 5 ⋅10
37. The number 1 raised to any power equals
1.
4
17. 8 = 8 ⋅ 8 ⋅ 8 ⋅ 8
2
38. 10 = 10 ⋅10 = 100
18. 26 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2
39. 103 = 10 ⋅10 ⋅10 = 1000
19. 48 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4
40. 104 = 10 ⋅10 ⋅10 ⋅10 = 10,000
20. 62 = 6 ⋅ 6
41. 105 = 10 ⋅10 ⋅10 ⋅10 ⋅10 = 100,000
31
Chapter 1
Whole Numbers
64. (8 + 4) ⋅ 6 + 8 = 12 ⋅ 6 + 8 = 72 + 8 = 80
9
42. 10 simplifies to a 1 followed by 9 zeros:
1,000,000,000.
65. 4 + 12 ÷ 3 = 4 + 4 = 8
43.
4 = 2 because 2 ⋅ 2 = 4.
44.
9 = 3 because 3 ⋅ 3 = 9.
45.
36 = 6 because 6 ⋅ 6 = 36.
46.
81 = 9 because 9 ⋅ 9 = 81.
2
2
69. 7 − 5 = 49 − 25 = 24
47.
100 = 10 because 10 ⋅ 10 = 100.
3
3
70. 3 − 2 = 27 − 8 = 19
48.
49 = 7 because 7 ⋅ 7 = 49.
71. (7 − 5) 2 = 22 = 4
49.
0 = 0 because 0 ⋅ 0 = 0.
72. (3 − 2)3 = 13 = 1
50.
16 = 4 because 4 ⋅ 4 = 16.
73. 100 ÷ 5 ⋅ 2 = 20 ⋅ 2 = 40
66. 9 + 15 ÷ 25 = 9 + 15 ÷ 5 = 9 + 3 = 12
67. 30 ÷ 2 ⋅ 9 = 30 ÷ 2 ⋅ 3 = 15 ⋅ 3 = 45
68. 55 ÷ 11 ⋅ 5 = 5 ⋅ 5 = 25
51. No, addition and subtraction should be
performed in the order in which they
appear from left to right.
74. 60 ÷ 3⋅ 2 = 20 ⋅ 2 = 40
52. No, multiplication and division should be
performed in the order in which they
appear from left to right.
76. 80 ÷ 2 ⋅ 2 = 40 ⋅ 2 = 80
75. 90 ÷ 3⋅ 3 = 30 ⋅ 3 = 90
77.
81 + 2(9 − 1) = 81 + 2 ⋅ 8
= 9 + 2⋅8
= 9 + 16
= 25
78.
121 + 3(8 − 3) = 121 + 3 ⋅ 5
= 11 + 3 ⋅ 5
= 11 + 15
= 26
53. 6 + 10 ⋅ 2 = 6 + 20 = 26
54. 4 + 3 ⋅ 7 = 4 + 21 = 25
2
55. 10 − 3 = 10 − 9 = 1
2
56. 11 − 2 = 11 − 4 = 7
79. 36 ÷ (22 + 5) = 36 ÷ (4 + 5) = 36 ÷ 9 = 4
57. (10 − 3) 2 = 7 2 = 49
80. 42 ÷ (32 − 2) = 42 ÷ (9 − 2) = 42 ÷ 7 = 6
58. (11 − 2) 2 = 92 = 81
81. 80 − (20 ÷ 4) + 6 = 80 − 5 + 6 = 75 + 6 =
81
59. 36 ÷ 2 ÷ 6 = 18 ÷ 6 = 3
60. 48 ÷ 4 ÷ 2 = 12 ÷ 2 = 6
82. 120 − (48 ÷ 8) − 40 = 120 − 6 − 40
= 114 − 40
= 74
61. 15 − (5 + 8) = 15 − 13 = 2
62. 41 − (13 + 8) = 41 − 21 = 20
63. (13 − 2) ⋅ 5 – 2 = 11 ⋅ 5 – 2 = 55 – 2 = 53
32
Section 1.7
Exponents, Square Roots, and the Order of Operations
92. 50 − 2 (36 ÷ 12 ⋅ 2 − 4) = 50 − 2 (3 ⋅ 2 − 4)
= 50 − 2 (6 − 4)
= 50 − 2 (2)
= 50 − 4
= 46
83. (43 − 26) ⋅ 2 − 42 = 17 ⋅ 2 − 42
= 17 ⋅ 2 − 16
= 34 − 16
= 18
84. (51 − 48) ⋅ 3 + 7 2 = 3 ⋅ 3 + 7 2
= 3 ⋅ 3 + 49
= 9 + 49
= 58
(
93. 16 + 5 (20 ÷ 4 ⋅ 8 − 3) = 16 + 5 (5 ⋅ 8 − 3)
= 16 + 5 (40 − 3)
= 16 + 5 (37)
= 16 + 185
= 201
)
85. (18 − 5) − 23 − 100 = 13 − (23 − 10)
86.
94. Mean =
= 13 − 13
=0
19 + 21 + 18 + 21 + 16 95
= = 19
5
5
105 + 114 + 123+ 101+ 100 + 111
6
654
=
= 109
6
( 36 + 11)− (31 − 16) = (6 + 11) − 15
95. Mean =
= 17 − 15
=2
2
2
2
87. 80 ÷ (9 − 7 ⋅11) = 80 ÷ (81 − 7 ⋅11)
= 80 ÷ (81 − 77)2
1480 + 1102 + 1032 + 1002
4
4616
=
= 1154
4
96. Mean =
= 80 ÷ 42
= 80 ÷ 16
=5
19 + 20 + 18 + 19 + 18 + 14
6
108
=
= 18
6
97. Average =
3
2
2
88. 108 ÷ (3 − 6 ⋅ 4) = 108 ÷ (27 − 6 ⋅ 4)
= 108 ÷ (27 − 24)2
= 108 ÷ 32
= 108 ÷ 9
= 12
98. Average =
83 + 95 + 87 + 91 356
=
= 89
4
4
69 + 74 + 49
3
192
=
= 64¢ per pound
3
89. 22 − 4 ( 25 − 3) 2 = 22 − 4 (5 − 3) 2
= 22 − 4 (2) 2
= 22 − 4 ⋅ 4
= 22 − 16
=6
99. Average =
7 + 10 + 8 + 7 32
=
4
4
= $8 per wash
100. Average =
90. 17 + 3 (7 − 9)2 = 17 + 3 (7 − 3)2
= 17 + 3 (4)2
= 17 + 3 ⋅ 16
= 17 + 48
= 65
118 + 123 + 122
3
363
=
= 121 mm per month
3
101. Average =
91. 96 − 3 (42 ÷ 7 ⋅ 6 − 5) = 96 − 3 (6 ⋅ 6 − 5)
= 96 − 3 (36 − 5)
= 96 − 3 (31)
= 96 − 93
=3
33
Chapter 1
Whole Numbers
107. 1562 = 24,336
9 + 20 + 22 + 16 + 13
5
80
=
= 16 in. per month
5
102. Average =
2
108. 4182 = 174,724
109. 125 = 248,832
2
103. 3[4 + (6 − 3) ] − 15 = 3[4 + 3 ] − 15
= 3[4 + 9] − 15
= 3[13] − 15
= 39 − 15
= 24
110. 354 = 1,500,625
111. 433 = 79, 507
112. 713 = 357, 911
104. 2[5(4 − 1) + 3] ÷ 6 = 2[5(3) + 3] ÷ 6
= 2[15 + 3] ÷ 6
= 2[18] ÷ 6
= 36 ÷ 6
=6
113. 8126 − 54,978 ÷ 561 = 8126 − 98 = 8028
114. 92,168 + 6954 × 29 = 92,168 + 201, 666
= 293,834
105. 5{21 − [32 − (4 − 2)]} = 5{21 − [32 − 2]}
= 5{21 − [9 − 2]}
= 5{21 − 7}
= 5{14}
= 70
3
115. (3548 − 3291) 2 = 257 2 = 66,049
116. (7500 ÷ 625)3 = 123 = 1728
3
106. 4{18 − [(10 − 8) + 2 ]} = 4{18 − [2 + 2 ]}
= 4{18 − [2 + 8]}
= 4{18 − 10}
= 4{8}
= 32
Section 1.8
117.
89,880
89,880
=
= 35
384 + 2184 2568
118.
54,137
54,137
=
= 43
3393 − 2134 1259
Problem-Solving Strategies
Section 1.8 Practice Exercises
9. 10 ⋅ 13 = 130
1. 4 ÷ 0
2. 89 − 66 = 23
10. 12 + 14 + 15 = 41
3. 71 + 14 = 85
11. 24 ÷ 6 = 4
4. 42 + 16 = 58
12. 78 − 41 = 37
5. 2 ⋅ 14 = 28
13. 5 + 13 + 25 = 43
6. 93 − 79 = 14
14. Answers may vary.
7. 102 − 32 = 70
15. For example: sum, added to, increased by,
more than, total of, plus
8. 60 ÷ 12 = 5
16. For example: product, times, multiply
34
Section 1.8
Problem-Solving Strategies
22. Given: Population of each country.
Find: Total population of 4 countries.
Operation: Addition
17. For example: difference, minus, decreased
by, less, subtract
18. For example: quotient, divide, per,
distributed equally, shared equally
11
1,339,000,000
127,000,000
140,000,000
+ 33,000,000
1,639,000,000
The population of China, Japan, Russia,
and Canada is 1,639,000,000 people.
19. Given: The height of each mountain
Find: The difference in height
Operation: Subtract
110 2 1110
20 , 3 2 0
− 14, 2 4 6
6, 0 7 4
Denali is 6,074 ft higher than White
Mountain Peak.
23. Given: The number of rows of pixels and
the number of pixels in each row.
Find: The number of pixels on the whole
screen.
Operation: Multiply
20. Given: The number of yearly subscriptions
Find: The difference in subscriptions
Operation: Subtract
5
2 13
126
96
1 756
11 340
12,096
There are 12,096 pixels on the whole
screen.
0 11 1110 11 10
12 , 2 1 2 , 000
− 3, 2 5 2 ,900
8, 9 5 9 ,100
Reader’s Digest has 8,959,100 more
subscriptions than Sports Illustrated.
×
21. Given: Oil consumption by country.
Find: Total oil consumption for
4 countries.
Operation: Addition
24. Given: The number of rows of tiles and
the number of tiles in each row.
Find: The number of tiles on the whole
floor.
Operation: Multiply
8, 220,000
4,360,000
4, 210,000
+ 2,170,000
18,960,000
The oil consumption of China, Japan,
Russia, and Canada is 18,960,000 barrels
per day.
1
62
× 38
11
496
1860
2356
There are 2,356 tiles.
35
Chapter 1
Whole Numbers
25. Given: Number of students and the
average class size.
Find: Number of classes offered
Operation: Division
120
25 3000
−25
50
−50
00
There will be 120 classes of Prealgebra.
29. Given: Yearly tuition for two schools
Find: Total tuition paid
Operation: Addition
1
39, 212
+ 3,024
42, 236
Jeannette will pay $42,236 for one year.
30. Given: Distances traveled in opposite
directions
Find: Total distance traveled
Operation: Addition
26. Given: Inheritance amount and number of
people to share equally
Find: Amount per person
Operation: Division
10 560
8 84, 480
−8
04 4
−4 0
48
−48
00
Each person will receive $10,560.
11
138
+ 96
234
They are 234 mi apart.
31. Given: Miles per gallon and number of
gallons
Find: How many miles
Operation: Multiplication
1
55
× 20
1,100
The Prius can go 1100 mi.
27. Given: 45 miles per gallon and driving
405 miles
Find: How many gallons used
Operation: Division
9
45 405
−405
0
There will be 9 gal used.
32. Given: Hours per week and number of
weeks.
Find: Total number of hours
Operation: Multiplication
1
16
×3
48
The class will meet for 48 hr during the
semester.
28. Given: 52 mph; 1352 mi
Find: How many hours
Operation: Divide
26
52 1352
−104
312
−312
0
They will travel for 26 hours.
33. Given: Number of rows and number of
seats in each row.
Find: Total number of seats
Operation: Multiplication
3
45
× 70
3150
The maximum capacity is 3150 seats.
36
Section 1.8
34. Given: Number of rows and number of
boxes in each row
Find: Total number of boxes
Operation: Multiplication
8
×8
64
There are 64 boxes in a checkerboard.
Problem-Solving Strategies
37. Given: Distance for each route and speed
traveled
Find: Time required for each route
Operations
(1) Watertown to Utica direct
Divide 80 ÷ 40 = 2 hr
(2) Watertown to Syracuse to Utica
Add distances 70 + 50 = 120 mi
Divide 120 ÷ 60 = 2 hr
Each trip will take 2 hours.
35. Given: total price: $16,540
down payment: $2500
payment plan: 36 months
Find: Amount of monthly payments
Operations
(1) Subtract
16,540
− 2 500
14,040
(2) Divide
390
36 14040
− 108
324
−324
00
Jackson’s monthly payments were $390.
38. Given: Distance for each route and speed
traveled
Find: Time required for each route
Operations
(1) Interstate:
Divide 220 ÷ 55 = 4 hr
(2) Back roads:
Divide 200 ÷ 40 = 5 hr
The interstate will take 4 hours and the
back roads will take 5 hours. The
interstate will take less time.
39. The distance around a figure is the
perimeter.
40. The amount of space covered is the area.
36. Given: total cost: 1170
down payment: 150
payment plan: 12 months
Find: Amount of monthly payments
Operations:
(1) Subtract
1170
− 150
1020
(2) Divide
85
12 1020
−96
60
−60
0
Lucio’s monthly payment was $85.
41. Given: The dimensions of a room and cost
per foot of molding
Find: Total cost
Operations:
(1) Add to find the perimeter, subtract
doorway.
11
46
12
−3
11
43 ft
+ 12
46
(2) Multiply to find the total cost.
43
× 2
86
The cost will be $86.
37
Chapter 1
Whole Numbers
42. Given: The dimensions of a yard and the
cost per foot of fence
Find: Total cost
Operations
(1) Add to find perimeter
45. Given: Starting balance in account and
individual checks written
Find: Remaining balance in account
Operations
(1) Add the individual checks
1
1
75
90
75
+ 90
330 ft
(2) Multiply the perimeter by cost per
foot.
330
× 5
1650
It will cost $1650.
82
159
+ 101
$242
(2) Subtract $242 from the initial balance
278
− 242
36
There will be $36 left in Gina’s account.
46. Given: Initial balance in account and
individual checks written
Find: The remaining balance
Operations
(1) Add the individual checks.
43. Given: dimensions of room and cost per
square yard
Find: total cost
Operations
(1) Multiply to find area
6 × 5 = 30 yd 2
(2) Multiply to find total cost
11
587
36
+ 156
$779
(2) Subtract $779 from the initial balance.
1
34
× 30
1020
The total cost is $1020.
2 13 14 15
34 55
− 7 79
26 76
There will be $2676 left in Jose’s account.
44. Given: Dimensions of room and cost per
foot
Find: Total cost
Operations
(1) Multiply to find area.
12
× 20
240
(2) Multiply to find total cost.
240
× 3
720
The total cost is $720.
47. Given: Number of computers and printers
purchased and the cost of each
Find: The total bill
Operations
(1) Multiply to find the amount spent on
computers, then printers.
11 5
33
2118
256
×
72
× 6
4 236
$1536
148 260
$152, 496
(2) Add to find the total bill.
1 11
152, 496
+ 1 536
154,032
The total bill was $154,032.
38
Section 1.8
48. Given: Price for children and adults, and
the number of children and adults
Find: Total cost for the trip
Operations
(1) Multiply to find the amount for
children and for adults.
2
Problem-Solving Strategies
Operations
7 10
(1) Multiply 89 (2) Subtract 4 8 0
−17 8
× 2
30 2
178
She will have $302 left.
4
51. Given: Number of field goals, three-point
shots and free throws and point values
Find: Total points scored
Operations
(1) Multiply
field goals
three-point shots
33
37
× 27
× 6
$222
231
+ 660
$891
(2) Add to find the total.
$ 891
+ 222
$1113
The amount of money required is $1,113.
1
12,192
×
2
24,384
(2) Add
49. Given: Amount to sell used CDs, amount
to buy used CDs and number of CDs sold
(a) Find: Money from selling 16 CDs
Operation: Multiply
16
× 3
48
Latayne will receive $48.
(b) Find: Number of used CDs to buy for
$48.
Operation: Division
48 ÷ 8 = 6
She can buy 6 CDs.
2
581
× 3
1,743
1 1 11
24 384
1 743
+ 7 327
33, 454
Michael Jordan scored 33,454 points with
the Bulls.
52. Given: Width of each picture and width of
the matte frame
Find: Space between each picture
Operations
(1) Multiply 5 × 5 = 25
(2) Subtract 37 − 25 = 12
(3) Divide
12 ÷ 6 = 2
There will be 2 in of matte between the
pictures.
50. Given: Wage per hour and number of
hours worked
(a) Find: Amount of weekly paycheck
Operation: Multiply
40
× 12
80
+ 400
$480
Shevona’s paycheck is worth $480.
(b) Given: Ticket price and number of
tickets
Find: Amount left over from paycheck
53. Given: Number of milliliters in the bottle
and the dosage
(a) Find: Days the bottle will last
Operation: Divide
60 ÷ 2 = 30
One bottle will last for 30 days.
(b) Find: Date to reorder
Operation: Subtract
30 − 2 = 28
The owner should order a refill no
later than September 28.
39
Chapter 1
Whole Numbers
54. Given: Number of male and female
doctors
(a) Find: Difference between male and
female doctors
Operation: Subtract
56. Given: Scale on a map
(a) Find: Actual distance between
Wichita and Des Moines
Operation: Multiply
40
× 8
320
The distance is 320 mi.
(b) Find: The distance between Seattle
and Sacramento on the map.
Operation: Divide
15
40 600
−40
200
−200
0
15 in. represents 600 mi.
9
2 10 13
6 3 0 , 300
− 2 0 5 , 900
4 2 4, 400
The difference between male and
female doctors is 424,400.
(b) Find: The total number of doctors
Operation: Add
1
630,300
+ 205,900
836,200
The total number of doctors is
836,200.
57. Given: Number of books per box and
number of books ordered
Find: Number of boxes completely filled
and number of books left over
Operation: Divide and find remainder
104 R 2
12 1250
−12
050
−48
2
104 boxes will be filled completely with
2 books left over.
55. Given: Scale on a map
(a) Find: Actual distance between Las
Vegas and Salt Lake City
Operation: Multiply
60
× 6
360
The distance is 360 mi.
(b) Find: Distance on map between
Madison and Dallas
Operation: Divide
14
60 840
−60
240
−240
0
14 in. represents 840 mi.
58. Given: Number of eggs in a container and
total number of eggs
Find: Number of containers filled and
number of eggs left over
Operation: Divide and find remainder
354 R 9
12 4257
−36
65
−60
57
−48
9
354 containers will be filled completely
with 9 eggs left over.
40
Section 1.8
59. Given: Total cost of dinner and type of bill
used
(a) Find: Number of $20 bills needed
Operation: Division
4 R4
20 84
−80
4
Four $20 bills are not enough so Marc
needs five $20 bills.
(b) Find: How much change
Operations: Multiply and subtract
20
100
× 5
− 84
100
16
He will receive $16 in change.
61. Given: Hourly wage and number of hours
worked
Find: Amount earned per week
Operations
(1) Multiply to find amount per job.
30 × 4 = 120
10 × 16 = 160
8 × 30 = 240
(2) Add to find total.
1
120
160
+ 240
520
He earned $520.
62. Given: Hourly wage and number of hours
worked
Find: Total paid to all four workers
Operations
(1) Multiply to find amount per worker
36 × 18 = 648
26 × 24 = 624
28 × 15 = 420
22 × 48 = 1056
(2) Add to find total paid.
60. Given: total cost of CDs and type of bill
used
(a) Find: How many $10 bills needed
Operation: Divide
5 R4
10 54
−50
4
Five $10 bills are not enough so
Shawn needs six $10 bills.
(b) Find: How much change
Operations: Multiply and subtract
10
60
× 6
− 54
60
6
He will receive $6 in change.
Chapter 1
Problem-Solving Strategies
1 11
648
420
624
+ 1056
2748
The total amount paid was $2748.
Review Exercises
7. Two hundred forty-five
Section 1.1
8. Thirty-thousand, eight hundred sixty-one
1. 10,024
Ten-thousands
2. 821,811
Hundred-thousands
9. 3602
10. 800,039
3. 92,046
11.
4. 503,160
12.
5. 3 millions + 4 hundred-thousands
+ 8 hundreds + 2 tens
2;
7;
13. 3 < 10 True
6. 3 ten-thousands + 5 hundreds + 5 tens
+ 4 ones
14. 10 > 12 False
41
Chapter 1
Whole Numbers
26. (a) Add the numbers for AA Auto.
31
25
+ 40
96 cars
(b) Add the numbers of Fords.
21
25
+ 20
66 Fords
Section 1.2
15. Addends: 105, 119; sum: 224
16. Addends: 53, 21; sum: 74
2
17.
18
24
+ 29
71
2
18.
27
9
+ 18
54
27.
35,377
+ 10,420
45,797 thousand seniors
28.
30
44
25
53
+ 25
177 m
1
19.
8 403
+ 9 007
17, 410
20.
68, 421
+ 2, 221
70,642
1
Section 1.3
29. minuend: 14
subtrahend: 8
difference: 6
21. (a) The order changed, so it is the
commutative property.
(b) The grouping changed, so it is the
associative property.
(c) The order changed, so it is the
commutative property.
30. minuend: 102
subtrahend: 78
difference: 24
22. 403 + 79; 482
31.
37
− 11
26
26 + 11 = 37
32.
61
− 41
20
20 + 41 = 61
1
403
+ 79
482
23. 44 + 92; 136
92
+ 44
136
9
1 10 10
33.
2 0 05
− 1 8 84
1 21
34.
13 89
− 2 99
10 90
24. 36 + 7 = 43
25. 23 + 6 = 29
2 18
42
Chapter 1
9 9
5 10 1010
35.
9
7 10 13
43.
86 , 0 0 0
− 54 9 8 1
31 ,0 1 9
9 9
6 10 1010
36.
44. 5,234,446
5,000,000
45. 9,332,945
9,330,000
37. 38 − 31; 7
38
− 31
7
38. 111 − 15; 96
10
0 0 11
46.
894,004 → 900,000
− 123,883 → 100,000
800,000
47.
330
489
123
+ 571
48.
140,041, 247 → 140,000,000
− 127,078,679 → 127,000,000
13,000,000
13,000,000 people
49.
96,050
+ 66,517
11 1
−1 5
96
4 11
25 1
−42
20 9
40. 90 − 52; 38
8 10
90
−5 2
3 8
300
500
100
600
1500
50. Factors: 32, 12
Product: 384
5 11
95 , 1 9 1,7 6 1
− 23, 2 9 9,3 2 3
71, 8 9 2, 4 3 8 tons
51. Factors: 33, 40
Product: 1320
52. (a) Yes
(b) Yes
(c) No
2 5,8 0 0, 0 0 0
− 1 8, 6 0 0, 0 0 0
$ 7, 200, 000
53. c
54. e
55. d
43
1
→
96,000
→ + 67,000
163,000 m3
Section 1.5
1 15
42.
→
→
→
→
3 10
39. 251 − 42; 209
41.
48 0 3
− 24 6 7
2,3 3 6 thousand visitors
Section 1.4
67 , 0 0 0
− 32 8 1 2
34 ,1 8 8
10 18
4 0 8 11
Review Exercises
Chapter 1
Whole Numbers
56. a
69. To check a division problem with no
remainder you multiply the quotient by the
divisor to get the dividend.
57. b
58.
1
1
70. To check a division problem with a
remainder you multiply the whole number
part of the quotient by the divisor and add
the remainder to get the dividend.
142
× 43
11
426
+ 5680
6106
58
71. 6 348
−30
48
−48
0
12
1024
59.
×
51
1 024
+ 51 200
52, 224
60.
6 000
5 00
30 00000
3,000,000
61.
26
+ 13
39
62.
3
551
× 7
3857
41 R 7
72. 11 458
−44
18
−11
7
39
× 11
39
390
$429
52 R 3
73. 20 1043
−100
43
−40
3
111
3857
× 2
7714 lb
74.
Section 1.6
3
58
× 6
348 
41
× 11
41
410
451
+ 7
458 
52
× 20
1040
+ 3
1043 
72
= 18
4
12
75. 9 108
−9
18
−18
0
63. 42 ÷ 6 = 7
divisor: 6, dividend: 42, quotient: 7
64. 4 52 = 13
divisor: 4, dividend: 52, quotient: 13
76. Divide 105 by 4.
26 R 1
4 105
−8
25
−24
1
26 photos with 1 left over
65. 3 ÷ 1 = 3 because 1 × 3 = 3.
66. 3 ÷ 3 = 1 because 1 × 3 = 3.
67. 3 ÷ 0 is undefined.
68. 0 ÷ 3 = 0 because 0 × 3 = 0.
44
Chapter 1
77. (a) Divide 60 by 15.
60 ÷ 15 = 4 T-shirts
92. mean =
(b) Divide 60 by 12.
60 ÷ 12 = 5 hats
80 + 78 + 101 + 92 + 94
5
445
=
5
= $89
5
78. 8 ⋅ 8 ⋅ 8 ⋅ 8 ⋅ 8 = 8
94. 6 + 9 + 11 + 13 + 5 + 4 = 8 houses per month
6
4 3
79. 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 5 ⋅ 5 = 2 ⋅ 5
Section 1.8
80. 53 = 5 × 5 × 5 = 25 × 5 = 125
95. Given: Number of animals and species at
two zoos
(a) Find: Which zoo has more animals
and how many more
Operation: Subtraction
17,000
− 4,000
13,000
The Cincinnati Zoo has 13,000 more
animals than the San Diego Zoo.
(b) Find: Which zoo has the most species,
and how many more
Operation: Subtract
81. 44 = 4 × 4 × 4 × 4 = 16 ×16 = 256
82. 17 = 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 ⋅ 1 = 1
83. 106 = 10 × 10 × 10 × 10 × 10 × 10 = 1,000,000
64 = 8 because 8 × 8 = 64.
85.
144 = 12 because 12 × 12 = 144.
7 + 6 + 12 + 5 + 7 + 6 + 13 56
= =8
7
7
93. Average =
Section 1.7
84.
Review Exercises
86. 14 ÷ 7 ⋅ 4 − 1 = 2 ⋅ 4 − 1 = 8 − 1 = 7
87. 10 2 − 5 2 = 100 − 25 = 75
7 10
800
− 750
50
The San Diego Zoo has 50 more
species than the Cincinnati Zoo.
88. 90 − 4 + 6 ÷ 3⋅ 2 = 90 − 4 + 2 ⋅ 2
= 90 − 4 + 4
= 86 + 4
= 90
96. Given: The distance traveled and the
number of trips
(a) Find: Number of miles traveled in one
week
Operations: Multiplication and
addition
15
5
+6
×3
21 miles per week
15
(b) Find: Number of miles traveled in
10 months with 4 weeks a month
Operation: Multiplication
21
84
×4
× 10
84 miles/month
840 miles/year
89. 2 + 3 ⋅ 12 ÷ 2 − 25 = 2 + 3 ⋅ 12 ÷ 2 − 5
= 2 + 36 ÷ 2 − 5
= 2 + 18 − 5
= 20 − 5
= 15
90. 62 − 42 + (9 − 7)3 = 62 − 42 + 23
= 36 − 16 + 8
= 20 + 8
= 28
91. 26 − 2(10 − 1) + (3+ 4 ⋅11)
= 26 − 2(9) + (3+ 44)
= 26 − 2(9) + 47
= 26 − 18 + 47
= 8 + 47
= 55
45
Chapter 1
Whole Numbers
97. Given: Contract: 252,000,000
Time period: 9 years
taxes: 75,600,000
Find: Amount per year after taxes
Operations
(1) Subtract
98. Given: dimensions of a rectangular garden
and size of division for plants
(a) Find: Number of plants
Operations
(1) Multiply
12 × 8 = 96
(2) Divide
96 ÷ 2 = 48
She should purchase 48 plants.
(b) Find: Cost of plants for $3 each
Operation: Multiply
14 11
1 4 1 10
252 , 000,000
− 75,600,000
176, 400,000
(2) Divide
19,600,000
9 176, 400,000
−9
86
−81
54
−54
0
He will receive $19,600,000 per year.
2
48
× 3
144
The plants will cost $144.
(c) Find: Perimeter of garden and cost of
fence
Operations
(1) Add
12 + 8 + 12 + 8 = 40
(2) Multiply
40 × 2 = $80
The fence costs $80.
(d) Find: Total cost of garden
Operations: Add
144
+ 80
224
Aletha’s total cost will be $224.
46
Chapter 1
Chapter 1
Test
492 hundreds
23,441 thousands
2,340,711 millions
340,592 ten-thousands
1. (a)
(b)
(c)
(d)
21 R9
10. 15 324
−30
24
−15
9
2. (a) 4,065,000
(b) Twenty-one million, three hundred
twenty-five thousand
(c) Twelve million, two hundred eightyseven thousand
(d) 729,000
(e) Eleven million, four hundred ten
thousand
9 9
2 10 10 12
11.
5.
6.
12.
51
+ 78
129
82
× 4
328
14.
154
− 41
113
5 00000
× 3 000
1,500,000,000
21
15.
34
89
191
+ 22
336
16. 403(0) = 0
17. 0 16 is undefined.
3
7
9.
10 ,984
− 2 881
8 103
20
13. 42 840
−84
00
227
7. 4 908
−8
10
−8
28
−28
0
8.
3 0 0 2
−2 4 5 6
5 4 6
0 10
3. (a) 14 > 6
(b) 72 < 81
4.
Test
18. (a) (11 ⋅ 6) ⋅ 3 = 11 ⋅ (6 ⋅ 3)
The associative property of
multiplication; the expression shows a
change in grouping.
58
× 49
522
2 320
2,842
11
149
+ 298
447
47
Chapter 1
Whole Numbers
26. (a) Subtract to find the change from year 2
to year 3.
(b) (11 ⋅ 6) ⋅ 3 = 3 ⋅ (11 ⋅ 6)
The commutative property of
multiplication; the expression shows a
change in the order of the factors.
2 9 11
21 3 , 0 1 5
− 2 1 2, 5 7 3
4 4 2 thousand subscribers
(b) The largest increase was from year 3
to year 4. The increase was 15,430
thousand.
19. (a) 4,850 → 4,900
(b) 12,493 → 12,000
(c) 7,963,126 → 8,000,000
20.
690,951 →
+ 739,117 →
1
690,000
740,000
1, 430,000
There were approximately 1,430,000
people.
27. Divide the number of calls by the number
of weeks.
North: 80 ÷ 16 = 5
South: 72 ÷ 18 = 4
East: 84 ÷ 28 = 3
The North Side Fire Department is the
busiest with an average of 5 calls per
week.
2
4
21. 8 ÷ 2 = 64 ÷ 16 = 4
22. 26 ⋅ 4 − 4(8 − 1) = 26 ⋅ 4 − 4 ⋅ 7
= 26 ⋅ 2 − 4 ⋅ 7
= 52 − 28
= 24
28. Add the sides.
1
15
31
32
15
32
+ 31
156 mm
23. 36 ÷ 3(14 − 10) = 36 ÷ 3(4) = 12(4) = 48
24. 65 − 2 (5 ⋅ 3 − 11) 2 = 65 − 2 (15 − 11) 2
= 65 − 2 (4) 2
= 65 − 2 ⋅ 16
= 65 − 32
= 33
29. Add to find the perimeter.
13
47
128
47
+ 128
350 ft
Multiply to find the area.
128
× 47
896
5120
2
6016 ft
25. Given: Quiz scores and number of quizzes
for Brittany and Jennifer
Find: Who has the higher average
Operations: Find the average of each
group.
Brittany:
29 + 28 + 24 + 27 + 30 + 30 168
=
= 28
6
6
Jennifer:
30 + 30 + 29 + 28 + 28 145
=
= 29
5
5
Jennifer has the higher average of 29.
Brittany has an average of 28.
3
30.
48
2379 →
2400
× 1872 → ×
1900
2 160 000
2 400 000
2
4,560,000 m
Chapter 2
Fractions and Mixed Numbers: Multiplication
and Division
Chapter Opener Puzzle
Section 2.1
Introduction to Fractions and Mixed Numbers
Section 2.1 Practice Exercises
1. (a) fractions
13. 2 ÷ 0; undefined
(b) numerator; denominator
14. 11 ÷ 0; undefined
(c) proper
15.
3
4
2
2.
7
16.
1
2
3. Numerator: 2; denominator: 3
17.
5
9
18.
3
5
19.
1
6
20.
4
7
21.
3
8
22.
2
3
(d) improper
(e) mixed
4. Numerator: 8; denominator: 9
5. Numerator: 12; denominator: 11
6. Numerator 1; denominator: 2
7. 6 ÷ 1; 6
8. 9 ÷ 1; 9
9. 2 ÷ 2; 1
10. 8 ÷ 8; 1
11. 0 ÷ 3; 0
12. 0 ÷ 7; 0
49
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
23.
3
4
43.
9
8
24.
1
4
44.
7
4
25.
1
8
45.
7 3
;1
4 4
26.
2
1
or
8
4
46.
13 1
;3
4 4
27.
41
103
47.
13 5
;1
8 8
28.
43
103
48.
5 1
;2
2 2
29.
10
21
3 4 ×1 + 3 7
49. 1 =
=
4
4
4
30.
10
63
1 6 × 3 + 1 19
50. 6 =
=
3
3
3
31. Proper
2 4 × 9 + 2 38
51. 4 =
=
9
9
9
32. Proper
1 3 × 5 + 1 16
52. 3 =
=
5
5
5
33. Improper
34. Improper
3 3 × 7 + 3 24
53. 3 =
=
7
7
7
35. Improper
36. Improper
2 8 × 3 + 2 26
54. 8 =
=
3
3
3
37. Proper
1 7 × 4 + 1 29
55. 7 =
=
4
4
4
38. Proper
39.
5
2
40.
4
3
41.
12
4
42.
27
9
3 10 × 5 + 3 53
56. 10 =
=
5
5
5
57. 11
5 11 × 12 + 5 137
=
=
12
12
12
1 12 × 6 + 1 73
58. 12 =
=
6
6
6
50
Section 2.1
3 21 × 8 + 3 171
59. 21 =
=
8
8
8
Introduction to Fractions and Mixed Numbers
2
70. 18 43
−36
7
1 15 × 2 + 1 31
60. 15 =
=
2
2
2
5
71. 9 52
−45
7
3 2 × 8 + 3 19
61. 2 =
=
8
8
8
19 eighths
5
72. 12 67
−60
7
3 2 × 5 + 3 13
62. 2 =
=
5
5
5
13 fifths
3 1× 4 + 3 7
63. 1 =
=
4
4
4
7 fourths
12
73. 11 133
−11
23
−22
1
2 5 × 3 + 2 17
64. 5 =
=
3
3
3
17 thirds
5
51
−50
1
4
65. 8 37
−32
5
5
4
8
74. 10
1
66. 7 13
−7
6
6
1
7
3
75. 6 23
−18
5
7
67. 5 39
−35
4
4
7
5
4
68. 4 19
−16
3
2
69. 10 27
−20
7
4
16
76. 7 115
−7
45
−42
3
3
4
77. 7
2
7
10
51
44
309
−28
29
−28
1
2
7
18
5
5
7
12
12
5
7
9
1
11
1
10
3
5
6
16
3
7
44
1
7
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
230
78. 4 921
−8
12
−12
1
−0
1
1056
79. 5 5281
−5
2
−0
28
−25
31
−30
1
901
80. 8 7213
−72
1
−0
13
−8
5
810
81. 11 8913
−88
11
−11
3
−0
3
185
82. 23 4257
−23
195
−184
117
−115
2
230
1056
12
83. 15 187
−15
37
−30
7
1
4
20
84. 34 695
−68
15
−0
15
1
5
85.
86.
901
5
8
87.
88.
89.
3
810
11
90.
91.
185
2
23
92.
93.
94.
52
12
7
15
20
15
34
Section 2.1
Introduction to Fractions and Mixed Numbers
95. False
97. True
96. True
98. True
Section 2.2
Prime Numbers and Factorization
Section 2.2 Practice Exercises
1. (a) factor
(b) prime
(c) composite
(d) prime
15.
Product
36
42
30
15
81
Factor
12
7
30
15
27
Factor
3
6
1
1
3
Sum
15
13
31
16
30
Product
36
42
45
72
24
Factor
9
7
15
18
8
Factor
4
6
3
4
3
Difference
5
13
12
14
5
2. c. Between 2 and 3
3.
8 4
;
12 12
4.
5 1
;
2 2
5.
16.
5 3
;
4 4
6.
6
; improper
5
17. A whole number is divisible by 2 if it is an
even number.
7.
7
; proper
12
18. A whole number is divisible by 10 if its
ones-place digit is 0.
8.
6
; improper
6
19. A whole number is divisible by 3 if the
sum of its digits is divisible by 3.
4
9. 5 23
−20
3
20. A whole number is divisible by 5 if its
ones-place digit is 5 or 0.
3
4
5
21. 45
(a)
(b)
(c)
(d)
2 6 × 7 + 2 44
10. 6 =
=
7
7
7
11. For example: 2 ⋅ 4 and 1 ⋅ 8
No; 45 is not even.
Yes; 4 + 5 = 9 is divisible by 3.
Yes; the ones-place digit is 5.
No; the ones-place digit is not 0.
22. 100
(a) Yes; 100 is even.
(b) No; 1 + 0 + 0 = 1 is not divisible by 3.
(c) Yes; the ones-place digit is 0.
(d) Yes; the ones-place digit is 0.
12. For example: 2 ⋅ 10 and 4 ⋅ 5
13. For example: 4 ⋅ 6 and 2 ⋅ 2 ⋅ 2 ⋅ 3
14. For example: 1 ⋅ 14 and 2 ⋅ 7
53
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
23. 137
(a) No; 137 is not even.
(b) No; 1 + 3 + 7 = 11 is not divisible by
3.
(c) No; the ones-place digit is not 0 or 5.
(d) No; the ones-place digit is not 0.
5
30. 22 110
−110
0
Yes, 110 is divisible by 22.
24. 241
(a) No; 241 is not even.
(b) No; 2 + 4 + 1 = 7 is not divisible by 3.
(c) No; the ones-place digit is not 0 or 5.
(d) No; the ones-place digit is not 0.
32. Prime
31. Prime
25. 108
(a) Yes; 108 is even.
(b) Yes; 1 + 0 + 8 = 9 is divisible by 3.
(c) No; the ones-place digit is not 0 or 5.
(d) No; the ones-place digit is not 0.
33. Composite
2 ⋅ 5 = 10
34. Composite
3 ⋅ 7 = 21
35. Composite
3 ⋅ 17 = 51
36. Composite
3 ⋅ 19 = 57
37. Prime
38. Prime
26. 1040
(a) Yes; 1040 is even.
(b) No; 1 + 0 + 4 + 0 = 5 is not divisible
by 3.
(c) Yes; the ones-place digit is 0.
(d) Yes; the ones-place digit is 0.
39. Neither
27. 3140
(a) Yes; 3140 is even.
(b) No; 3 + 1 + 4 + 0 = 8 is not divisible
by 3.
(c) Yes; the ones-place digit is 0.
(d) Yes; the ones-place digit is 0.
43. Prime
28. 2115
(a) No; 2115 is not even.
(b) Yes; 2 + 1 + 1 + 5 = 9 is divisible by
3.
(c) Yes; the ones-place digit is 5.
(d) No; the ones-place digit is not 0.
47. There are two whole numbers that are
neither prime nor composite, 0 and 1.
40. Neither
41. Composite
11 ⋅ 11 = 121
42. Composite
3 ⋅ 23 = 69
44. Prime
45. Composite
3 ⋅ 13 = 39
46. Composite
7 ⋅ 7 = 49
48. False; the square of any prime number is
divisible by that prime number.
49. False; 9 is not prime.
50. False; 2 is not composite.
3
29. 28 84
−84
0
Yes, 84 is divisible by 28.
51. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47
52. 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37,
41, 43, 47, 53, 59, 61, 67, 71, 73, 79
54
Section 2.2
53. No, 9 is not a prime number.
Prime Numbers and Factorization
11
64. 7 77
54. No, 8 is not a prime number.
55. Yes
3 231
56. Yes
11
65. 7 77
7
57. 5 35
2 ⋅ 5 ⋅ 7 = 70
2 308
2 616
3 ⋅ 3 ⋅ 5 ⋅11 = 32 ⋅ 5 ⋅11 = 495
13
66. 7 91
3 165
3 495
13
59. 5 65
2 ⋅ 2 ⋅ 2 ⋅ 7 ⋅11 = 23 ⋅ 7 ⋅11 = 616
2 154
2 70
11
58. 5 55
3 ⋅ 7 ⋅ 11 = 231
2 ⋅ 2 ⋅ 7 ⋅13 = 22 ⋅ 7 ⋅13 = 364
2 182
2 364
2 ⋅ 2 ⋅ 5 ⋅13 = 22 ⋅ 5 ⋅13 = 260
67. 47 is prime.
2 130
68. 41 is prime.
2 260
7
5
35
60.
69. 1, 2, 3, 4, 6, 12
70. 1, 2, 3, 6, 9, 18
5 ⋅ 5 ⋅ 7 = 52 ⋅ 7 = 175
71. 1, 2, 4, 8, 16, 32
5 175
7
61. 7 49
72. 1, 5, 11, 55
73. 1, 3, 9, 27, 81
3 ⋅ 7 ⋅ 7 = 3 ⋅ 72 = 147
74. 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, 60
3 147
17
62. 3 51
75. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
76. 1, 2, 3, 4, 6, 8, 9, 12, 18, 24, 36, 72
2 ⋅ 3 ⋅ 17 = 51
77. No; 30 is not divisible by 4.
2 102
23
63. 3 69
78. No; 46 is not divisible by 4.
79. Yes; 16 is divisible by 4.
2 ⋅ 3 ⋅ 23 = 138
80. Yes; 64 is divisible by 4.
2 138
81. Yes; 32 is divisible by 8.
82. Yes; 520 is divisible by 8.
55
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
83. No; 126 is not divisible by 8.
88. No; 1 + 5 + 8 + 7 = 21 is not divisible by
9.
84. No; 58 is not divisible by 8.
89. Yes; 522 is even and 5 + 2 + 2 = 9 is
divisible by 3.
85. Yes; 3 + 9 + 6 = 18 is divisible by 9.
86. Yes; 4 + 1 + 4 = 9 is divisible by 9.
90. Yes; 546 is even and 5 + 4 + 6 = 15 is
divisible by 3.
87. No; 8 + 4 + 5 + 3 = 20 is not divisible by
9.
91. No; 5917 is not even.
Section 2.3
92. No; 6 + 3 + 9 + 4 = 22 is not divisible by
3.
Simplifying Fractions to Lowest Terms
Section 2.3 Practice Exercises
1. lowest
5
8. 3 15
2. (a) No
(b) Yes
(c) Yes
(d) No
29
3. 5 145
19
4. 3 57
2 30
2 60
2 120
5 ⋅ 29 = 145
13
9. 5 65
2 ⋅ 3 ⋅ 19 = 114
5
10. 3 15
2
2 ⋅ 2 ⋅ 23 = 2 ⋅ 23 = 92
3 45
2 92
17
6. 3 51
2 90
2
3 ⋅ 3 ⋅17 = 3 ⋅17 = 153
2 180
3 153
17
7. 5 85
3 ⋅ 5 ⋅ 13 = 195
3 195
2 114
23
5. 2 46
2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 23 ⋅ 3 ⋅ 5 = 120
11.
5 ⋅ 17 = 85
56
2⋅2⋅3⋅3⋅5 = 22 ⋅32 ⋅5 = 180
Section 2.3
Simplifying Fractions to Lowest Terms
12.
23. 8 × 27  9 × 20
216 ≠ 180
8 20
≠
9 27
13.
24. 5 × 18  6 × 12
90 ≠ 72
5 12
≠
6 18
14.
15. False; 5 × 5 ≠ 4 × 4
16. Two fractions are equivalent if they both
represent the same part of a whole.
17. 2 × 5  3 × 3
10 ≠ 9
2 3
≠
3 5
18. 1 × 9  4 × 2
9≠8
1 2
≠
4 9
19. 1 × 6  2 × 3
6=6
1 3
=
2 6
20. 6 × 8  16 × 3
48 = 48
6 3
=
16 8
21. 12 × 4  16 × 3
48 = 48
12 3
=
6 4
22. 4 × 15  5 × 12
60 = 60
4 12
=
5 15
57
25.
12
2⋅2⋅3
1
=
=
24 2 ⋅ 2 ⋅ 2 ⋅ 3 2
26.
15
3 ⋅5
5
=
=
18 2 ⋅ 3 ⋅ 3 6
27.
6
2⋅3
1
=
=
18 2 ⋅ 3 ⋅ 3 3
28.
21
3 ⋅7
7
=
=
24 2 ⋅ 2 ⋅ 2 ⋅ 3 8
29.
36 2 ⋅ 2 ⋅ 3 ⋅ 3 9
=
=
20
2 ⋅ 2 ⋅5
5
30.
49
7 ⋅7
7
=
=
42 2 ⋅ 3 ⋅ 7 6
31.
15
3 ⋅5
5
=
=
12 2 ⋅ 2 ⋅ 3 4
32.
30 2 ⋅ 3 ⋅ 5 6
=
=
25
5 ⋅5
5
33.
20 2 ⋅ 2 ⋅ 5 4
=
=
25
5 ⋅5
5
34.
8
8
1
=
=
16 2 ⋅ 8 2
35.
14
=1
14
36.
8
=1
8
37.
50 2 ⋅ 25
=2
=
25
25
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
38.
24 4 ⋅ 6
=
=4
6
6
53.
4 2 ⋅2 2
6−2
=
=
=
10 + 4 14 2 ⋅7 7
39.
9
=1
9
54.
9 −1 8 2 ⋅2⋅2 4
= =
=
15 + 3 18 2 ⋅3⋅3 9
40.
2
=1
2
55.
5−5 0
= =0
7−2 5
41.
105
3⋅ 5 ⋅ 7
3
=
=
140 2 ⋅ 2 ⋅ 5 ⋅ 7 4
56.
11 − 11 0
= =0
4 + 7 11
42.
84
2 ⋅2⋅ 3 ⋅ 7 2
=
=
126 2 ⋅ 3 ⋅ 3 ⋅ 7 3
57.
7−2 5
= = undefined
5−5 0
43.
33 3 ⋅ 11
=
=3
11
11
58.
4 + 7 11
= = undefined
11 − 11 0
65 5 ⋅ 13
44.
=
= 13
5
5
59.
8−2 6
2 ⋅3 3
=
=
=
8 + 2 10 2 ⋅ 5 5
77 7 ⋅ 11
7
=
=
45.
110 10 ⋅ 11 10
60.
15 + 3 18 6 ⋅ 3 3
=
=
=
15 − 3 12 6 ⋅ 2 2
85
5 ⋅ 17
5
=
=
46.
153 3 ⋅ 3 ⋅ 17 9
61.
120 12
2 ⋅ 2 ⋅3
3
=
=
=
160 16 2 ⋅ 2 ⋅ 2 ⋅ 2 4
62.
720 72 8 ⋅ 9
9
=
=
=
800 80 8 ⋅ 10 10
63.
3000 30 2 ⋅ 3 ⋅ 5 5
=
=
=
1800 18 2 ⋅ 3 ⋅ 3 3
64.
2000 20 2 ⋅ 2 ⋅ 5 4
=
=
=
1500 15
3⋅ 5
3
65.
42, 000 42 2 ⋅ 21 21
=
=
=
22, 000 22 2 ⋅ 11 11
66.
50, 000 50 2 ⋅ 5 ⋅ 5 10
=
=
=
65, 000 65
5 ⋅ 13 13
67.
5100
51
3 ⋅ 17
17
=
=
=
30,000 300 3 ⋅ 100 100
68.
9800
98
2 ⋅ 7 ⋅7
7
=
=
=
28,000 280 2 ⋅ 2 ⋅ 2 ⋅ 5 ⋅ 7 20
130
2 ⋅ 5 ⋅ 13 13
47.
=
=
150 2 ⋅ 3 ⋅ 5 ⋅ 5 15
70
2 ⋅ 5 ⋅7
7
48.
=
=
120 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 12
385 5 ⋅ 7 ⋅ 11 77
49.
=
=
195 3 ⋅ 5 ⋅ 13 39
50.
51.
52.
39
3 ⋅ 13
3
=
=
130 2 ⋅ 5 ⋅ 13 10
34 2 ⋅ 17 2
=
=
85 5 ⋅ 17 5
69
3 ⋅ 23
3
=
=
92 2 ⋅ 2 ⋅ 23 4
58
Section 2.3
20
2 ⋅ 2 ⋅5
5
=
=
48 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 12
Tails: 48 − 20 = 28
28
2 ⋅ 2 ⋅7
7
=
=
48 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 12
Simplifying Fractions to Lowest Terms
76. (a) 15
=
27
16
=
(b)
36
69. Heads:
3 ⋅5
5
=
3 ⋅3⋅3 9
2 ⋅ 2 ⋅2⋅2 4
=
2 ⋅ 2 ⋅3⋅3 9
77. (a) 300,000,000
(b) 36,000,000
36, 000 , 000
36
(c)
=
300, 000 , 000 300
2 ⋅ 3 ⋅ 2 ⋅3
3
=
=
2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 5 25
70 2 ⋅ 5 ⋅ 7 2
70.
=
=
105 3 ⋅ 5 ⋅ 7 3
6
2 ⋅3 3
=
=
26 2 ⋅ 13 13
(b) 26 − 6 = 20
20 2 ⋅ 2 ⋅ 5 10
=
=
26
2 ⋅ 13 13
71. (a)
78. (a) 300,000,000
(b) 75,000,000
300, 000 , 000 300
(c)
=
75, 000 , 000
75
2⋅2⋅ 3 ⋅ 5 ⋅ 5 4
=
=
1
3⋅5 ⋅5
(d) 4 times greater
12
2 ⋅ 2 ⋅3
3
=
=
88 2 ⋅ 2 ⋅ 2 ⋅ 11 22
36 2 ⋅ 2 ⋅ 3 ⋅ 3
9
=
=
(b)
88 2 ⋅ 2 ⋅ 2 ⋅ 11 22
72. (a)
73. (a) Jonathan: 25 5 ⋅ 5 5
=
=
35 5 ⋅ 7 7
24 2 ⋅ 2 ⋅ 2 ⋅ 3 6
=
=
Jared:
28
2 ⋅ 2 ⋅7
7
(b) Jared sold the greater fractional part
6 5
because > .
7 7
6 9 12
, ,
8 12 16
2 3 4
80. For example, , ,
6 9 12
79. For example,
3 ⋅5
5
74. (a) Lynette: 15
=
=
24 2 ⋅ 2 ⋅ 2 ⋅ 3 8
14
2 ⋅7
7
=
=
Lisa:
16 2 ⋅ 2 ⋅ 2 ⋅ 2 8
(b) Lisa has completed more of her course
7 5
because > .
8 8
75. (a) Raymond:
720 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 10
=
=
792
2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 11 11
540 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 9
Travis:
=
=
660
2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 11 11
(b) Raymond read the greater fractional
10 9
part because
> .
11 11
59
81. For example,
6 4 2
, ,
9 6 3
82. For example,
40 8 4
, ,
50 10 5
83.
792 8
=
891 9
84.
728 13
=
784 14
85.
779 41
=
969 51
86.
462 21
=
220 10
87.
493 29
=
510 30
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
88.
871 13
=
469 7
89.
969 3
=
646 2
Section 2.4
90.
713 31
=
437 19
Multiplication of Fractions and Applications
Section 2.4 Practice Exercises
1. (a) one-tenth
1
(b) bh
2
9.
2
5
33
(b)
8
2. (a) 3
10.
3. Numerator: 10; denominator: 14
10 2 ⋅ 5 5
=
=
14 2 ⋅ 7 7
4. Numerator: 32; denominator: 36
32 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 8
=
=
36
2 ⋅ 2 ⋅3⋅3
9
5. Numerator: 25; denominator: 15
25 5 ⋅ 5 5
=
=
15 3 ⋅ 5 3
6. Numerator: 2100; denominator: 7000
2100 21
3⋅ 7
3
=
=
=
7000 70 2 ⋅ 5 ⋅ 7 10
7.
8.
11.
1 1 1 ⋅1 1
⋅ =
=
2 4 2⋅4 8
12.
2 1 2 ⋅1 2
⋅ =
=
3 5 3 ⋅ 5 15
13.
3
3 8 24
⋅8 = ⋅ =
=6
4
4 1 4
14.
2
2 20 40
⋅ 20 = ⋅
=
=8
5
5 1
5
15.
1 3 1× 3 3
× =
=
2 8 2 × 8 16
16.
2 1 2 ×1 2
× =
=
3 3 3× 3 9
17.
14 1 14 ⋅ 1 14
⋅ =
=
9 9 9 ⋅ 9 81
18.
1 9 1⋅ 9 9
⋅ =
=
8 8 8 ⋅ 8 64
 12  2  12 × 2 24
=
19.    =
 7  5  7 × 5 35
60
Section 2.4
Multiplication of Fractions and Applications
 9  7  9 × 7 63
=
20.    =
 10  4  10 × 4 40
3⋅4⋅5
1
 12  5  12 ⋅ 5
=
=
36.    =
45
4
45
4
3
3
5
4
3
⋅
⋅
⋅
⋅
  
 1  8 1 8 ⋅1 8
=
21. 8 ⋅   = ⋅ =
 11  1 11 1 ⋅11 11
 17  72  17 ⋅ 72 17 ⋅ 8 ⋅ 9 8
=
= =8
37.    =
1
9 ⋅ 17
 9  17  9 ⋅17
 2  3 2 3⋅ 2 6
=
22. 3 ⋅   = ⋅ =
 7  1 7 1⋅ 7 7
 39  11  39 ⋅11 3 ⋅ 13 ⋅ 11 3
=
= =3
38.    =
1
11 ⋅ 13
 11  13  11 ⋅13
23.
4
4 6 4 ⋅ 6 24
⋅6 = ⋅ =
=
5
5 1 5 ⋅1 5
39.
21 16 3 ⋅ 7 4 ⋅ 4 12
⋅ =
⋅
= = 12
4 7
4
7
1
24.
5
5 5 5 ⋅ 5 25
⋅5 = ⋅ =
=
8
8 1 8 ⋅1 8
40.
85 12 5 ⋅ 17 2 ⋅ 2 ⋅ 3 17
⋅ =
⋅
=
= 17
6 10 2 ⋅ 3
2⋅5
1
25.
26.
27.
13 5 13 × 5 65
× =
=
9 4 9 × 4 36
41. 12 ×
6 7 6 × 7 42
× =
=
5 5 5 × 5 25
42. 4 ×
2 3
2
3 2
× =
× =
9 5 3 ⋅ 3 5 15
28.
1 4
1
4 1
× =
× =
8 7 2 ⋅ 4 7 14
29.
5 3
5
3 5
× =
× =
6 4 2⋅ 3 4 8
30.
7 18
7
2 ⋅ 3 ⋅ 3 21
× =
×
=
12 5 2 ⋅ 2 ⋅ 3
5
10
31.
21 25 3 ⋅ 7 5 ⋅ 5
35
⋅
=
⋅
=
5 12
5 2⋅2⋅ 3 4
43.
44.
45.
16 15 16 3 ⋅ 5
3
⋅ =
⋅
=
32.
25 32 5 ⋅ 5 2 ⋅ 16 10
46.
24 5 2 ⋅ 2 ⋅ 2 ⋅ 3 5 8
33.
⋅ =
⋅ =
15 3
3⋅5
3 3
34.
49 6
7 ⋅7
2⋅3 7
⋅ =
⋅
=
24 7 2 ⋅ 2 ⋅ 2 ⋅ 3
7
4
 6  22  6 ⋅ 22 2 ⋅ 3 ⋅ 2 ⋅ 11 4
=
=
35.    =
5
11 ⋅ 3 ⋅ 5
 11  15  11 ⋅ 15
47.
61
15 2 ⋅ 2 ⋅ 3
3 ⋅5
30
=
×
=
42
1
2 ⋅ 3 ⋅7 7
8
2⋅2
2⋅2⋅2
8
=
×
=
92
1
2 ⋅ 2 ⋅ 23 23
9 16 25
× ×
15 3 8
3 ⋅ 3 2 ⋅ 2 ⋅ 2 ⋅2
5 ⋅5
=
×
×
3⋅5
3
2⋅2⋅2
10
=
= 10
1
49 4 20
7 ⋅7
2 ⋅ 2 2 ⋅2⋅ 5
× ×
=
×
×
8 5 7
2⋅2⋅2
5
7
14
= = 14
1
5 10 7 5 2 ⋅ 5 7 5
× × = ×
× =
2 21 5 2 3 ⋅ 7 5 3
55 18 24
×
×
9 32 11
2 ⋅3⋅3
2 ⋅ 2 ⋅ 2 ⋅3
5⋅ 11
=
×
×
3 ⋅ 3 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅2
11
15
=
2
7 3
7
3
5 3
⋅ ⋅5 =
⋅
⋅ =
10 28
2⋅ 5 2⋅2⋅ 7 1 8
Chapter 2
48.
49.
50.
Fractions and Mixed Numbers: Multiplication and Division
11 2
11
2
3 ⋅ 5 11
⋅ ⋅ 15 =
⋅
⋅
=
18 20
2 ⋅ 3 ⋅3 2⋅2⋅ 5
1
12
 2  5 2 
60.  5 ⋅  =  ⋅  = 23 = 8
 5  1 5 
100
14 2 ⋅ 2 ⋅ 5 ⋅ 5 3 ⋅ 7 2 ⋅ 7
× 21 ×
=
×
×
49
7⋅7
5⋅5
25
1
24
=
= 24
1
1 1
1
1 3
1
61.  ⋅  =   = ⋅ =
15 15 225
 15 
9 5
1
2
2
3
38
5 2 ⋅ 19 11 5 5
×
×
= =5
× 11× =
22
19 2 ⋅ 11 1 19 1
1
2
 10 1   1  2 1 1
1
62. 
⋅
 =  = ⋅ =
30 30 900
 3 100   30 
10
3
1 1 1
1
 1
51.   = ⋅ ⋅ =
 10  10 10 10 1000
3
2
2
1
1
1  21 8  1 6
63. ⋅ 
⋅ = ⋅ =2
3  4 7 3 1
4
1 1 1 1
1
 1 
52.   = ⋅ ⋅ ⋅ =
10 10 10 10 10,000
 10 
6
3
1 1 1 1 1 1
1
53.   = ⋅ ⋅ ⋅ ⋅ ⋅
 10  10 10 10 10 10 10
1
=
1,000,000
 1
54.  
 10 
3
3
1
6
3
1  24 30  1 18
64.
⋅
⋅
=3
= ⋅
6  5
8  6 1
1
1
3
1
2
16  1 
16 1 2
⋅ =
⋅  =
65.
9 2
9 8 9
9
1
1 1 1 1 1 1 1 1 1
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
10 10 10 10 10 10 10 10 10
1
=
1,000,000,000
2
66.
7
28  3 
28 9 21
⋅ =
⋅  =
6 2
6 4 2
2
2
1 1 1
1
55.   = ⋅ =
9 9 81
9
67.
2
1 1 1
1
56.   = ⋅ =
4
4 4 16
 
68.
3
3 3 3 27
3
57.   = ⋅ ⋅ =
2 2 2 8
2
3
4 4 4 64
4
58.   = ⋅ ⋅ =
3 3 3 27
3
3
69.
3
 3  4 3 
59.  4 ⋅  =  ⋅  = 33 = 27
 4  1 4 
62
3
1
Section 2.4
70.
Multiplication of Fractions and Applications
1
23 3 23 2
⋅ =
ft
80. A = l × w =
24 4 32
8
4
1
1
1 11 8
71. A = bh = (11)(8) = ⋅ ⋅ = 44 cm 2
2
2
2 1 1
1
81. A = (8)(4) + (8)(4) = 32 + 4 ⋅ 4 = 32 + 16
2
= 48 yd 2
1
1
1
1 15 12
72. A = bh = (15)(12) = ⋅ ⋅
2
2
2 1 1
6
1
82. A = (8)(3) + (8)(3) = 24 + 4 ⋅ 3 = 24 + 12
2
2
= 36 m
1
= 90 in.2
1  7 1  2
2
7
83. A = (6)   + (6)   = 3⋅ + 3⋅
2  3 2  3
3
3
3 7 3 2
= ⋅ + ⋅ = 7 + 2 = 9 cm 2
1 3 1 3
4
1
1
1 8 8
73. A = bh = (8)(8) = ⋅ ⋅ = 32 m2
2
2
2 1 1
1
1  9  1  15 
15
9
(8)   + (8)   = 4 ⋅ + 4 ⋅
2  4 2  4 
4
4
4 9 4 15
= ⋅ + ⋅ = 9 + 15 = 24 m 2
1 4 1 4
1
17
1 7 1 7
74. A = bh =   (1) = ⋅ ⋅ = ft 2
2
2 4
2 4 1 8
1
84. A =
4
1
1 8 1 5 8
75. A = bh = (5)   = ⋅ ⋅ = 4 yd 2
2
2 5 2 1 5
1
76. A =
1
1  16 
1
bh = (3)  
2  9
2
1
=
2
5
5 16
85. ⋅ 16 = ⋅
= 10
8
8 1
1
The amount left is 10 gal.
2750
3 11,000
3
⋅11,000 = ⋅
= 8250
86.
4
4
1
8
1 3 16 8
2
⋅ ⋅
= or 2 mm 2
3
3
2 1 9
1
1
3
The cost is $8250.
1 1 1
87.
⋅ =
4 2 8
1
Trey ate of the pizza for breakfast.
8
1
77. A = l × w =
3 1 1
⋅ = cm 2
4 3 4
1
1
1
1 2 1
⋅ =
88.
4 5 10
8
8 3
78. A = l × w = ⋅ 3 = ⋅ = 8 m2
3
3 1
2
1
1
of the sample has O-negative blood.
10
13 15 195
79. A = l × w = ⋅ =
in.2
16 16 256
63
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
1
89. 3 1 3 11 33
⋅5 = ⋅ =
= 4 Corrine will
4 2 4 2
8
8
1
prepare 4 lb.
8
90.
(b)
2
2 2 4
2
96. (a)   = ⋅ =
7 7 49
7
3
2 3 422 1 211 211
3
⋅ 140 = ⋅
= ⋅
=
= 52 ;
8
3 8 3
4 1
8
4
3
52 lb must be destroyed.
4
(b)
3, 275, 000
91.
2
2 9,825,000
⋅ 9,825,000 = ⋅
3
1
3
1
= 6,550,000
There are 6,550,000 viewers.
1
3 3 3 9
92. 3⋅ = ⋅ = or 2
4
4 1 4 4
9
1
or 2 hr a day.
Nancy spends
4
4
400
1
1 1 1
=
⋅ =
25
5 5 5
98.
1
1 1
1
=
⋅ =
100
10 10 10
99.
64
8 8 8
=
⋅ =
81
9 9 9
100.
9
3 3 3
=
⋅ =
4
2 2 2
1
300
1
1 1200
= $300
Second place: ⋅ 1200 = ⋅
4
4
1
4
2 2 2
=
⋅ =
49
7 7 7
97.
101.
2
2 1200
= $800
93. First place: ⋅ 1200 = ⋅
3
3
1
102.
1
1 1
1 1
1 1
1
, =
, =
,
=
2 4 2 ⋅ 2 8 4 ⋅ 2 16 8 ⋅ 2
1
1
The next number is
= .
16 ⋅ 2 32
2 2
2 2
2
, =
,
=
3 9 3 ⋅ 3 27 9 ⋅ 3
The next number is
100
Third place:
1
1 1 1
=
⋅ =
36
6 6 6
1
1 1200
⋅ 1200 =
⋅
= $100
12
1
12
1
103.
12
2
2 40 36
94.
= 960
⋅ (40)(36) = ⋅ ⋅
3
3 1 1
1
40 × 36 = 1440
1440 − 960 = 480
Frankie mowed 960 yd 2 . He has 480 yd 2
left to mow.
104.
2
1 1 1
1
95. (a)   = ⋅ =
6 6 36
6
64
11 1
 =
2  8  16
1 1  1
 =
8  2  16
They are the same.
2 1 2 1
 = =
3  4  12 6
1 2 2 1
 = =
4  3  12 6
They are the same.
2
2
= .
27 ⋅ 3 81
Section 2.5
Division of Fractions and Applications
Section 2.5 Practice Exercises
1. reciprocals
1
6
1
(d) No,
is undefined.
0
(c) Yes,
2. 22 • 33
2
9 22 18
×
=
3.
5
11 5
1
3
1
24 7
⋅ =3
4.
7 8
1
1
2
1
34 5
⋅
=2
5.
5 17
1
13.
8
7
14.
6
5
15.
9
10
16.
5
14
17.
1
4
18.
1
9
1
1
7 3 7 7
6. 3 ⋅   = ⋅ =
6 1 6 2
2
1
5
 5  8 5
=
7. 8 ⋅   = ⋅
24
1
24 3
 
19. No reciprocal exists.
 2   7  14
=1
8.     =
 7   2  14
21.
1
3
22.
1
5
20. No reciprocal exists.
3
 9   5  45
=1
9.     =
 5   9  45
10.
11.
23. multiplying
24. multiplying
1
1 10 10
× 10 = ⋅ =
=1
10
10 1 10
1
1 3 3
×3 = ⋅ = =1
3
3 1 3
2
=2
1
3
(b) Yes,
5
25.
2 5
2 12
2 2⋅2⋅ 3 8
÷ = ⋅ =
⋅
=
15 12 15 5 3 ⋅ 5
5
25
26.
11 6 11 5 55
÷ = ⋅ =
3 5 3 6 18
27.
7 2 7 5 35
÷ = ⋅ =
13 5 13 2 26
12. (a) Yes,
65
Chapter 2
28.
Fractions and Mixed Numbers: Multiplication and Division
8 3 8 10 80
÷ = ⋅ =
7 10 7 3 21
7
14 6 14 5 35
÷ =
⋅ =
29.
3 5 3 6 9
1
4 1 4 3
42.
÷ = ⋅ =4
3 3 3 1
1
1
1
15 3 15 2
⋅ =5
÷ =
2 3
2 2
1
1
1
1
5
1
4
4 12 4
43. 12 ⋅ =
⋅ = 16
3 1 3
1
9 9 9 2 1
⋅ =
÷ =
32.
10 2 10 9 5
34.
3
5 24 5
44. 24 ⋅ =
⋅ = 15
8
1 8
1
3 3 3 4 12
÷ = ⋅ = =1
4 4 4 3 12
10
9
13
9 1000 90
÷
=
⋅
=
45.
100 1000 100 13
13
6 6 6 5 30
÷ = ⋅ =
=1
5 5 5 6 30
35. 7 ÷
1
2 7 3 21
= ⋅ =
3 1 2 2
100
1000 10 1000 3 300
46.
÷ =
⋅
=
17
3
17 10 17
3 4 5 20
36. 4 ÷ = ⋅ =
5 1 3 3
1
3
37.
1
1
2
33.
5
2
11 3 11 4 22
÷ = ⋅ =
30.
2 4 2 3 3
31.
1
10 1 10 18
÷ = ⋅
= 20
41.
9 18 9 1
3
5
2
30 15 30 8
2
÷ =
=
⋅
40.
40 8
5
40 15
4
5
1
1
36 25
47.
= 20
⋅
9
5
12
12 1 3
÷4=
⋅ =
5
5 4 5
1
2
4
13 10 26
48.
⋅
=
5 17 17
20
20 1 4 2
38.
÷5 =
⋅ = =
6
6 5 6 3
1
1
1
1
2
2
1
7 1 7 4 7
49.
÷ = ⋅ =
8 4 8 1 2
9 18
9 25 1
÷
=
⋅
=
39.
50 25 50 18 4
2
1
7 5 7 3 7
50.
÷ =
⋅ =
12 3 12 5 20
4
66
Section 2.5
1
62.
5 2
5
51.
⋅ =
8 9 36
Division of Fractions and Applications
40 18 5 ⋅ 8 3 ⋅ 6 48
⋅
=
⋅
=
21 25 3 ⋅ 7 5 ⋅ 5 35
4
1
16 8 3
3
63. 8 ÷ = ⋅
=
3 1 16 2
1
1 4 1
52.
⋅ =
16 3 12
2
4
1
15 5 4
4
64. 5 ÷ = ⋅
=
4 1 15 3
2
4 6 4
53. 6 ⋅ = ⋅ = 8
3 1 3
3
1
65.
2
5 12 5
54. 12 ⋅ =
⋅ = 10
6
1 6
2
2
6
2
by , and ÷ 6
⋅ 6 multiplies
3
3
1
3
2
2
1
2
2 6
multiplies
by . So ⋅ 6 = ⋅ = 4
3
6
3
3 1
1
1
2
1
16
16 1 2
55.
÷8 =
⋅ =
5
5 8 5
2
2 1 1
and ÷ 6 = ⋅ = .
3
3 6 9
1
3
6
2
2
2
multiplies 8 by , and 8 ÷
3
3
3
3
2 8 2 16
multiplies 8 by . So 8 ⋅ = ⋅ =
2
3 1 3 3
66. 8 ⋅
42
42 1 6
56.
÷7 =
⋅ =
11
11 7 11
1
4
8
2 8 3
and 8 ÷ = ⋅ = 12.
3 1 2
16 2 16 5 40
57.
÷ =
⋅ =
3 5
3 2
3
1
1
1
2
7
1
27 1 3
=
⋅ =
7 9 7
1
1 16
=2
⋅ 16 = ⋅
8
8 1
1
3
60.
1
3
2
59.
27
67. 54 ÷ 2 ÷ 9 = 54 ⋅ 3 ÷ 9 = 27 ÷ 9
21 3
7
21 2
17 1 17 4 17
58.
÷ = ⋅ =
8 4 8 1
2
Ź
1
16
1
7
1
48 8
16
48 3
÷ ÷8=
⋅ ÷8= ÷8
68.
7
56 8
56 3
2
2 9
⋅9 = ⋅ = 6
3
3 1
1
2
16 1 2
=
⋅ =
7 8 7
22 5 2 ⋅ 11 5
55
⋅ =
⋅
=
61.
7 16
7
2 ⋅ 8 56
1
67
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
1
1
1
2
1
70.
1
2
3 3 8 9 8
= ⋅ ⋅ = ⋅ = 18
2 2 1 4 1
5 35 1 5 16 1 2 1 1
÷ ⋅ = ⋅
⋅ = ⋅ =
8 16 4 8 35 4 7 4 14
7
1
1
2
2
20 15  2 2  20
15  2 
⋅  ÷
= ⋅ ⋅ ÷
77.
21 16  3 3  21
16  3 
2
 3
9 3 3 9
9
9
÷
71.   ÷ = ⋅ ÷ =
14 8 8 14 64 14
 8
9 14
9
2 ⋅7 7
= ⋅ =
⋅
=
64 9 2 ⋅ 32 9
32
15 4 20 3 ⋅5 4
20
⋅ ÷
=
⋅
÷
16 9 21 4 ⋅ 4 3 ⋅3 21
5 20 5 21
= ÷
= ⋅
12 21 12 20
5 3 ⋅7 7
=
⋅
=
3 ⋅ 4 4 ⋅ 5 16
=
2
7  1
7  1 1 7 1
÷  = ÷ ⋅  = ÷
72.
8  2
8  2 2 8 4
1
7 4 7
= ⋅ =
8 1 2
2
78.
2
1
2
73.
2
2
1
2
3
2
2




 
76. 25 ÷ 50 ⋅8 =  25 ⋅ 9  ⋅8 = 3 ⋅8
 3
 2 
9 
 3 50 
3 6 5 3 7 5 7 5 7
÷ ⋅ = ⋅ ⋅ =
⋅ =
69.
5 7 3 5 6 3 10 3 6
2
8 9 13
8
9
13
⋅ ÷ =
⋅
÷
27 16 18 3⋅ 9 2 ⋅ 8 18
1 13 1 18 1 3⋅ 6
3
= ÷ = ⋅ = ⋅
=
6 18 6 13 6 13 13
=
2
 2 8
 2 3
 3
3 3
 5 ÷ 3  =  5 ⋅ 8  =  20  = 20 ⋅ 20
4
=
9
400
2
1 2
2
9 1 9 8
79.
÷ = ⋅ = 18
4 8 4 1
1
2


74.  5 ÷ 2  = 5 ⋅ 3 =  5  = 5 ⋅ 5


 12 3 
 8 
8 8
 12 2 
2
4
=
2
8  3
13 8  3 3  13
⋅  ÷ =
⋅ ⋅ ÷
27  4 
18 27  4 4  18
4 1 4 6
÷ = ⋅ =8
80.
3 6 3 1
25
64
1
7
1
18
2
2 36 3
81. 36 ÷ =
⋅ = 54
3
1 2
2
 63 4 
 63 9 
7
⋅  ⋅4 =  ⋅4
75.  ÷  ⋅ 4 = 
 8 4
2
 8 9
2
1
1
Li wrapped 54 packages.
1
20
7 7 4 49 4
= ⋅ ⋅ =
⋅ = 49
2 2 1 4 1
3 60 4
82. 60 ÷ =
⋅ = 80
4
1 3
1
1
She can sell 80 parcels of land.
68
Section 2.5
Division of Fractions and Applications
(c) $240,000 − $24,000 = $216,000
He will have to finance $216,000.
8
3 1 3 16
83.
= 24 cups of juice
÷ = ⋅
2 16 2 1
1
90. (a)
25
5
1
5 100
84.
= 125 cm
÷
= ⋅
4 100 4 1
1
1
1 19,560
⋅ 19,560 = ⋅
12
12
1
19,560
=
12
= 1630
The down payment is $1630.
4
815
3 16 3
85. 16 ⋅ =
⋅ = 12
4
1 4
(b)
1
1
The stack will be 12 in. high.
$815 = $815
Althea will have to pay $815.
6
86. 24 ⋅
5 24 5
=
⋅ = 30
4
1 4
(c) $19,560 − $1630 = $17,930
She will have to finance $17,930.
1
Yes, the books will take up only 30 in.
3
1 9 3
91. (a)
⋅ =
3 4 4
9
87. (a) 18 ÷
1
1 1630
⋅ 1630 = ⋅
= 815 $1630 −
2
2
1
2 18 3
=
⋅ = 27
3
1 2
1
1
She plans to sell
27 commercials in 1 hr
(b) 27 × 24 = 648
648 commercials in 1 day
(b) She keeps
1 20 2
=
⋅ = 40
2 1 1
40 commercials in 1 hr
3
1
2
2
of the land.
3
1
2 9 3
⋅ = or 1 acres
2
3 4 2
88. (a) 20 ÷
(b) 40 × 24 = 960
960 commercials in 1 day
89. (a)
1
3
acre.
4
7
1
1
1 42
92. (a) ⋅ (24 + 18) = ⋅ (42) = ⋅
=7
6
6
6 1
1
1 240,000
⋅ 240,000 = ⋅
10
10
1
240,000
=
10
= 24,000
The down payment is $24,000.
1
Josh has read 7 pages.
(b) (24 + 18) − 7 = 42 − 7 = 35
He still must read 35 pages.
2
8000
7 1 7 8
93.
÷ = ⋅ = 14
4 8 4 1
2
2 24,000
= 16,000
⋅ 24,000 = ⋅
1
3
3
1
1
She can prepare 14 samples.
Ricardo’s mother will pay $16,000.
(b) $24,000 − $16,000 = $8000
Ricardo will have to pay $8000.
69
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
97. The product will be less than 47 because
3
is less than one.
5
2
7 1 7 16
94.
= 14
÷ = ⋅
8 16 8 1
1
Tony must make 14 strikes.
98. The product will be less than 81 because
4
is less than one.
7
95. The length is 12 ft, because
5 30 2 5 ⋅ 6 2 12
30 ÷ = ⋅ =
⋅ =
= 12
1 5 1
2 1 5
99. The quotient will be more than 25 because
2
is between zero and one.
3
4
m, because
7
8 1
2 ⋅4 1
4
8 ÷ 14 = ⋅ =
⋅
=
1 2 ⋅7 7
1 14
96. The width is
100. The quotient will be more than 41 because
2
is between zero and one.
11
Problem Recognition Exercises: Multiplication and Division of
Fractions
1. (a)
8 6 8 3 ⋅ 2 16
⋅ = ⋅
=
3 5 3 5
5
(c)
12 ÷
(b)
6 8 3 ⋅ 2 8 16
⋅ =
⋅ =
5 3
5 3 5
(d)
9
9 1 3 ⋅3 1
3
÷ 12 = ⋅ =
⋅
=
8
8 12
8 3 ⋅ 4 32
(c)
8 6 8 5 2 ⋅4 5
20
÷ = ⋅ =
⋅
=
3 5 3 6
3 2 ⋅3 9
(d)
6 8 6 3 2 ⋅3 3
9
÷ = ⋅ =
⋅
=
5 3 5 8
5 2 ⋅ 4 20
2. (a)
10 12 10 3 ⋅ 4 40
⋅ = ⋅
=
3 7
7
3 7
(b)
12 10 3 ⋅ 4 10 40
⋅ =
⋅ =
7
7 3
7
3
(c)
10 12 10 7
2 ⋅5 7
35
÷ = ⋅ =
⋅
=
3 2 ⋅6 18
3 7
3 12
(d)
12 10 12 3 2 ⋅ 6 3
18
÷ = ⋅ =
⋅
=
7 2 ⋅5 35
7
3
7 10
3 15 3 3⋅ 5 3 9
⋅ = =9
4. (a) 15⋅ = ⋅ =
1 5 1
5 1 5
(b)
3
3 15 3 3⋅ 5 9
⋅15 = ⋅ = ⋅
= =9
5
5 1 5 1
1
3 15 5 3 ⋅ 5 5 25
⋅ =
= 25
(c) 15 ÷ = ⋅ =
5 1 3
1 3 1
(d)
5. (a)
(b)
9 12 9 3 ⋅ 4 9
27
⋅
=
3. (a) 12 ⋅ = ⋅ =
8 1 8
1 2⋅ 4
2
(c)
9
9 12
9 3⋅ 4 27
⋅12 = ⋅ =
⋅
=
8
8 1 2⋅ 4 1
2
(d)
(b)
9 12 8 3 ⋅ 4 8
32
= ⋅ =
⋅
=
1 3 ⋅3 3
8 1 9
70
3
3 1 3 1
1
÷ 15 = ⋅ = ⋅
=
5
5 15 5 3 ⋅ 5 25
5 5 25
⋅ =
6 6 36
5 6 1
⋅ = =1
6 5 1
5 5 5 6 1
÷ = ⋅ = =1
6 6 6 5 1
5 6 5 5 25
÷ = ⋅ =
6 5 6 6 36
Problem Recognition Exercises: Multiplication and Division of Fractions
9
⋅0 = 0
8
9
(b) 0 ⋅ = 0
8
9
(c)
÷ 0 = Undefined
8
9
8
(d) 0 ÷ = 0 ⋅ = 0
8
9
6. (a)
(d)
10. (a)
(b)
1 2 16
1 2 4 ⋅4
8
⋅ ⋅ =
⋅ ⋅
=
12 3 21 3 ⋅ 4 3 21 189
1 2 16 1 2 21
⋅ ÷ = ⋅ ⋅
(b)
12 3 21 12 3 16
1 2 3 ⋅7 7
=
= ⋅ ⋅
12 3 2 ⋅ 8 96
7. (a)
(c)
(d)
1 2 16
1
3 4 ⋅ 2 ⋅2
÷ ⋅ =
⋅ ⋅
12 3 21 3 ⋅ 4 2
21
2
=
21
1 2 16 1 3 21
÷ ÷ = ⋅ ⋅
(d)
12 3 21 12 2 16
1 3 21 21
⋅ ⋅ =
=
3 ⋅ 4 2 16 128
(c)
8. (a)
(b)
9
1 9 1 4
÷6÷ = ⋅ ⋅
10
4 10 6 1
3⋅ 3
1
2⋅2 3
⋅
⋅
=
=
1
5
2 ⋅5 2 ⋅ 3
4 1
2⋅ 2
1
10 2
⋅ ⋅10 =
⋅
⋅
=
5 20
5 2 ⋅ 10 1
5
4 1
4 1 1
1
⋅ ÷ 10 = ⋅
⋅ =
5 20
5 4 ⋅5 10 250
4 1
4 20 2 ⋅ 5 160
÷ ⋅ 10 = ⋅ ⋅
=
5 20
1
1
5 1
= 160
4 1
4 20 1
÷ ÷ 10 = ⋅ ⋅
5 20
5 1 10
2⋅ 2 4⋅ 5 1
8
=
⋅
⋅
=
1 2 ⋅5 5
5
2
2
⋅1 =
3
3
2 2
(b) 1⋅ =
3 3
2
2
(c)
÷1=
3
3
2
3 3
(d) 1 ÷ = 1⋅ =
3
2 2
11. (a)
1 7 2
7
⋅ ⋅ =
2 9 3 27
1 7 2 1 7 3 1 7 3 7
⋅ ÷ = ⋅ ⋅ = ⋅
⋅ =
2 9 3 2 9 2 2 3 ⋅ 3 2 12
12. (a)
1 7 2 1 9 2 1 3 ⋅3 2 3
÷ ⋅ = ⋅ ⋅ = ⋅
⋅ =
2 9 3 2 7 3 2 7 3 7
1 7 2 1 9 3 27
(d)
÷ ÷ = ⋅ ⋅ =
2 9 3 2 7 2 28
9
1 9 6 1
9. (a)
⋅6⋅ = ⋅ ⋅
10
4 10 1 4
9 2 ⋅3 1
27
= ⋅
⋅
=
10 1
2 ⋅ 2 20
9
1 9 6 4
(b)
⋅6 ÷ = ⋅ ⋅
10
4 10 1 1
9
2 ⋅ 3 4 108
=
⋅
⋅ =
5
2 ⋅5 1 1
9
1 9 1 1
÷6⋅ = ⋅ ⋅
(c)
10
4 10 6 4
3⋅ 3 1 1 3
=
⋅
⋅ =
10 2 ⋅ 3 4 80
6 1
2 ⋅3 1
3
6 ÷ 10 = ⋅ =
⋅
=
1 2 ⋅5 5
1 10
10 1 2 ⋅5 1
5
⋅ =
⋅
=
1 2 ⋅3 3
1 6
(c) 6 ⋅10 = 60
(d) 10 ⋅ 6 = 60
(c)
(b) 10 ÷ 6 =
1
= 8 ⋅ 4 = 32
4
1 8
(b) 8 ⋅ = = 2
4 4
(c) 8 ÷ 4 = 2
(d) 8 ⋅ 4 = 32
13. (a)
14. (a)
(b)
71
8÷
1
1 1 1
÷2= ⋅ =
7
7 2 14
1
1 2 2
⋅2 = ⋅ =
7
7 1 7
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
2
1 1 1
⋅ =
7 2 14
1 1 1 2 2
÷ = ⋅ =
7 2 7 1 7
(c)
(d)
16. (a)
2
 1
2 1 1 3 3
(b)   ÷ = ⋅ ⋅ =
3 2 2 2 8
 2
1
1
1 2 ⋅8 1
42 ⋅ = 4 ⋅ 4 ⋅ = 16 ⋅ =
⋅
6
6
6
1 2 ⋅3
8
=
3
1
1
6
(b) 42 ÷ = 4 ⋅ 4 ÷ = 16 ⋅ = 16 ⋅ 6 = 96
6
6
1
2
15. (a)
(c)
1  2
1 2 2 2
⋅
= ⋅ ⋅ =
2  3 
2 3 3 9
2
(d)
2
(c)
 1 2 1 1 2 1
 2  ⋅ 3 = 2 ⋅ ⋅ 3 = 6
2
 1
4 1 1 4
4
1
4⋅  = ⋅ ⋅ =
=
=
6
1
6
6
36
9
 
4 ⋅9
1  2
1  2 2 1 4
÷  = ÷ ⋅  = ÷
2  3
2  3 3 2 9
1 9 9
= ⋅ =
2 4 8
2
 1
4  1 1 4  1 
(d) 4 ÷   = ÷  ⋅  = ÷  
1  6 6  1  36 
 6
4 36
= ⋅ = 144
1 1
Section 2.6
Multiplication and Division of Mixed Numbers
Section 2.6 Practice Exercises
4
1. improper
52
52 1
4 2
7.
÷ 13 =
⋅
= =
18
18 13 18 9
1
5 2
5
2.
⋅ =
6 9 27
1
8. 1. Multiply the whole number by the
denominator.
2. Add the result to the numerator.
3. Write the result from step 2 over the
denominator.
3
2
13 10 26
3.
⋅
=
5 9
9
1
2
1
3
1
2 3 × 5 + 2 17
9. 3 =
=
5
5
5
20 10 20 3 2
÷ =
⋅
=
4.
9
3
9 10 3
10. 2
6
42 7 42 2 12
5.
÷ =
⋅ =
11 2 11 7 11
7 2 × 10 + 7 27
=
=
10
10
10
4 1 × 7 + 4 11
11. 1 =
=
7
7
7
1
8
1 4 × 8 + 1 33
12. 4 =
=
8
8
8
32
32 1 8
6.
÷4=
⋅ =
15
15 4 15
1
72
Section 2.6
12
13. 6 77
−6
17
−12
5
12
1
5
6
1 5 7 5 5
19. 2 ⋅ = ⋅ =
3 7 3 7 3
1
5
14. 11 57
−55
2
5
1
3 5
−3
2
2
11
=1
2
3
7
1
1 4 49 4 7
20. 6 ⋅ =
⋅ =
8 7 2
8 7
2
9
15. 4 39
−36
3
16. 2
Multiplication and Division of Mixed Numbers
15
31
−2
11
−10
1
9
3
4
15
1
2
3
2 7
−6
1
=3
1
1
2
1
2
38 9
21. 4 ⋅ 9 = ⋅ = 38
9
9 1
1
2
1
10 6
22. 3 ⋅ 6 = ⋅ = 20
3
3 1
1
 2  1  12 37 37
17.  2  3  =
⋅
=
5
 5  12  5 12
1
1
7
5 37
−35
2
=7
1
 3   1  83 16 83
⋅
=
23.  5   5  =
3
 16   3  16 3
2
5
1
13
3
1
2
27
3 83
−6
23
−21
2
 1  3  26 15 39
18.  5  3  =
⋅
=
2
 5  4  5 4
2
19
39
−2
19
−18
1
= 19
1
2
= 27
2
3
2
9
1
1
 2   1  26 27
⋅
= 18
24.  8   2  =
3 13
 3   13 
5
 1
29 10 145
⋅
=
25.  7  ⋅10 =
4 1
2
 4
2
73
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
72
2 145
−14
5
−4
1
1
8 1 53 4 53 3 53
5
35. 5 ÷ 1 = ÷ = ⋅ =
=4
9 3 9 3 9 4 12
12
1
= 72
2
3
1
4
3 64 13 64 5 64
12
36. 12 ÷ 2 =
÷ =
⋅ =
=4
5
5 5 5
5 13 13
13
1
1
8 3
 2
26.  2  ⋅ 3 = ⋅ = 8
3 1
 3
8
1
1
1 5 17 5 16 40
6
37. 2 ÷ 1 = ÷ = ⋅
=
=2
2 16 2 16 2 17 17
17
1
5
27. 4 ⋅ 0 = 0
8
28. 0 ⋅ 6
2
3
7 38 19 38 12 24
4
38. 7 ÷ 1 =
÷ =
⋅
=
=4
5 12 5 12
5 19
5
5
1
=0
10
1
1
 1   1  7 15 15
1
29.  3   2  = ⋅ = = 7
2
2
 2  7 2 7
1
2
1
1
1
1 9 9 9 4
39. 4 ÷ 2 = ÷ = ⋅ = 2
2
4 2 4 2 9
1
1
5
 3   1  13 5 13
5
⋅ = =1
30.  1   1  =
8
 10   4  10 4 8
1
5
1 35 7 35 3 5
1
40. 5 ÷ 2 =
÷ =
⋅ = =2
6
3 6 3
6 7 2
2
2
2
1
 2   2   4  27 2 9 54
4
⋅ ⋅ =
=2
31.  5     1  =
25
 5   9   5  5 9 5 25
41. 0 ÷ 6
1
7
42. 0 ÷ 1
1
 1   3   8  49 11 8 77
1
⋅ ⋅ =
= 19
32.  6   2    =
8 4 7
4
4
 8  4   7 
1
1
7
=0
12
9
=0
11
1
5 1 17 1 17 6
43. 2 ÷ = ÷ = ⋅ = 17
6 6 6 6 6 1
1
1
2
7
3 17 11 17 4 34
⋅ =
33. 1 ÷ 2 = ÷ =
10
4 10 4 10 11 55
1
1 1 13 1 13 2
44. 6 ÷ = ÷ = ⋅ = 13
2 2 2 2 2 1
5
1
17
2
1 3 51 3 51 4 34
4
34. 5 ÷ = ÷ =
⋅ =
=6
10 4 10 4 10 3 5
5
5
2
1 2 4 2 4 7 14
2
45. 1 ÷ = ÷ = ⋅ = = 4
3 7 3 7 3 2 3
3
1
1
74
Section 2.6
Multiplication and Division of Mixed Numbers
3
3
7 3 7 1 7
54. 1 ÷ 3 = ÷ = ⋅ =
4
4 1 4 3 12
7
Each child will inherit $
million.
12
1 5 15 5 15 13 39
4
46. 2 ÷ = ÷ =
⋅ =
=5
7 13 7 13 7 5
7
7
1
1
7 2 7 1 7
3
47. 3 ÷ 2 = ÷ = ⋅ = = 1
2
2 1 2 2 4
4
7
1
71 14
= 497
55. (a) Lucy: 35 × 14 = ⋅
2
2 1
2
14 3 14 1 14
5
48. 4 ÷ 3 = ÷ = ⋅ = = 1
3
3 1 3 3 9
9
1
5
1
85 10
Ricky: 42 × 10 = ⋅
= 425
2
2 1
2
3
19 8
49. 4 ⋅ 8 = ⋅ = 38
4
4 1
1
497 − 425 = 72
Lucy earned $72 more than Ricky.
1
Tabitha earned $38.
(b) 497 + 425 = 922
Together they earned $922.
3500
2
8 10,500
50. 2 ⋅ 10,500 = ⋅
= 28,000
1
3
3
17 28 41 28 24 672
=
÷
=
⋅ =
24 1 24 1 41 41
16
= 16
41
16
The roll is 16
ft long.
41
56. 28 ÷ 1
1
The land will cost Kurt $28,000.
5
7
257 25 1285
1
⋅
=
= 642
51. 25 ⋅ 25 =
10
2
2
10 1
2
1
2
1
1 11 11 11 10
=2
⋅
57. 2 ÷ 1 = ÷ =
5 10 5 10 5 11
1
Average Americans consume 642 lb.
2
1
1
4
52. 12 ÷
3 12 4 16
=
⋅ = = 16
4 1 3 1
5
3 5 15 11 55
7
58. 3 ⋅ 1 =
⋅ =
=6
4 6
4 6
8
8
1
2
Kayla will have 16 doses.
2
1
1 6 9 6 8 16
1
59. 6 ÷ 1 = ÷ = ⋅ = = 5
8 1 8 1 9 3
3
3 1 7 1 7 4
53. (a) 1 ÷ = ÷ = ⋅ = 7 weeks old
4 4 4 4 4 1
3
1
1 8 7 8 3 24
3
60. 8 ÷ 2 = ÷ = ⋅ =
=3
3 1 3 1 7 7
7
1 1 17 1
(b) 2 ÷ =
÷
8 4 8 4
1
1
17 4 17
1
= ⋅ =
= 8 weeks old
2
2
8 1
2
9
2 7 2 27 9
4
61.
⋅2 = ⋅
= =1
3 10 3 10 5
5
1
1
75
5
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
1
4 1 4 41 41
5
62.
⋅5 = ⋅ =
=6
3 8 3 8
6
6
2
63. 4
1
⋅0 = 0
12
=
3
1
1
1
3
1
57 3 4 19
3
=
⋅ ⋅ =
=2
8 4 9 8
8
7
1
21 9 21 1 7
1
65. 10 ÷ 9 = ÷ =
⋅ = =1
2
2 1
2 9 6
6
1
5
5 25 40 21
76. 3 ÷ 5 ÷ 1 =
÷
÷
8
7 16 8
7 16
3
2 8 2 17 34
⋅1 = ⋅ =
7 9 7 9 63
5
1
2
25 7 16 10 5
=
⋅
⋅
=
=
8 40 21 24 12
2
67. 0 ÷ 9 = 0
3
1
8
3
77. The perimeter of the garden is
2(20) + 2(15) = 40 + 30 = 70 ft.
1
3
1 3 5 3 2
3
÷2 = ÷ = ⋅ =
8
2 8 2 8 5 20
14
1 70 5 70 4
70 ÷ 1 =
÷ =
⋅ = 56
4 1 4
1 5
4
1
3
56 bricks will be needed.
56 × $3 = $168
The total cost is $168.
1 12 1 3
1
69. 12 ⋅ =
⋅ = =1
8 1 8 2
2
2
3
4
70. 20 ⋅
1
62 31
4
=
=3
18 9
9
19
1
68.
2
1 1
1 57 4 9
75. 7 ÷ 1 ÷ 2 =
÷ ÷
8 3
4 8 3 4
2
1
16 6
64. 5 ⋅ 6 = ⋅ = 32
3
3 1
66.
1
 1   4   14  31 11 14
74.  5   1    = ⋅
⋅
 6   7   33  6 7 33
1
1
1 129 43 129 2
=3
÷ =
⋅
78. 64 ÷ 21 =
2
2
2
2
2 43
2 20 2 8
2
⋅
=
= =2
15
1 15 3
3
1
3
1
It takes 3 gallons of gas for Sara to get to
and from work.
3 × $5 = $15
It costs Sara $15 each day.
8
71. 6 ÷ 0 is undefined.
9
1
72. 0 ⋅ 2 = 0
8
1
2
1
1
79. 12 ⋅ 25 = 318
3
8
4
3
 2   7   3  17 7 15 21
73.  3     3  =
⋅
⋅
=
5 34 4
8
 5   34   4 
1
=2
1
1
1
80. 38 ÷ 12 = 3
3
2
15
2
5
8
76
Section 2.6
Multiplication and Division of Mixed Numbers
5
1
18
81. 56 ÷ 3 = 17
6
6
19
1
5
404
84. 106 ÷ 41 = 2
9
6
753
1
1
1
82. 25 ⋅18 = 466
5
2
5
1
3
1
85. 11 ⋅ 41 = 480
2
4
8
83. 32
7
1
99
÷ 12 = 2
12
6
146
Chapter 2
8
1
5
86. 9 ⋅ 28 = 280
9
3
27
Review Exercises
Section 2.1
1.
1
2
2.
4
7
5
11. 9 47
−45
2
12.
5
3
(b) Improper
6.
23
7
or 2
8
8
7.
23
2
=1
21 21
134
16. 7 941
−7
24
−21
31
−28
3
1
4. (a)
6
(b) Proper
7
15
2
9
13−15.
3. (a)
5.
5
17. 26
7
1
or 1
6
6
1 6 × 7 + 1 43
8. 6 =
=
7
7
7
60
1582
−156
22
−0
22
134
60
3
7
22
11
= 60
26
13
Section 2.2
2 11 × 5 + 2 57
9. 11 =
=
5
5
5
18. 21, 51, 1200
1
19. 55, 140, 260, 1200
1 1 17 1 17 4
10. 4 ÷ = ÷ = ⋅ = 17
4 4 4 4 4 1
20. 58, 124, 140, 260, 1200
1
21. Prime
77
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
31. 15 × 14  21× 10
210 = 210
15 10
=
21 14
22. Composite 44 = 4 × 11
23. Neither
24. Neither
2
25. 2 4
2 8
2 16
2 32
32.
5
5
1
=
=
20 4 ⋅ 5 4
33.
14 2 ⋅ 7 2
=
=
49 7 ⋅ 7 7
34.
24 3 ⋅ 8 3
=
=
16 2 ⋅ 8 2
35.
63 9 ⋅ 7 7
=
=
27 9 ⋅ 3 3
36.
17
=1
17
37.
42 2 ⋅ 21
=
=2
21
21
38.
120 12 3 ⋅ 4 4
=
=
=
150 15 3 ⋅ 5 5
39.
1400 14
2 ⋅7
7
=
=
=
2000 20 2 ⋅ 10 10
2 64
2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 = 26 = 64
11
26. 5 55
3 165
2 330
2 ⋅ 3 ⋅ 5 ⋅ 11 = 330
3
27. 3 9
5 45
5 225
2 450
40.
2 900
2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 5 = 22 ⋅ 32 ⋅ 52 = 900
42 3 ⋅ 14 14
=
=
45 3 ⋅ 15 15
45 − 42 = 3
1
3
3
1
=
=
45 3 ⋅ 15 15
28. 1, 2, 3, 4, 6, 8, 12, 16, 24, 48
1
29. 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
Section 2.3
30. 3 × 9  6 × 5
18 ≠ 30
3 5
≠
6 9
41. (a)
6 2 ⋅3 3
=
=
10 2 ⋅ 5 5
(b)
6 2⋅ 3 2
=
=
15 3 ⋅ 5 5
Section 2.4
42.
78
3 2 6
× =
5 7 35
Chapter 2
43.
Review Exercises
3
4 8 32
× =
3 3 9
1
17 6 17
 17 
54. A = (12)   = 6 ⋅ = ⋅ = 51 ft 2
2
2 1 2
 2
1
7
44. 14 ⋅
9 14 9
=
⋅ = 63
2 1 2
2
5 8 10
1
55. A = lw = ⋅ =
or 3 m 2
4 3 3
3
1
1
3
45. 33 ⋅
5 33 5
=
⋅
= 15
11 1 11
56. A =
1
20
1 20
⋅3+ ⋅ ⋅6
3
2 3
1
1
1
2 5 36 1
⋅ ⋅
=
46.
9 8 25 5
1
1
5
1
3
4
5
1
2
lumber.
900
2
2  1  2 2  1 1 
49.   ⋅   =  ⋅  ⋅  ⋅ 
 5   10   5 5   10 10 
1
1 3600
58.
⋅ 3600 = ⋅
= 900
1
4
4
1
1
4
1
= ⋅
25 100
There are 900 African American students.
25
300
1
1 3600
= 300
⋅ 3600 =
⋅
59.
1
12
12
1
=
625
1
3
1
There are 300 Asian American students.
3
 3 2  1 
1 1 1
1
⋅  =  = ⋅ ⋅ =
50. 
 20 3   10  10 10 10 1000
10
60.
1
1
3
7
1
or 3 yd of
2
2
Maximus requires
4
1 1 1 1
1
 1 
48.   = ⋅ ⋅ ⋅ =
10 10 10 10 10,000
 10 
1
1
7 4 7 7
1
57. 4 ⋅ = ⋅ = or 3
8 1 8 2
2
7
2
1
1
45 6 28 12
47.
=
⋅
⋅
7 10 63 7
1
2
= 20 + 20
= 40 yd 2
5
4
1
10
20 3 1 20 6
=
⋅ + ⋅
⋅
3 1 2 3 1
41
1
1000 1
 1   1000 
⋅
=
51.   
=
17
 10   17  1000 17
1 1
1 1 3600 3600
⋅ ⋅ 3600 = ⋅ ⋅
=
= 300
2 6
2 6 1
12
There are 300 Hispanic female students.
300
1 5
1 5 3600 1500
⋅ ⋅ 3600 = ⋅
⋅
=
= 750
61.
2 12
2 12
1
2
1
1
1
52. A = bh
2
There are 750 Caucasian male students.
53. A = lw
79
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
Section 2.5
1
1
1
3 4
62.
⋅ =1
4 3
1
1
1
1
65.
1
7
2
1
1
1
27
9
1
81 3 3 81 11 2 18
÷ ÷ =
⋅
⋅ =
78.
55 11 2 55 3 3 5
67. 6
5
1
5
1
1
3
2
28 21 28 20 4 ⋅ 7 4 ⋅ 5 16
÷
=
⋅
=
⋅
=
15 20 15 21 3 ⋅ 5 3 ⋅ 7 9
1 1 1
⋅ =
26 2 52
=
4
7 35 7 63 7 7 ⋅ 9 7
71.
÷
= ⋅ = ⋅
=
9 63 9 35 9 7 ⋅ 5 5
4
4 20
80.
⋅ 20 = ⋅
= 16
5
5 1
1
1
72.
6
6 1
1
=
÷ 18 = ⋅
7
7 18 21
9
2 18 3
81. 18 ÷ =
⋅ = 27
3
1 2
3
1
73.
1
1
3 9
3 5 1
⋅ =
÷ =
10 5 10 9 6
2
12
2 24 3
82. 24 ÷ =
⋅ = 36
3
1 2
3
1
74.
1
 
79. 4 ⋅ 1 ÷ 2 = 4 ⋅ 1 ÷ 2 = 1 ÷ 2
26
13  2 
13 8
69. multiplying
70.
1
36 ⋅ 4 5
4
=
⋅
=
5 ⋅5 36 5
66. Reciprocal does not exist.
68.
4
 12 
36 144 36 144 5
77.   ÷
=
÷
=
⋅
5
25
5
25 36
 5
1
2
7
3
1 1 1 1
⋅ ⋅ =
4 4 4 64
=
1
1 12
=1
⋅12 =
⋅
63.
12
12 1
64.
1
3
3
76.  2 ÷ 8  =  2 ⋅ 19  =  1 


 19 19 
 4 
 19 8 
1
1
200 25 200 17 25 ⋅ 8 17 8
÷
=
⋅ =
⋅
=
51 17 51 25 17 ⋅ 3 25 3
1
36 bags of candy
8
1
4
4 40
83.
= 32 hr
⋅ 40 = ⋅
5
5 1
2
6 12 7
75. 12 ÷ =
⋅ = 14
7
1 6
1
32 × $18 = $576
Amelia earned $576.
1
80
Chapter 2
84.
4 4 16
⋅ =
3 3 9
90. 45
Review Exercises
5
⋅0 = 0
13
4
16
16 10 12 640
⋅ 10 ⋅ 12 = ⋅ ⋅
=
9
9 1 1
3
3
3
2
640
1
The area is
or 213 ft 2 .
3
3
1
2
5
4 38 19 38 5 10
92. 3 ÷ 3 =
⋅
=
÷ =
11
5 11 5 11 19 11
3
3 9 8
85. 9 ÷ = ⋅ = 24
8 1 3
1
1
1
5 7 14 7 9
9
1
93. 7 ÷ 1 = ÷ = ⋅
= =4
9 1 9 1 14 2
2
Yes, he will have 24 pieces, which is more
than enough for his class.
2
Section 2.6
25
6
50 2 50 1 25
3
94. 4 ÷ 2 =
÷ =
⋅ =
=2
11
11 1 11 2 11
11
 2  2  11 32 352
86.  3  6  = ⋅ =
15
 3  5  3 5
23
7
15 352 = 23
15
−30
52
−45
7
1
3
1
51 17 51 1
3
95. 10 ÷ 17 = ÷ =
⋅
=
5
5 1
5 17 5
1
96. 0 ÷ 3
1
1
1
5
=0
12
1 1 5 5 25
1
97. 2 ⋅ 1 = ⋅ =
=3
2 4 2 4 8
8
1
It will take 3 gal.
8
2
 1  3  34 71 71
87. 11  2  =
⋅
= = 23
3
3
 3  34  3 34
8
5
1 3 13 16
88. 6 ⋅ 1 =
⋅
=8
2 13 2 13
1
2
1 1 25 5 25 4
98. 12 ÷ 1 =
÷ =
⋅ = 10
2 4 2 4
2 5
1
1
There will be 10 pieces.
1
1
 5  4 45 45
89. 4 ⋅  5  = ⋅
=
= 22
2
2
 8 1 8
2
Chapter 2
1
5
7 69 23 69 8
3
1
91. 4 ÷ 2 =
÷
=
⋅
= =1
16
8 16 8
2
16 23 2
Test
5
8
(b) Proper
7
3
(b) Improper
1. (a)
2. (a)
81
1
Chapter 2
3.
4.
Fractions and Mixed Numbers: Multiplication and Division
13. (a) No; 1155 is not even.
(b) Yes; 1 + 1 + 5 + 5 = 12 is divisible by
3.
(c) Yes; the digit in the ones-place is a 5.
(d) No; the digit in the ones-place is not 0.
11 1
;5
2 2
7
is an improper fraction because the
7
numerator is greater than or equal to the
denominator.
3
5. (a) 12 44
−36
8
3
14. 15 × 4  12 × 5
60 = 60
15 5
=
12 4
8
2
=3
12
3
15. 2 × 25  5 × 4
50 ≠ 20
2 4
≠
5 25
7 3 × 9 + 7 34
(b) 3 =
=
9
9
9
6.
16.
150 5 ⋅ 5 ⋅ 2 ⋅ 3 10
3
=
=
or 1
105
5 ⋅ 3 ⋅7
7
7
17.
1, 200 , 000 12 2 ⋅ 6 6
=
=
=
1, 400 , 000 14 2 ⋅ 7 7
7.
8.
15 3 ⋅ 5 3
=
=
25 5 ⋅ 5 5
16 4 ⋅ 4 4
=
=
Brad:
20 4 ⋅ 5 5
(b) Brad has the greater fractional part
4 3
completed since > .
5 5
18. (a) Christine:
9.
10. (a)
(b)
(c)
(d)
(e)
(f)
Composite
Neither
Prime
Neither
Prime
Composite
15 = 3 × 5
19.
39 = 3 × 13
1
75 4 75 3 ⋅ 25
20.  75 
⋅ =
=
 24  ⋅ 4 =
3 ⋅2
24 1 6
11. (a) 1, 3, 5, 9, 15, 45
(b)
2 57
2 3 ⋅ 19 19
×
=
⋅
=
9 46 3 ⋅ 3 2 ⋅ 23 69
3
39
6
=
5 45
3⋅3⋅5 = 32 ⋅5 = 45
21.
12. (a) Add the digits of the number. If the
sum is divisible by 3, then the original
number is divisible by 3.
(b) Yes; 1 + 9 + 8 + 1 + 0 + 1 + 1 = 21
and 21 is divisible by 3.
82
25
1
or 12
2
2
28 21 28 8
÷ =
⋅
24 8 24 21
2 ⋅ 2 ⋅ 7 2 ⋅2⋅2 4
=
⋅
=
9
2 ⋅ 2 ⋅ 2 ⋅3 3⋅ 7
Chapter 2
21
2
  2 

 52 2
52
1
8
1
52
8
27.
÷   ⋅  =
÷ ⋅ =
÷
72  2  3  72  4 3  72 3


105
105 1 21 1
22.
÷5 =
⋅ =
=
42
42 5 42 2
1
23.
1
2 9 40
2
3 ⋅ 3 2 ⋅2⋅2⋅ 5
× ×
=
⋅
⋅
18 25 6
2 ⋅ 3 ⋅ 3 5 ⋅5
2 ⋅3
4
=
15
26
24
600 50 13 1 5 ⋅10 13
÷
÷ = ÷
÷
1200 65 15 2 5 ⋅13 15
=
25.
4
1 8 11 44
2
or 14 cm 2
= ⋅ ⋅ =
2 1 3
3
3
3
1 13 15 3
⋅
⋅
=
2 10 13 4
2
1
1
5
1 20 1
29. 20 ⋅ =
⋅ =5
4
1 4
10
1 10 25
÷4 = ÷
21
6 21 6
10 6
= ⋅
21 25
2⋅ 5 2⋅ 3
=
⋅
3 ⋅7 5 ⋅5
4
=
35
1
1 20 1 20 4
=
÷ =
⋅ = 80
4 1 4 1 1
1
20 ÷ is greater.
4
20 ÷
1 12 1 12 4
= ÷ = ⋅ = 48
4 1 4 1 1
48 quarter-pounders
30. 12 ÷
2
4
4 72 34 144 3 ⋅ 48
26. 4 ⋅ 2 =
⋅
=
=
3 ⋅5
17 15 17 15
15
1
1
=
1
1
1  11 
28. A = bh = (8)  
2
2 3
1
1
1
52 3 26 13
=
⋅ =
=
72 2 24 12
1
24.
Test
15
1 5
1 5 120 15
⋅ ⋅ 120 =
⋅ ⋅
= =5
31.
15 8
3
15 8 1
48
3
=9
5
5
3
1
5 dogs are female pure breeds.
32.
1 4 4 2
⋅ =
=
2 5 10 5
They can build on a maximum of
83
2
acre.
5
Chapter 2
Fractions and Mixed Numbers: Multiplication and Division
Chapters 1–2
Cumulative Review Exercises
1.
Mountain
Mt. Foraker
(Alaska)
Mt. Kilimanjaro
(Tanzania)
El Libertador
(Argentina)
Mont Blanc
(France-Italy)
Height (ft)
Words
Standard
Form
17,400
Seventeen thousand, four hundred
19,340
Nineteen thousand, three hundred forty
22,047
Twenty-two thousand, forty-seven
15,771
Fifteen thousand, seven hundred seventy-one
3,000,000
×
40,000
120,000,000,000
2.
432
+ 998
1430
8.
3.
572
− 433
139
9. 1007
− 823
184
4.
4122
52
×
8 244
206 100
214,344
10.
11. 6 + 2 ⋅ 8 = 6 + 16 = 22
2
2
12. 5 − 3 = 25 − 9 = 16
24
5. 16 384
−32
64
−64
0
6.
48
=6
8
13. (5 − 3) 2 = 22 = 4
14. d
15. c
16. b
23
× 81
23
1840
1863
17. e
18. a
4
7
1
7
(b)
or 2
3
3
19. (a)
18 R 2
7. 4 74
−4
34
−32
2
84
Chapters 1–2
Cumulative Review Exercises
1
20. (a) Proper
(b) Improper
(c) Improper
1 2
26. Yes;  ⋅
2 9
1
 5 1 5 5
 ⋅ 3 = 9 ⋅ 3 = 27 and

5
21. (a) 1, 2, 3, 5, 6, 10, 15, 30
1  2 5  1 10
5
= .
⋅ ⋅  = ⋅
2  9 3  2 27 27
3
(b) 3 15
1
2 30
1
2
2
2
 5 12 
2 2
2 4 2
⋅
÷
=
÷
27. 

  ÷ =
6
3
5
3
25 3
25 
 

1
5
2 ⋅ 3 ⋅ 5 = 30
144 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 12
5
or 1
=
=
84
2 ⋅ 2 ⋅ 3 ⋅7
7
7
60 , 000
6 2⋅ 3 2
(b)
= =
=
150 , 000 15 5 ⋅ 3 5
22. (a)
2
=
4 3
6
⋅ =
25 2 25
1
1
35 51
5 ⋅ 7 3 ⋅ 17 119
23.
⋅ =
⋅
=
27 95 3 ⋅ 3 ⋅ 3 5 ⋅ 19 171
11 5 11
2
or 1 m2
28. A = lw = ⋅ =
5 9 9
9
1
1
2
4 17 34 17 5
5
24. 5 ÷ 6 = ÷
=
⋅
=
3
5 3
5
3 34 6
1
29. A = bh
2
1  25 
1 25 2 ⋅ 2 ⋅ 2
=   (8) = ⋅ ⋅
= 50 ft 2
2 2 
2 2
1
2
1
1
8 5
5
5 8
5
25. Yes;
and
=
⋅
⋅ = .
13 16 26
16 13 26
2
30.
2
85
1 3 3 3
of the students are males
⋅ =
10 4 40 40
from out of state
Chapter 3
Fractions and Mixed Numbers: Addition
and Subtraction
Chapter Opener Puzzle
Section 3.1
Addition and Subtraction of Like Fractions
Section 3.1 Practice Exercises
1. like
9.
2. 9 cm + 11 cm – 20 cm
3. 3 ft + 5 ft = 8 ft
4. 7 chairs + 2 chairs = 9 chairs
5. 7 m + 13 m = 20 m
10.
6. 8 thirds + 2 thirds = 10 thirds
7. 1 fourth + 6 fourths = 7 fourths
8.
86
2 7
× is multiplication in which we
5 5
multiply the numerators and multiply the
denominators:
2 7 14
× = .
5 5 25
2 7
The expression + is addition in which
5 5
we add the numerators but keep the
denominator:
2 7 9
+ = .
5 5 5
Section 3.1
11.
6 7 6 + 7 13
+ =
=
11 11
11
11
12.
5 2 5+2 7
+ =
=
3 3
3
3
24.
1 3 1+ 3 4
+ =
= =1
4 4
4
4
16.
1 3 1+ 3 4 1
+ =
= =
8 8
8
8 2
17.
2 4 2+4 6 2
+ =
= =
9 9
9
9 3
18.
3 5 3+5 8
+ =
= =4
2 2
2
2
19.
20.
26. 52 cards − 13 cards = 39 cards
27. 7 fifths − 1 fifth = 6 fifths
3
4 3+ 4 7
+ =
=
10 10
10
10
15.
28. 18 tenths − 11 tenths = 7 tenths
29.
30.
3
8 15 3+ 8 + 15 26
+
+
=
=
20 20 20
20
20
2 ⋅13 13
=
=
2 ⋅10 10
5 4 9 5 + 4 + 9 18 2 ⋅ 9 9
+ + =
= =
=
8 8 8
8
8 2 ⋅4 4
21.
18 11 6 18 + 11 + 6 35 5 ⋅ 7 5
+ + =
=
=
=
14 14 14
14
14 2 ⋅ 7 2
22.
23.
7 1 7 +1 8 4
1
+ =
= = or 1
6 6
6
6 3
3
4
1
Austin traveled a total of
or 1 mi.
3
3
25. 15 baskets − 4 baskets = 11 baskets
6 3 6+3 9
13.
+ =
=
5 5
5
5
14.
Addition and Subtraction of Like Fractions
7 22 10 7 + 22 + 10 39
+
+ =
=
18 18 18
18
18
3 ⋅13 13
=
=
3 ⋅6
6
1 9 1 + 9 10 5
1
+ =
= = or 2
4 4
4
4 2
2
5
1
Bethany has
or 2 cups of bleach and
2
2
water mixture.
87
31.
9 6 9−6 3
− =
=
8 8
8
8
32.
7 6 7−6 1
− =
=
9 9
9
9
33.
9 6 9−6 3
− =
=
2 2
2
2
34.
10 5 10 − 5 5
− =
=
4 4
4
4
35.
13 7 13 − 7 6
− =
= =2
3 3
3
3
36.
13 3 13 − 3 10
− =
=
=1
10
10
10 10
37.
23 15 23 − 15 8 2 ⋅ 4 2
− =
=
=
=
12 12
12
12 3 ⋅ 4 3
38.
13 5 13 − 5 8 4 ⋅ 2 4
− =
= =
=
6 6
6
6 3⋅ 2 3
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
39.
28 14 4 28 − 14 − 4 10 2 ⋅ 5 2
−
−
=
=
=
=
25 25 25
25
25 5 ⋅ 5 5
54.
8 2 1 8 + 2 −1 9
+ − =
= =3
3 3 3
3
3
40.
34 6 3 34 − 6 − 3 25 5 ⋅ 5 5
− − =
=
=
=
15 15 15
15
15 3 ⋅ 5 3
55.
19 5 7 19 − 5 + 7 21 3 ⋅ 7 7
− + =
=
=
=
12 12 12
12
12 3 ⋅ 4 4
41.
10 1
5 10 − 1 − 5 4 1
− − =
=
=
16 16 16
16
16 4
56.
7 13 5 7 + 13 − 5 15 3 ⋅ 5 5
+ − =
= =
=
18 18 18
18
18 3 ⋅ 6 6
42.
31 14 12 31 − 14 − 12 5 1
−
−
=
=
=
40 40 40
40
40 8
43.
2
2
9 9
81
 11 2 
 9
57.  −  =   = ⋅ =
10 10 100
 10 10 
 10 
5 3 5−3 2
− =
= =
8 8
8
8
1
g is left.
4
3
3
2 2 2 8
7 5  2
58.  −  =   = ⋅ ⋅ =
3 3 3 27
 3 3  3
1
11 3 11 − 3 8
44.
− =
= =2
4 4
4
4
Jason has 2 acres left.
59. 5 ÷ 3 + 5 = 5 ⋅ 2 + 5 = 5 + 5 = 5 + 5
4 2 6 4 3 6 6 6
6
2
=
7 5 7 + 5 12 4 ⋅ 3 3
+ =
= =
=
45.
8 8
8
8 4 ⋅2 2
10 5
=
6 3
3
1 2 5 1 21 5 3 5 8
60.
÷ + = ⋅
+ = + = =4
7 21 2 7 2 2 2 2 2
1 13 1 + 13 14 2 ⋅ 7 2
46.
+
=
=
=
=
21 21
21
21 3 ⋅ 7 3
1
47.
14 2 14 − 2 12
− =
=
5 5
5
5
61.
6 7 4 6+7−4 9
+ − =
=
5 5 5
5
5
48.
5 2 5−2 3
− =
= =1
3 3
3
3
62.
10 2 5 10 − 2 + 5 13
− + =
=
3 3 3
3
3
49.
6 7 6 + 7 13
+ =
= =1
13 13
13
13
50.
20 12 20 + 12 32
+
=
=
35 35
35
35
51.
1
3 13
3 13 2 3 13 16
63.
⋅ = + =
+ ⋅2 = +
7
7 14
7 14 1 7 7
7
1
13 5
13 5 3 13 5 8 4
64.
⋅ = − = =
− ⋅3 = −
6 18
6 18 1 6 6 6 3
14 2 4 14 + 2 − 4 12 4 ⋅ 3 4
+ − =
= =
=
15 15 15
15
15 5 ⋅ 3 5
52.
19 11 5 19 − 11 + 5 13
− + =
=
6 6 6
6
6
53.
7 3 1 7 − 3 +1 5
− + =
=
2 2 2
2
2
6
1
 2 11  1 13 1 13 7 13
65.  +  ÷ = ÷ =
⋅ =
 21 21  7 21 7 21 1 3
3
88
Section 3.1
1
1
5
1
2
5 6 1
 17 12  5 5 5
66.  −  ÷ =
⋅ =
÷ =
 30 30  6 30 6 30 5 5
1 2 3 1 1 6 2
76. ⋅  + +  = ⋅ =
3 7 7 7 3 7 7
1
The deer ate
10
1
17 1 7 17 7 10
− ⋅ =
−
=
=
=
67.
30 2 15 30 30 30 3 ⋅ 10 3
68.
77. (a)
4
4
1
5 1 1 5 1
− ⋅ = − =
=
=
12 2 6 12 12 12 3 ⋅ 4 3
69. Perimeter =
5 5 2 12
5
+ + =
or 1 m
7 7 7 7
7
70. Perimeter =
11 20 23 54
+
+
=
= 6 ft
9
9
9
9
Addition and Subtraction of Like Fractions
(b)
15 13 15 13 56
+ + + =
16 16 16 16 16
7⋅ 8 7
1
=
= or 3 in.
2⋅ 8 2
2
71. Perimeter =
72. Perimeter =
4 8 4 8 24
+ + + =
= 8 yd
3 3 3 3 3
78. (a)
1 7 3 8 3 5 1
73.  +  − = − = =
 10 10  10 10 10 10 2
1
There was gal left over.
2
3 4 1 7 1 6 3
74.  +  − = − = =
8 8 8 8 8 8 4
3
cup was left over.
4
(b)
3
1  5 7  1 12 3
75.
⋅ +  = ⋅
=
4 8 8 4 8 8
1
2
of the garden.
7
4 7
9
5 13 17
+ + + + +
10 10 10 10 10 10
4 + 7 + 9 + 5 + 13 + 17 55
=
=
10
10
5 ⋅11 11
1
=
= =5
5 ⋅2
2
2
1
Thilan walked 5 mi total.
2
11
11 6 11 1 11
÷6 = ÷ = ⋅ =
2
2 1 2 6 12
He walked an average of
11
mi per day.
12
9 17 2 + 7 + 9 + 17
2 7
+ + +
=
10
10 10 10 10
35 5 ⋅ 7 7
=
=
=
10 2 ⋅ 5 2
1
=3
2
1
The total amount of rain is 3 in.
2
35
35 4 35 1 35
÷4=
÷ = ⋅ =
10
10 1 10 4 40
5 ⋅7 7
=
=
5 ⋅8 8
7
in. per week.
The average is
8
3 5 3 5 16
79. Perimeter = + + + = = 2 ft
8 8 8 8 8
3 5 15 2
Area = ⋅ =
ft
8 8 64
3
He used L.
8
89
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
80. Perimeter =
30 25 750 375 ⋅ 2
⋅ =
=
4 4
16
8⋅ 2
375
7 2
or 46 in.
=
8
8
15 7 15 7
+ + +
8 8 8 8
Area =
11
1
44 11
=
=
or 5 m
8
2
2
2
83.
15 7 105
41 2
Area = ⋅ =
or 1
m
8 8 64
64
4
5 7 12 4
84.
=
+ ;
3
9 9 9
13 22 13 22
+
+ +
3
3
3
3
70
1
or 23 yd
=
3
3
22 13 286
7
Area =
⋅ =
or 31 yd 2
3 3
9
9
81. Perimeter =
82. Perimeter =
3 2 5
+ ; =1
5 5 5
3
30 25 30 25
+
+
+
4
4
4
4
85.
11 8 3 1
− ;
=
15 15 15 5
86.
5 2 3
− ;
7 7 7
55
1
110 55
=
or 27 in.
=
4
2
2
2
Section 3.2
Least Common Multiple
Section 3.2 Practice Exercises
1. (a) multiple
(b) least common multiple
(c) least common denominator (LCD)
5
31 2 8 31 + 2 − 8 25 5
+ − =
=
=
5.
15 15 15
15
15 3
3
2 1 3
+ =
5 5 5
2 1 2
(b) ⋅ =
5 5 25
2 1
(c)
÷ =2
5 5
2 1 1
(d) − =
5 5 5
2. (a)
8 12 20
+ =
=4
5 5
5
7.
11 7 18
+ = =6
3 3 3
8.
5
2
3
−
=
19 19 19
9. (a) 48, 72, 240
19 16 3 1
3.
− = =
6 6 6 2
4.
6.
(b) 4, 8, 12
10. (a) 90, 120, 60
28 22 6 3
−
= =
4
4 4 2
(b) 15, 3, 5
90
Section 3.2
11. (a) 72, 360, 108
(b) 6, 12, 9
Least Common Multiple
23. 15 = 3 ⋅ 5
25 = 5 ⋅ 5
LCM: 3 ⋅ 5 ⋅ 5 = 75
12. (a) 56, 140, 280
(b) 7, 4, 2
24. 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 = 48
13. 10: 10, 20, 30, 40, 50
25: 25, 50, 75
LCM: 50
25. 24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
30 = 2 ⋅ 3 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 120
14. 21: 21, 42, 63
14: 14, 28, 42, 56
LCM: 42
26. 14 = 2 ⋅ 7
35 = 5 ⋅ 7
LCM: 2 ⋅ 5 ⋅ 7 = 70
15. 16: 16, 32, 48, 64
12: 12, 24, 36, 48
LCM: 48
27. 42 = 2 ⋅ 3 ⋅ 7
70 = 2 ⋅ 5 ⋅ 7
LCM: 2 ⋅ 3 ⋅ 5 ⋅ 7 = 210
16. 20: 20, 40, 60, 80
12: 12, 24, 36, 48, 60
LCM: 60
28. 6 = 2 ⋅ 3
21 = 3 ⋅ 7
LCM: 2 ⋅ 3 ⋅ 7 = 42
17. 8: 8, 16, 24, 32, 40, 48, 56, 64, 72, 80, 88,
96, 104, 112, 120
10: 10, 20, 30, 40, 50, 60, 70, 80, 90, 100,
110, 120
12: 12, 24, 36, 48, 60, 72, 84, 96, 108, 120
LCM: 120
29. 20 = 2 ⋅ 2 ⋅ 5
18 = 2 ⋅ 3 ⋅ 3
27 = 3 ⋅ 3 ⋅ 3
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3 ⋅ 5 = 540
18. 4: 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, 44,
48, 52, 56, 60, 64, 68, 72, 76, 80, 84
6: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60, 66,
72, 78, 84
14: 14, 28, 42, 56, 70, 84
LCM: 84
30. 9 = 3 ⋅ 3
15 = 3 ⋅ 5
42 = 2 ⋅ 3 ⋅ 7
LCM: 2 ⋅ 3 ⋅ 3 ⋅ 5 ⋅ 7 = 630
31. 12 = 2 ⋅ 2 ⋅ 3
15 = 3 ⋅ 5
20 = 2 ⋅ 2 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 = 60
19. 18 = 2 ⋅ 3 ⋅ 3
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 = 72
20. 9 = 3 ⋅ 3
30 = 2 ⋅ 3 ⋅ 5
LCM: 2 ⋅ 3 ⋅ 3 ⋅ 5 = 90
32. 20 = 2 ⋅ 2 ⋅ 5
30 = 2 ⋅ 3 ⋅ 5
40 = 2 ⋅ 2 ⋅ 2 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 120
21. 12 = 2 ⋅ 2 ⋅ 3
15 = 3 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 = 60
33. 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
30 = 2 ⋅ 3 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 240
22. 27 = 3 ⋅ 3 ⋅ 3
45 = 3 ⋅ 3 ⋅ 5
LCM: 3 ⋅ 3 ⋅ 3 ⋅ 5 = 135
91
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 120
It will take 120 hr (5 days) for the
satellites to be lined up again.
34. 20 = 2 ⋅ 2 ⋅ 5
42 = 2 ⋅ 3 ⋅ 7
35 = 5 ⋅ 7
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 7 = 420
42. Find the LCM of 3, 7, and 12.
3=3
7=7
12 = 2 ⋅ 2 ⋅ 3
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 7 = 84
It will take 84 months (7 years) for the
planets to be aligned again.
35. 6 = 2 ⋅ 3
12 = 2 ⋅ 2 ⋅ 3
18 = 2 ⋅ 3 ⋅ 3
20 = 2 ⋅ 2 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180
36. 21 = 3 ⋅ 7
35 = 5 ⋅ 7
50 = 2 ⋅ 5 ⋅ 5
75 = 3 ⋅ 5 ⋅ 5
LCM: 2 ⋅ 3 ⋅ 5 ⋅ 5 ⋅ 7 = 1050
37. 5 = 5
15 = 3 ⋅ 5
18 = 2 ⋅ 3 ⋅ 3
20 = 2 ⋅ 2 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 180
38. 28 = 2 ⋅ 2 ⋅ 7
10 = 2 ⋅ 5
21 = 3 ⋅ 7
35 = 5 ⋅ 7
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 7 = 420
39. Find the LCM of 10, 12, and 15.
10 = 2 ⋅ 5
12 = 2 ⋅ 2 ⋅ 3
15 = 3 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 = 60
The shortest length of floor space is 60 in.
40. (a) Find the LCM of 4, 5, and 12.
4=2⋅2
5=5
12 = 2 ⋅ 2 ⋅ 3
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 = 60
The shortest length before all are done
at the same time is 60 hr.
(b) Wednesday at 6:00 p.m.
41. Find the LCM of 6, 8, 10, and 15.
6=2⋅3
8=2⋅2⋅2
10 = 2 ⋅ 5
15 = 3 ⋅ 5
92
43.
2 2 ⋅ 7 14
=
=
3 3 ⋅ 7 21
44.
7 7 ⋅ 8 56
=
=
4 4 ⋅ 8 32
45.
5 5 ⋅ 2 10
=
=
8 8 ⋅ 2 16
46.
2 2⋅3 6
=
=
9 9 ⋅ 3 27
47.
3 3 ⋅ 4 12
=
=
4 4 ⋅ 4 16
48.
3
3 ⋅ 5 15
=
=
10 10 ⋅ 5 50
49.
4 4 ⋅ 3 12
=
=
5 5⋅ 3 15
50.
3 3⋅10 30
=
=
7 7 ⋅10 70
51.
7 7 ⋅ 7 49
=
=
6 6 ⋅ 7 42
52.
10 10 ⋅ 6 60
=
=
3
3⋅ 6 18
53.
11 11⋅11 121
=
=
9
9 ⋅11 99
54.
7 7 ⋅ 7 49
=
=
5 5⋅ 7 35
Section 3.2
55.
5
5⋅ 3 15
=
=
13 13⋅ 3 39
56.
6
6 ⋅ 2 12
=
=
17 17 ⋅ 2 34
57.
11 11 ⋅ 1000 11,000
=
=
4 4 ⋅ 1000
4000
58.
18 18 ⋅ 100 1800
=
=
7
7 ⋅ 100
700
59.
3
3⋅5 15
=
=
14 14 ⋅5 70
60.
5
5⋅ 3
15
=
=
66 66 ⋅ 3 198
61.
65.
66.
67.
3 3⋅ 2 6
=
=
4 4⋅2 8
7 6
7 3
> so >
8 8
8 4
7
7 ⋅ 4 28
62.
=
=
15 15 ⋅ 4 60
11 11 ⋅ 3 33
=
=
20 20 ⋅ 3 60
28 33
7 11
so
<
< .
60 60
15 20
68.
13 13 ⋅ 3 39
63.
=
=
10 10 ⋅ 3 30
22 22 ⋅ 2 44
=
=
15 15 ⋅ 2 30
39 44
13 22
so
<
<
30 30
10 15
69.
64.
15 15 ⋅ 3 45
=
=
4
4 ⋅ 3 12
21 21 ⋅ 2 42
=
=
6
6 ⋅ 2 12
45 42
15 21
so
>
>
12 12
4
6
93
Least Common Multiple
3
3⋅ 2
6
=
=
12 12 ⋅ 2 24
2 2⋅3 6
=
=
8 8 ⋅ 3 24
6
6
3 2
so
=
=
24 24
12 8
5
5 ⋅ 4 20
=
=
20 20 ⋅ 4 80
4
4 ⋅ 5 20
=
=
16 16 ⋅ 5 80
20 20
5
4
so
=
=
80 80
20 16
5
5⋅ 3 15
=
=
18 18 ⋅ 3 54
8
8 ⋅ 2 16
=
=
27 27 ⋅ 2 54
15 16
5
8
so
<
<
54 54
18 27
9
9⋅7
63
=
=
24 24 ⋅ 7 168
8
8 ⋅8
64
=
=
21 21⋅8 168
63
9
64
8
so
<
<
168 168
24 21
2 2 ⋅ 8 16
=
=
3 3 ⋅ 8 24
7 7 ⋅ 3 21
=
=
8 8 ⋅ 3 24
5 5 ⋅ 4 20
=
=
6 6 ⋅ 4 24
1 1 ⋅ 12 12
=
=
2 2 ⋅ 12 24
21 7
= has the greatest value.
24 8
Chapter 3
70.
71.
Fractions and Mixed Numbers: Addition and Subtraction
1 1 ⋅ 30 30
=
=
6 6 ⋅ 30 180
1 1 ⋅ 45 45
=
=
4 4 ⋅ 45 180
2
2 ⋅ 12
24
=
=
15 15 ⋅ 12 180
2 2 ⋅ 20 40
=
=
9 9 ⋅ 20 180
24
2
has the least value.
=
180 15
7 7 ⋅ 3 21
=
=
8 8 ⋅ 3 24
2 2 ⋅ 8 16
=
=
3 3 ⋅ 8 24
3 3 ⋅ 6 18
=
=
4 4 ⋅ 6 24
73.
75.
2 3 7
, ,
3 4 8
76.
5
5 ⋅ 2 10
=
=
12 12 ⋅ 2 24
3 3⋅3 9
=
=
8 8 ⋅ 3 24
2 2 ⋅ 8 16
=
=
3 3 ⋅ 8 24
3 5 2
9 , 10 , 16 so , ,
8 12 3
77.
16 , 18 , 21 so
72.
74.
5
5
=
16 16
3 3⋅ 2 6
=
=
8 8 ⋅ 2 16
1 1⋅ 4 4
=
=
4 4 ⋅ 4 16
1 5 3
4 , 5 , 6 so , ,
4 16 8
2 2 ⋅ 6 12
=
=
5 5 ⋅ 6 30
3
3⋅3
9
=
=
10 10 ⋅ 3 30
5 5 ⋅ 5 25
=
=
6 6 ⋅ 5 30
3 2 5
9 , 12 , 25 so
, ,
10 5 6
4 4 ⋅ 20 80
=
=
3 3 ⋅ 20 60
13 13 ⋅ 5 65
=
=
12 12 ⋅ 5 60
17 17 ⋅ 4 68
=
=
15 15 ⋅ 4 60
13 17 4
65 , 68 , 80 so
, ,
12 15 3
5 5 ⋅ 15 75
=
=
7 7 ⋅ 15 105
11 11 ⋅ 5 55
=
=
21 21 ⋅ 5 105
18 18 ⋅ 3 54
=
=
35 35 ⋅ 3 105
18 11 5
54 , 55 , 75 so
, ,
35 21 7
3 3⋅ 4 12
=
=
4 4 ⋅ 4 16
11 11
=
16 16
7 7 ⋅ 2 14
=
=
8 8 ⋅ 2 16
14 7
= in - above the
16 8
11
left eye. The shortest cut is
in - right
16
hand.
The longest cut is
94
Section 3.2
78.
3 3⋅ 4 12
=
=
4 4 ⋅ 4 16
5 5⋅ 2 10
=
=
8 8 ⋅ 2 16
11 11
=
16 16
81.
12 3
= in . The
16 4
10 5
shortest screw is
= in .
16 8
The longest screw is
79.
2 2 ⋅ 40 80
=
=
3 3 ⋅ 40 120
3 3 ⋅ 24 72
=
=
5 5 ⋅ 24 120
5 5 ⋅ 15 75
=
=
8 8 ⋅ 15 120
80 2
= lb of tur120 3
72 3
key. The least amount is
= lb of ham.
120 5
3
=
4
7
=
8
4
=
5
82.
3 ⋅ 10 30
=
4 ⋅ 10 40
7 ⋅ 5 35
=
8 ⋅ 5 40
4 ⋅ 8 32
=
5 ⋅ 8 40
30 3
= lb of cheddar;
40 4
35 7
the greatest amount is
= lb of Swiss.
40 8
The least amount is
Section 3.3
1 1⋅ 6
6
=
=
4 4 ⋅ 6 24
5 5⋅ 4 20
=
=
6 6 ⋅ 4 24
5
5 ⋅ 2 10
=
=
12 12 ⋅ 2 24
2 2 ⋅ 8 16
=
=
3 3 ⋅ 8 24
1 1⋅ 3 3
=
=
8 8 ⋅ 3 24
6 10 16 20
3
<
<
<
<
24 24 24 24 24
10 5
16 2
and
= (a and b) are
=
24 12
24 3
1
5
between
and .
4
6
The greatest amount is
80.
1 1⋅ 5 5
=
=
3 3 ⋅ 5 15
11 11
=
15 15
2 2 ⋅ 5 10
=
=
3 3 ⋅ 5 15
4 4 ⋅ 3 12
=
=
5 5 ⋅ 3 15
2 2⋅3 6
=
=
5 5 ⋅ 3 15
5
6 10 11 12
< < < <
15 15 15 15 15
6 2
10 2
= and
= (a and c) are between
15 5
15 3
1
11
and
.
3
15
Addition and Subtraction of Unlike Fractions
Section 3.3 Practice Exercises
1. (a) is
(b) is not
2.
Least Common Multiple
3 3⋅3 9
=
=
5 5 ⋅ 3 15
95
3.
6 6 ⋅ 2 12
=
=
7 7 ⋅ 2 14
4.
4 4 ⋅ 4 16
=
=
9 9 ⋅ 4 36
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
5.
2 2 ⋅ 7 14
=
=
3 3 ⋅ 7 21
6.
3 3 ⋅ 10 30
=
=
1 1 ⋅ 10 10
18.
19.
5 5 ⋅ 5 25
7.
=
=
1 1⋅ 5 5
8.
4 4 ⋅ 12 48
=
=
1 1 ⋅ 12 12
9.
2 2⋅4 8
=
=
1 1⋅ 4 4
10.
3 3⋅3 9
=
=
4 4 ⋅ 3 12
11.
4 4 ⋅ 20 80
=
=
5 5 ⋅ 20 100
12.
3 3 ⋅ 9 27
=
=
2 2 ⋅ 9 18
13.
1 1⋅ 5 5
=
=
8 8 ⋅ 5 40
20.
21.
22.
23.
14. To multiply two fractions, multiply their
numerators and multiply their
denominators. Then simplify to lowest
terms. To add two fractions, rewrite the
fractions so that they have a common
denominator. Then add the numerators
and keep the common denominator.
Simplify the answer to lowest terms.
24.
17.
1
3
1⋅ 2
3
2
3 2+3
+
=
+
=
+
=
10 20 10 ⋅ 2 20 20 20
20
5
5
1
=
=
=
20 4 ⋅ 5 4
4 2 4 2⋅3 4 6 4 + 6
+ = +
= + =
15 5 15 5⋅ 3 15 15
15
10 2
=
=
15 3
5 8 5⋅ 7 8 ⋅ 6 35 48
+ =
+
=
+
6 7 6 ⋅ 7 7 ⋅ 6 42 42
35 + 48 83
=
=
42
42
2 4 2 ⋅5 4 ⋅11 10 44
+ =
+
= +
11 5 11⋅5 5⋅11 55 55
10 + 44 54
=
=
55
55
7 1 7 ⋅ 5 1 ⋅ 8 35 8 35 − 8 27
− =
−
=
−
=
=
8 5 8 ⋅ 5 5 ⋅ 8 40 40
40
40
9 1 9 ⋅ 3 1 ⋅ 10 27 10
− =
−
=
−
10 3 10 ⋅ 3 3 ⋅ 10 30 30
27 − 10 17
=
=
30
30
25. 13 3 13 3⋅ 3 13 9 13 − 9
− =
−
=
−
=
12
12 4 12 4 ⋅ 3 12 12
2 1 2 ⋅ 8 1 ⋅ 3 16 3 16 + 3 19
15. + =
+
=
+
=
=
3 8 3 ⋅ 8 8 ⋅ 3 24 24
24
24
16.
5 3 5⋅ 4 3⋅ 3 20 9
+ =
+
=
+
6 8 6 ⋅ 4 8 ⋅3 24 24
20 + 9 29
=
=
24
24
1
=
3 2 3 ⋅ 5 2 ⋅ 4 15 8 15 + 8 23
+ =
+
=
+
=
=
4 5 4 ⋅ 5 5 ⋅ 4 20 20
20
20
4
12
=
1
3
3
1 1
1⋅ 2
1⋅3
2
3
+ =
+
=
+
15 10 15⋅ 2 10 ⋅3 30 30
2+3 5 1
=
=
=
30
30 6
26.
96
29 7 29 7 ⋅ 3 29 21
− =
−
=
−
30 10 30 10 ⋅ 3 30 30
29 − 21 8
4
=
=
=
30
30 15
Section 3.3
27.
10 5 10 ⋅ 4 5⋅ 3 40 15
−
=
−
=
−
9 12 9 ⋅ 4 12 ⋅ 3 36 36
40 − 15 25
=
=
36
36
28.
7 1 7 ⋅5 1⋅ 2 35 2 33 11
− =
−
=
−
=
=
6 15 6 ⋅5 15⋅ 2 30 30 30 10
29.
5 0 5
5
− = −0=
8 11 8
8
40.
41.
42.
7 0 7
7
30.
− = −0=
12 5 12
12
31. 2 +
32. 3 +
9 2 8 9 16 9 16 + 9 25
= ⋅ + = + =
=
8 1 8 8 8 8
8
8
11 3 9 11 27 11 27 + 11
= ⋅ + =
+ =
9 1 9 9
9 9
9
38
=
9
33. 4 −
4 4 3 4 12 4 12 − 4 8
= ⋅ − = − =
=
3 1 3 3 3 3
3
3
34. 2 −
3 2 8 3 16 3 16 − 3 13
= ⋅ − = − =
=
8 1 8 8 8 8
8
8
35.
36.
43.
44.
14
14 3 14 + 3 17
+1 = + =
=
3
3 3
3
3
12
12 2 5 12 10 12 + 10
+2= + ⋅ = + =
5
5 1 5 5 5
5
22
=
5
37.
16
16 2 7 16 14 16 − 14 2
−2= − ⋅ = − =
=
7
7 1 7 7 7
7
7
38.
15
15 3 4 15 12 15 − 12 3
−3= − ⋅ = − =
=
4
4 1 4 4 4
4
4
39.
Addition and Subtraction of Unlike Fractions
45.
3 27
3⋅10
27
30 27
+
=
+
=
+
10 100 10 ⋅10 100 100 100
30 + 27 57
=
=
100
100
1
9
1⋅10
9
10
9
−
=
−
=
−
10 100 10 ⋅10 100 100 100
10 − 9
1
=
=
100
100
3
21
3 ⋅ 10
21
−
=
−
100 1000 100 ⋅ 10 1000
30
21
30 − 21
=
−
=
1000 1000
1000
9
=
1000
9
1
3
+
+
10 100 1000
3 ⋅ 100
9 ⋅ 10
1
=
+
+
10 ⋅ 100 100 ⋅ 10 1000
300
90
1
=
+
+
1000 1000 1000
300 + 90 + 1 391
=
=
1000
1000
3
7
1
+
+
10 100 1000
1 ⋅ 100
3 ⋅ 10
7
=
+
+
10 ⋅ 100 100 ⋅ 10 1000
100
30
7
=
+
+
1000 1000 1000
100 + 30 + 7 137
=
=
1000
1000
5 7 5 5⋅8 7 ⋅ 4 5⋅3
− + =
−
+
3 6 8 3⋅8 6 ⋅ 4 8 ⋅3
40 28 15 40 − 28 + 15
=
−
+
=
24 24 24
24
9
7 19
7 ⋅10 19
70 19
+
=
+
=
+
10 100 10 ⋅10 100 100 100
70 + 19 89
=
=
100
100
=
27
24
8
97
=
9
8
Chapter 3
46.
47.
48.
49.
50.
Fractions and Mixed Numbers: Addition and Subtraction
7 2 5
7 ⋅ 15 2 ⋅ 12 5 ⋅ 10
− + =
−
+
12 15 18 12 ⋅ 15 15 ⋅ 12 18 ⋅ 10
105 24 50
=
−
+
180 180 180
105 − 24 + 50 131
=
=
180
180
53.
1
54. 3 ÷ 6 − 2 = 3 ⋅ 7 − 2 = 7 − 2 = 7 − 2 ⋅ 2
5 7 5 5 6 5 10 5 10 5⋅ 2
1 5 7
1 ⋅ 6 5 ⋅ 15 7 ⋅ 5
+ −
=
+
−
20 8 24 20 ⋅ 6 8 ⋅ 15 24 ⋅ 5
6
75 35
=
+
−
120 120 120
6 + 75 − 35 46 23
=
=
=
120
120 60
2
=
55.
5 3 1 5 ⋅ 15 3 ⋅ 12
1 ⋅ 10
+ − =
+
−
8 10 12 8 ⋅ 15 10 ⋅ 12 12 ⋅ 10
75 36 10
=
+
−
120 120 120
75 + 36 − 10 101
=
=
120
120
56.
1 1 1 1 1⋅ 8 1⋅ 4 1⋅ 2 1
+ − − =
+
−
−
2 4 8 16 2 ⋅ 8 4 ⋅ 4 8 ⋅ 2 16
8 4 2 1
= + − −
16 16 16 16
8+ 4 − 2 −1 9
=
=
16
16
2
2
2
7
4 7−4 3
− =
=
10 10
10
10
3 4
5 3 1 5 3 4 5
+ ÷ = + ⋅ = +
⋅
6 8 4 6 8 1 6 2⋅ 4 1
5 3 5 3⋅ 3 5 9
= + = +
= +
6 2 6 2⋅3 6 6
5 + 9 14 7
=
=
=
6
6 3
11 1 7 11 1 9 11 1
+ ÷ = + ⋅ = +
12 9 9 12 9 7 12 7
11⋅7 1⋅12 77 12
=
+
=
+
12 ⋅7 7 ⋅12 84 84
77 + 12 89
=
=
84
84
 7 1  8  7 1⋅ 2  8
57.  −  ⋅ =  −
⋅
 10 5  3  10 5 ⋅ 2  3
 7 2  8 7−2 8
=  − ⋅ = 
⋅
 10 10  3  10  3
1 1 1
1 1 ⋅ 27 1 ⋅ 9 1 ⋅ 3
1
− +
− =
−
+
−
3 9 27 81 3 ⋅ 27 9 ⋅ 9 27 ⋅ 3 81
27 9 3 1
=
− + −
81 81 81 81
27 − 9 + 3 − 1 20
=
=
81
81
 1 1
 1⋅ 3 1⋅ 2 
 3 2
51.  −  = 
−
= − 

 2 3
 2 ⋅3 3⋅ 2 
 6 6
2 1 3 2 2 3 4 3 4 ⋅ 4 3⋅ 3
÷ − = ⋅ − = − =
−
3 2 4 3 1 4 3 4 3⋅ 4 4 ⋅ 3
16 9 16 − 9 7
= − =
=
12 12
12
12
4
5
2
3
 2 9  5  2⋅2 9  5
+
⋅
58.  +  ⋅ = 
 5 10  6  5⋅ 2 10  6
 4 9  5  4 + 9 5
⋅
= + ⋅ =
 10 10  6  10  6
2
 3− 2
 1
1 1 1
=
=  = ⋅ =

6 6 36
 6 
 6
1
2
2
 2 1
 2 ⋅ 2 1
 4 1
52.  +  = 
+  = + 
 3 6
 3⋅ 2 6 
 6 6
2
4
5 8 20 4
=
⋅ =
=
10 3 15 3
13 5 13
=
⋅ =
10 6 12
2
2
2
 4 + 1
 5
5 5 25
=
=  = ⋅ =

6 6 36
 6 
 6
98
Section 3.3
1
2
2
 1 1  2 
3
64.  +  ⋅   +
10
 2 3  5 
 1⋅3 1⋅ 2   2 2  3
=
+
⋅ ⋅ +
 2 ⋅3 3⋅ 2   5 5  10
 3+ 2   4  3
=
⋅
+
 6   25  10
59. 4 + 5 ⋅ 16 = 4 + 2 = 4 ⋅7 + 2 ⋅5
5 8 35 5 7 5⋅7 7 ⋅5
1
=
7
28 10 28 + 10 38
+ =
=
35 35
35
35
1
Addition and Subtraction of Unlike Fractions
2
60. 1 + 3 ⋅ 14 = 1 + 2 = 1⋅5 + 2 ⋅6
6 7 15 6 5 6 ⋅5 5⋅6
1
=
5
5 12 5 + 12 17
+
=
=
30 30
30
30
2
3
5
2 ⋅ 2 3⋅3
4
9
+
=
+
15⋅ 2 10 ⋅3 30 30
4 + 9 13
=
=
30
30
=
3
 2
1 2 2 2 1 5
8
5
= ⋅ ⋅ + ⋅ =
+
61.   +
25 5 5 5 25 5 125 125
 5
8 + 5 13
=
=
125 125
65.
3
 3
5 3 3 3 5 2 27 10
−
62.   − = ⋅ ⋅ − ⋅ =
4 2 2 2 4 2 8
8
 2
27 − 10 17
=
=
8
8
66.
2
 1
 5 2 7
63.   ÷  −  +
 4
 6 3  12
 1 1  5 2⋅2 7
= ⋅  ÷ −
+
 4 4   6 3⋅ 2  12
1  5− 4 7
1 1 7
=
÷
+ =
÷ +

16  6  12 16 6 12
1
1
3
2 3
5 4
= ⋅
+ = +
6 25 10 15 10
67.
3
68.
6 7 3 7
=
⋅ + = +
16 1 12 8 12
8
3⋅ 3 7 ⋅ 2
9 14
+
=
+
8⋅ 3 12 ⋅ 2 24 24
9 + 14 23
=
=
24
24
=
3 3 3⋅ 2 3 6 3 6 + 3 9
+ =
+ = + =
=
4 8 4⋅2 8 8 8
8
8
1
Inez added 1 cup.
8
7 1 7 1 ⋅ 4 7 4 7 + 4 11
+ = +
= + =
=
8 2 8 2⋅4 8 8
8
8
3
A screw that is at least 1 in. long is
8
needed.
9 1 9 1⋅ 4 9
4 9−4 5
− =
−
=
−
=
=
32 8 32 8 ⋅ 4 32 32
32
32
5
The storm delivered
in. of rain.
32
9 5 9 ⋅ 2 5 18 5 18 − 5 13
− =
− = − =
=
8 16 8 ⋅ 2 16 16 16
16
16
13
The garden needs
in. more water.
16
3 3 5 4 3
3 5 3 3
+ = − +
69. 5 − 4 ⋅ + = − ⋅
8 2 1 1 4 ⋅2 2 1 2 2
5⋅ 2 3 3 10 3 3
=
− + = − +
1⋅ 2 2 2 2 2 2
10 − 3+ 3 10
=
=
=5
2
2
The trough now holds the original amount
of 5 gal.
99
Chapter 3
70.
Fractions and Mixed Numbers: Addition and Subtraction
Now find the perimeter.
1 1 3 3 5 7
+ + + + +
4 2 8 8 8 8
1⋅ 2 1⋅ 4 3 3 5 7
=
+
+ + + +
4⋅2 2⋅4 8 8 8 8
2 4 3 3 5 7
= + + + + +
8 8 8 8 8 8
24
=
= 3 ft
8
3 3 3 ⋅ 8 3 ⋅ 5 24 15 39
+ =
+
=
+
=
5 8 5 ⋅ 8 8 ⋅ 5 40 40 40
The job did not get completed. There is
1
still
of the job left.
40
71. (a)
(b)
72. (a)
7 1 7 1⋅ 6 7
6 13
+ =
+
=
+
=
36 6 36 6 ⋅ 6 36 36 36
13 5 13 5 ⋅ 2 13 10 23
+ =
+
=
+
=
36 18 36 18 ⋅ 2 36 36 36
76. First find the missing dimensions.
7 1 7 1⋅ 2 7 2 5
x= − = −
= − = m
8 4 8 4⋅ 2 8 8 8
9 3 6 3
y= − = = m
8 8 8 4
27 1 27 1⋅12 27 12
+ =
+
=
+
48 4 48 4 ⋅12 48 48
13
=
39
48
=
13
16
Now find the perimeter.
16
(b)
73.
9 1 6 5 3 7
+ + + + +
8 4 8 8 8 8
9 1⋅ 2 6 5 3 7
= +
+ + + +
8 4⋅2 8 8 8 8
9 2 6 5 3 7
= + + + + +
8 8 8 8 8 8
32
=
=4m
8
1
5
1⋅ 2
5
2
5
7
+
=
+
=
+
=
24 48 24 ⋅ 2 48 48 48 48
2 9 2 9 2⋅2 9 2⋅2 9
+ + + =
+ +
+
5 10 5 10 5 ⋅ 2 10 5 ⋅ 2 10
4 9 4 9
= + + +
10 10 10 10
26 13
3
=
=
or 2 m
10 5
5
74. 1+
77. The LCD is 60.
3 3 ⋅ 15 45
=
=
4 4 ⋅ 15 60
7
7 ⋅ 6 42
=
=
10 10 ⋅ 6 60
5 5 ⋅ 10 50
=
=
6 6 ⋅ 10 60
1 1 ⋅ 30 30
=
=
2 2 ⋅ 30 60
42
30
is closest to
, so choice b is
60
60
1
closest to .
2
11 5 12 11 5⋅ 2 12 11 10
+ = + +
= + +
12 6 12 12 6 ⋅ 2 12 12 12
33 11
3
=
=
or 2 yd
12 4
4
75. First find the missing dimensions.
5 1 5 1⋅ 2 5 2 3
a= − = −
= − = ft
8 4 8 4⋅2 8 8 8
7 1 7 1⋅ 4 7 4 3
b= − = −
= − = ft
8 2 8 2⋅4 8 8 8
100
Section 3.3
78. The LCD is 24.
5 5 ⋅ 3 15
=
=
8 8 ⋅ 3 24
7
7 ⋅ 2 14
=
=
12 12 ⋅ 2 24
5 5 ⋅ 4 20
=
=
6 6 ⋅ 4 24
Section 3.4
Addition and Subtraction of Unlike Fractions
3 3 ⋅ 6 18
=
=
4 4 ⋅ 6 24
20
18
is closest to
, so choice c is closest to
24
24
3
.
4
Addition and Subtraction of Mixed Numbers
Section 3.4 Practice Exercises
1.
8 11 8 + 11 19
+ =
=
15 15
15
15
2.
3 7
3⋅3 7 ⋅ 4
9 28 37
+ =
+
=
+
=
16 12 16 ⋅ 3 12 ⋅ 4 48 48 48
9.
1
11
3
+5
11
4
7
11
2
13
25 23 25⋅ 3 23 75 23 52
−
=
−
=
−
=
3.
8 24 8 ⋅3 24 24 24 24
10.
6
=
4. 4 −
5.
6.
7.
8.
13
6
15 4 7 15 28 15 13
= ⋅ − =
− =
7 1 7 7
7
7
7
11.
9
9 3 5 9 15 24
+3= + ⋅ = + =
5
5 1 5 5 5
5
23 5 2 23 5 2 ⋅ 2 23 + 5 − 4
+ − =
+ −
=
6
6 6 3 6 6 3⋅ 2
24
=
=4
6
12.
125 51 58 16 1
−
−
=
=
32 32 32 32 2
17 23 321
−
+
10 100 1000
17 ⋅100 23⋅10
321
=
−
+
10 ⋅100 100 ⋅10 1000
1700 − 230 + 321 1791
=
=
1000
1000
13.
101
2
7
3
+4
7
5
9
7
5
1
14
5
+3
14
6
3
15 = 15
14
7
12
3
20
7
+ 17
20
10
1
18 = 18
20
2
1
5
5
=
4
16
16
1
4
+ 11
= + 11
4
16
9
15
16
4
Chapter 3
14.
15.
16.
Fractions and Mixed Numbers: Addition and Subtraction
26. Estimate
2
2
=
21
9
9
1
3
+ 10
= + 10
3
9
5
31
9
21
2
2 ⋅5
=
6
=
3
3⋅5
1
1⋅ 3
+4
= +4
=
5
5⋅3
6
9
+ 14
23
10
15
3
+4
15
13
10
15
6
27. Estimate
15
+8
23
1
1⋅ 4
4
= 7
=
7
6
6⋅4
24
15
5
5⋅3
+3 = +3
= +3
24
8
8⋅3
19
10
24
7
28. Estimate
Exact
3
8
5
4
+ 13
5
2
2
7
21 = 21 + 1 = 22
5
5
5
Exact
7
7
= 14
14
8
8
1
2
+8
= +8
4
8
1
9
22 = 23
8
8
Exact
3
6
=
21
5
10
9
9
+ 24
= + 24
10
10
1
15
45 = 46
2
10
21
22
+ 25
47
17. 5
18. 3
29. Estimate
19. 2
7
21
=
3
16
48
11
44
+ 15
= + 15
12
48
17
65
18 = 19
48
48
3
4
+ 16
20
20. 6
6
1
1
21. 2 = 2 + 1 = 3
5
5
5
8
1
1
22. 4 = 4 + 1 = 5
7
7
7
30. Estimate
5
2
2
23. 7 = 7 + 1 = 8
3
3
3
8
+9
17
9
4
4
24. 1 = 1 + 1 = 2
5
5
5
25. Estimate
7
+8
15
Exact
Exact
7
14
=
7
7
9
18
5
15
+8
= +8
6
18
11
29
15 = 16
18
18
7
7
31. 3 + 6 = 9
8
8
Exact
3
6
4
3
+7
4
2
1
6
13 = 13 + 1 = 14
4
2
4
32. 5 + 11
1
1
= 16
13
13
2
2
33. 32 + 10 = 42
7
7
102
Section 3.4
34. 2
35.
36.
37.
18
18
+ 16 = 18
37
37
41.
9
10
3
− 10
10
3
6
11 = 11
5
10
21
42.
2
3
1
−4
3
1
15
3
19
9
15
7
−3
15
2
2
15
40.
5
10
= 11
7
14
5
5
−9
= −9
14
14
5
2
14
11
9
11
13
−2
22
5
43.
3
3
44.
5
5
45.
12
12
46.
6
6
5
18
22
13
= −2
22
5
3
22
=
5
47. Estimate:
25
− 14
11
11
38. 33
12
5
− 14
12
1
6
19 = 19
2
12
39.
Addition and Subtraction of Mixed Numbers
Exact:
1
1 4
5
= 24 +
=
25
24
4
4 4
4
3
3
3
− 13
= − 13
= − 13
4
4
4
1
2
11 = 11
2
4
5
5
= 18
6
6
2
4
−6
= −6
3
6
1
12
6
18
48. Estimate:
36
− 13
23
17
17
=
21
20
20
1
2
− 20
= − 20
10
20
3
15
1 =1
4
20
21
Exact:
1
1 5
6
36
=
35 +
=
35
5
5 5
5
3
3
3
− 12
= − 12
= − 12
5
5
5
3
23
5
103
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
49. Estimate:
17
− 15
2
53. Estimate: Exact:
6
5
6
5
1
6−2 =5 −2 =3
−3
6
6
6
6
3
Exact:
1
2
2 12
14
17 = 17 = 16 + = 16
6
12
12 12
12
5
5
5
5
−15 = −15 = −15
= −15
12
12
12
12
3
9
1 =1
4
12
54. Estimate: Exact:
9
1
2
1
1
− 5 9−4 =8 −4 = 4
2
2
2
2
4
55. Estimate: Exact:
12
2
9
2
7
12 − 9 = 11 − 9 = 2
−9
9
9
9
9
3
50. Estimate:
22
−11
11
56. Estimate: Exact:
10
1
3
1 2
10 − 9 = 9 − 9 =
−9
3
3
3 3
1
Exact:
5
5
5 18
23
= 22 = 21 + = 21
18
18
18 18
18
7
14
14
14
-10 = -10 = -10
= -10
9
18
18
18
1
9
11 = 11
2
18
51. Estimate:
46
−39
7
22
57. Estimate: Exact:
5
3
3
5 −3= 2
−3
17
17
2
58. Estimate: Exact:
16
4
4
16 − 5 = 11
−5
11
11
11
59. Estimate: Exact:
23
5
5
23 − 17 = 6
− 17
14
14
6
Exact:
3
6
6 14
20
46 = 46 = 45 + = 45
7
14
14 14
14
1
7
7
7
−38 = −38 = −38
= −38
2
14
14
14
13
7
14
60. Estimate: Exact:
22
3
3
21 − 10 = 11
− 10
4
4
12
2
5 8 37 8 ⋅8 37 ⋅ 3
61. 2 + 4 = +
=
+
3
8 3 8 3⋅8 8 ⋅ 3
64 111 175
7
=
+
=
=7
24 24
24
24
1
1 21 7 21 7 ⋅ 2
62. 5 − 3 =
− =
−
4
2 4 2 4 2⋅2
21 14 7
3
=
−
= =1
4
4 4
4
52. Estimate:
24
− 19
5
Exact:
1
13
13 26
39
23 = 23 = 22 +
= 22
2
26
26 26
26
10
20
20
20
−18 = −18 = −18
= −18
13
26
26
26
19
4
26
104
Section 3.4
63. 1
2 26 22 26 22 ⋅ 3
11
+4 =
+
=
+
5 15 5 15 5⋅ 3
15
26 66 92
2
=
+
=
=6
15 15 15
15
64. 2
71. 6
1 32 5 32 ⋅ 2 5⋅11
10
+2 =
+ =
+
2 11 2 11⋅ 2 2 ⋅11
11
64 55 119
9
=
+
=
=5
22 22 22
22
72. 8
3 31 51 31⋅ 2 51
7
65. 3 − 3 = − =
−
16 8 16 8 ⋅ 2 16
8
62 51 11
=
− =
16 16 16
69. 9
70. 4
74. 5
1 49 46 49 ⋅ 3 46 ⋅ 4
1
+5 =
+
=
+
9 12 9 12 ⋅ 3 9 ⋅ 4
12
147 184 331
7
=
+
=
=9
36
36
36
36
68. 10
11
1 95 25 95⋅ 3 25⋅7
+4 = +
=
+
14
6 14 6 14 ⋅3 6 ⋅ 7
285 175 460
40
=
+
=
= 10
42 42
42
42
20
= 10
21
3
1 179 17 179 ⋅ 2 17 ⋅11
+4 =
+
=
+
22
4 22
4
22 ⋅ 2
4 ⋅11
358 187 545
17
=
+
=
= 12
44
44
44
44
1
2 61 79 61⋅ 7 79 ⋅5
73. 12 − 11 = −
=
−
5
7 5
7
5⋅ 7
7 ⋅5
427 395 32
=
−
=
35
35 35
23 19 47 19 ⋅ 4 47
1
66. 3 − 1 = −
=
−
24 6 24 6 ⋅ 4 24
6
76 47 29
5
=
−
=
=1
24 24 24
24
67. 4
Addition and Subtraction of Mixed Numbers
11
3 161 23 161⋅ 2 23⋅15
+5 =
+
=
+
30
4 30 4
30 ⋅ 2 4 ⋅15
322 345 667
7
=
+
=
= 11
60
60
60
60
1
17 81 53 81⋅ 9 53⋅ 4
75. 10 − 2 = − =
−
8
18 8 18 8 ⋅ 9 18 ⋅ 4
729 212 517
13
=
−
=
=7
72
72
72
72
2
13 252 153
−7 =
−
25
20 25 20
252 ⋅ 4 153⋅5
=
−
25⋅ 4
20 ⋅5
1008 765 243
43
=
−
=
=2
100 100 100
100
76. 3
8
8 71 62 71⋅ 3 62 ⋅7
+6 = +
=
+
21
9 21 9 21⋅ 3 9 ⋅7
213 434 647
17
=
+
=
= 10
63
63
63
63
1
1 45 7 45 7 ⋅ 2 45 14
77. 11 − 3 =
− =
−
=
−
4
2 4 2 4 2⋅2 4
4
31
3
=
= 7 in.
4
4
1 293 33 293 33⋅8
5
−8 =
− =
−
4 32
4
32
4 ⋅8
32
293 264 29
=
−
=
32
32 32
1
1
2
1
1
78. 5 − 2 = 5 − 2 = 3 in.
2
4
4
4
4
7 163 23 163 23⋅5
3
−2 =
−
=
−
8 40 8
40 8 ⋅5
40
163 115 48
8
1
=
−
=
=1 =1
40 40 40
40
5
79. The index finger is longer.
1
1
80. 11 − 6 = 5 in.
4
4
A Belted Kingfisher is 5
105
1
in. longer.
4
Chapter 3
81.
82.
Fractions and Mixed Numbers: Addition and Subtraction
1 1 1 17 5 5 17 ⋅ 2 5 5
− −
87. 8 − 1 − 1 = − − =
2 4 4 2 4 4 2⋅2 4 4
34 5 5 24
=
− − =
= 6 in.
4 4 4 4
2
8
=
8
3
12
1
6
4
4
=
2
12
3
9
+3
= +3
4
12
11
23
15 = 16
12
12
11
The total is 16
hr.
12
8
1
=
2
1
3 =
8
1
+4
= +
3
2
88.
12
24
3
3
24
8
4
24
23
9
24
2
The total amount is 9
89.
23
tons.
24
3
1 23 7 23 ⋅ 3 7 ⋅ 4
− =
−
83. 5 − 2 =
4
3 4 3 4 ⋅3 3⋅ 4
69 28 41
5
=
−
= = 3 ft.
12 12 12
12
84.
90.
6
7 13
3 7 3⋅ 2 7
+
=
+ =
+
=
in.
8 16 8 ⋅ 2 16 16 16 16
91.
3
1 19 9 19 9 ⋅ 2
85. 1 − 1 = − = −
16 8 16 8 16 8 ⋅ 2
19 18 1
= − = in.
16 16 16
Divide by 2 to get the thickness:
1 1 1
1
÷2= ⋅ =
in
16 2 32
16
86.
92.
15 1 1 15 1 ⋅ 4 1 ⋅ 4
− − = −
−
16 4 4 16 4 ⋅ 4 4 ⋅ 4
15 4 4 7
= − − = in.
16 16 16 16
106
1
4
12
=
32
=
31
2
8
8
7
7
7
− 25
= − 25
= − 25
8
8
8
5
6
8
5
The water rose 6 in.
8
32
1
2
8
=
=
5
4
3
6
6
1
3
3
−2
= −2
= −2
2
6
6
5
2
6
5
There is 2 hr remaining.
6
5
1
1
1 5
10
4
19
+ 10 + 3 =
+ 10 + 3 = 13
4
2
5 20
20
20
20
19
The total distance is 13
mi.
20
3
9
=
3
4
12
5
5
−3
= −3
12
12
4 1
=
12 3
1
The blind will hang ft below the
3
window.
3
1
2
14
=
=
9
8
6
12
12
9
9
3
−3
= −3
= −3
12
12
4
5
5
12
5
He worked 5 hr more on Monday.
12
9
Section 3.4
1 7 3 1 5 7 3 1
93. (a) 1 + + + = + + +
4 8 4 2 4 8 4 2
5 ⋅ 2 7 3⋅ 2 1⋅ 4
=
+ +
+
4 ⋅2 8 4 ⋅2 2⋅4
10 7 6 4
= + + +
8 8 8 8
27
3
=
=3 L
8
8
3
greater than the last;
4
1 3
2 3
5
1
3 + =3 + =3 =4
2 4
4 4
4
4
97. Each number is
3
8
3 5
(b) 4 − 3 = 3 − 3 = L
8
8
8 8
1
94. Each number is greater than the last;
3
1 1
2
2 + =2
3 3
3
95. Each number is
Addition and Subtraction of Mixed Numbers
3
greater than the last;
4
1 3
3 + =4
4 4
2
greater than the last;
6
5 2
7
1
1 + =1 = 2
6 6
6
6
96. Each number is
98.
43
23 17 211
+
=
or 1
168
42 24 168
99.
14 9 11
+
=
75 50 30
100.
31 14 37
−
=
44 33 132
101.
29 7 137
−
=
68 92 391
102. 32
7
2 2509
25
+ 14
=
or 46
18
27
54
54
103. 21
3
31 2171
71
+4
=
or 25
28
42
84
84
104. 7
11
10 402
17
−2 =
or 5
21
33 77
77
105. 5
14
47 213
9
−2
=
or 3
17
68 68
68
Problem Recognition Exercises: Operations on Fractions and Mixed
Numbers
1
4 5 4 ⋅ 2 5 8 5 13
+ =
+ = + =
or 2
6
3 6 3⋅ 2 6 6 6 6
4 5 4 ⋅2 5 8 5 3 1
(d) − =
− = − = =
3 6 3⋅ 2 6 6 6 6 2
4
7 2 9
+ = or 1
5
5 5 5
7 2 14
(b)
× =
5 5 25
1
7 2 7 5 7
(c) ÷ = ⋅ = or 3
2
5 5 5 2 2
7 2 5
(d)
− = =1
5 5 5
(c)
1. (a)
2. (a)
(b)
3
1 11 3 11 3⋅ 2 11 6
+1 = + = +
= +
4
2 4 2 4 2⋅2 4 4
17
1
=
or 4
4
4
3
1 11 3 11 3 ⋅ 2
(b) 2 − 1 = − = −
4
2 4 2 4 2⋅2
11 6 5
1
= − = or 1
4 4 4
4
3. (a) 2
5
10
1
4 5 2 ⋅2
× =
×
=
or 1
9
3 6
3
2 ⋅3 9
3
4 5 4 6 4 2⋅ 3 8
÷ = ⋅ = ⋅
= or 1
5
5
3 6 3 5 3 5
107
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
1 2 21 2 21 ⋅ 3 2 ⋅ 5
(b) 4 + =
+ =
+
5 3 5 3 5⋅3 3⋅5
63 10 73
13
=
+
=
or 4
15 15 15
15
1 2 21 2 7 ⋅ 3 2 14
4
⋅ =
or 2
(c) 4 ⋅ = ⋅ =
5 3 5
5 3 5 3
5
1 2 21 3 63
3
(d) 4 ÷ = ⋅ =
or 6
5 3 5 2 10
10
3
1 11 3 11 2
11 2
÷1 = ÷ = ⋅ =
⋅
4
2 4 2 4 3 2⋅ 2 3
11
5
=
or 1
6
6
3
1 11 3 33
1
(d) 2 × 1 = ⋅ =
or 4
4
2 4 2 8
8
(c) 2
1
5 13 17 221
5
4. (a) 4 × 2 = ⋅ =
or 12
3
6 3 6
18
18
1
5 13 17 13 6
(b) 4 ÷ 2 = ÷ = ⋅
3
6 3 6
3 17
9
13 2 ⋅ 3 26
= ⋅
=
or 1
17
17
17
3
1
5 13 17 13⋅ 2 17
−
(c) 4 − 2 = − =
3
6 3 6
3⋅ 2 6
26 17 9 3
1
=
− = = or 1
6
6 6 2
2
1
5 13 17 13⋅ 2 17
=
+
(d) 4 + 2 = +
3
6 3 6
3⋅ 2 6
26 17 43
1
=
+ =
or 7
6
6
6
6
25 1 25
7
25
÷2=
⋅ =
or 1
9 2 18
18
9
25 2 50
5
25
(b)
⋅2 =
⋅ =
or 5
9 1 9
9
9
25 2 ⋅ 9 25 18 7
25
(c)
−2=
−
=
−
=
9 1⋅ 9
9
9 9
9
25
25 2 ⋅ 9 25 18
+2=
+
=
+
9 1⋅ 9 9
9
(d) 9
43
7
=
or 4
9
9
8. (a)
4 5 9 5 1
9. (a) 1 ⋅ = ⋅ = = 1
5 9 5 9 1
4 5 9 5 9⋅9 5⋅5
(b) 1 + = + =
+
5 9 5 9 5⋅9 9⋅5
81 25 106
16
=
+
=
or 2
45 45 45
45
4 5 9 9 81
6
(c) 1 ÷ = ⋅ =
or 3
5 9 5 5 25
25
4 5 9 5 9⋅9 5⋅5
(d) 1 − = − =
−
5 9 5 9 5⋅9 9⋅5
81 25 56
11
=
−
=
or 1
45 45 45
45
3 4 3 4 ⋅ 8 3 32 3
= − =
− =
−
8 1 8 1⋅ 8 8 8 8
29
5
=
or 3
8
8
3 4
3
3
1
= or 1
(b) 4 × = ×
8 1 2⋅ 4 2
2
3 4 8 32
2
(c) 4 ÷ = × =
or 10
8 1 3 3
3
3
3
35
(d) 4 + = 4 or
8
8
8
5. (a) 4 −
2
11 1 11
5
6. (a) 3 ÷ 2 = × =
or 1
3
3 2 6
6
2
5
2
(b) 3 − 2 = or 1
3
3
3
2
2
17
(c) 3 + 2 = 5 or
3
3
3
2
11 2 22
1
(d) 3 ⋅ 2 = ⋅ =
or 7
3
3 1 3
3
7 3 7 3 1
⋅ = ⋅ = =1
3 7 3 7 1
7 3 7 ⋅ 7 3 ⋅ 3 49 9 49 + 9
+
=
+ =
(b) + =
3 7 3 ⋅ 7 7 ⋅ 3 21 21
21
58
16
=
or 2
21
21
7 3 7 ⋅ 7 3 ⋅ 3 49 9 49 − 9
−
=
− =
(c) − =
3 7 3 ⋅ 7 7 ⋅ 3 21 21
21
40
19
=
or 1
21
21
7 3 7 7 7 ⋅ 7 49
4
(d) ÷ = ⋅ =
or 5
=
3 7 3 3 3⋅3 9
9
10. (a)
1 2 21 2 21 ⋅ 3 2 ⋅ 5
7. (a) 4 − = − =
−
5 3 5 3 5⋅3 3⋅5
63 10 53
8
=
−
=
or 3
15 15 15
15
108
Section 3.5
Section 3.5
Order of Operations and Applications of Fractions and Mixed Numbers
Order of Operations and Applications of Fractions and
Mixed Numbers
Section 3.5 Practice Exercises
3
4
10
4
6
1. 4 − 1 = 3 − 1 = 2
7
7
7
7
7
2. 7
13.
3
14
9
28
37
7
+ 2 = 7 + 2 = 9 = 10
10
15
30
30
30
30
5
5 29
−25
4
7
9
7
2
3. 16 − 3 = 15 − 3 = 12
9
9
9
9
14.
5
5 1 45 19 95
7
4. 5 ⋅ 2 =
⋅ =
= 11
8 9
8 9
8
8
1
8
1
3
1
3
12
32
=
=
24
23
5
20
20
3
15
15
− 14
= − 14
= − 14
4
20
20
17
9
20
1
19 30
−19
11
16.
2
121
13
 5
 11 
7.  1  =   =
=3
36
36
 6
6
8. 13
2 5 × 13 + 2 67
=
=
13
13
13
10. 2
7 2 × 11 + 7 29
=
=
11
11
11
2
2
2

 4 1
 3
1
3 3
17.  2 −  =  −  =   = ⋅
2
2 2

 2 2
 2
9
1
= =2
4
4
2
2
2

 15 2 
 13 
2
13 13
18.  3 −  =  −  =   = ⋅
5
5 5

 5 5
 5
169
19
=
=6
25
25
9 3 × 10 + 9 39
11. 3 =
=
10
10
10
12. 1
25
1
=3
8
8
3
8 25
−24
1
1
5
1
10
11
+ 4 = 13 + 4 = 17
14
7
14
14
14
9. 5
7
50
−49
1
30
11
15. 19 = 119
24
2
50
1
=7
7
7
7
1
2 64 8 64 3 8
2
5. 7 ÷ 2 =
÷ =
⋅ = =2
9
3 9 3
9 8 3
3
6.
29
4
=5
5
5
15 1 × 16 + 15 31
=
=
16
16
16
109
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
1 11 5 5
5 1
19. 1 ⋅ 2 ÷ 1 = ⋅ ÷
4 6 2 4
6 2
1
2
1
1
 1
5 2  2
5 2
25.  3 + 1  ⋅ 2 =  3 + 1  ⋅ 2
8 3
 4 8 3  8
 7 2
= 4  ⋅2
 8 3
1
11 5 4 11
2
= ⋅ ⋅ = =3
6 2 5 3
3
3
1
1
2
1
1
3 37 7 7
+ ÷
21. 6 + 2 ÷ 1 =
6
3 4 6 3 4
1
1
73
1
1
3
2
2
 1  7
5   6   16 17 
27.  1  ⋅  1 − 1  =   ⋅  − 
 5   9 12   5   9 12 
36  64 51 
= ⋅ − 
25  36 36 
37 8 45
3
1
=
+ =
=7 =7
6 6 6
6
2
5
7
1 1 79 13 10 79 65
22. 8 + 2 ⋅ 3 =
+ ⋅
=
+
9
9
9
6 3 9 6 3
1
3
36 13 13
=
⋅
=
25 36 25
144
= 16
9
1
12
3
1 1
36 1
12
⋅ = 6−
23. 6 − 5 ⋅ = 6 −
7 3
7
7 3
3
 1  7
2   4   25 5 
28.  1  ÷  2 + 1  =   ÷  + 
3   3   9 3
 3  9
64  25 15 
=
÷
+
27  9 9 
64 40
=
÷
27 9
1
42 12 30
2
− =
=4
7
7
7
7
1 1
19 7
24. 11− 6 ÷ 1 = 11− ÷
3 6
3 6
8
2
=
19 6
38
= 11− ⋅ = 11−
7
3 7
64
27
3
1
=
1
146 35 73
1
=
⋅
=
= 24
3
3
35 6
37 7 4 37 4
=
+ ⋅ =
+
6 3 7
6 3
=
1
 3
4  5  8 18  35
26.  1 + 2  ⋅5 =  +  ⋅
7 6  5 7  6
 5
 56 90  35
= + ⋅
 35 35  6
15 3 7
9
1
=
⋅ ⋅
= =1
8
7 4 10 8
=
1
39 8
=
⋅ = 13
8 3
1 1 7 15 4 7
20. 2 ÷ 1 ⋅ = ÷ ⋅
3 10 7 3 10
7
3
13
77 38 39
4
−
=
=5
7
7
7
7
110
1
⋅
9
40
5
=
8
15
Section 3.5
Order of Operations and Applications of Fractions and Mixed Numbers
3

11 
1 
34.  4 + 2  ÷  2 − 1 
36 
9 

 36 19   72 47 
= +  ÷ − 
9   36 36 
 9
55 25
= ÷
9 36
3
 3
 27 17   3 
1  1 
29.  6 − 2  ÷  1  =  −  ÷  
8   2
8  2 
 4
 4
 54 17   27 
= −  ÷ 
8  8
 8
1
37 8 37
10
=
⋅
=
=1
8 27 27
27
2
 1 7  1
 5 15   8 
30.  2 + 1  ⋅  1  =  +  ⋅  
 2 8  7
 2 8   7
 20 15  64
= + ⋅
 8 8  49
5
8
1
7
11
4
1
5
55 36 44
4
=
⋅
=
=8
9 25
5
5
1
2
7
2
7
4
3
−3 =3 −3 =
10
5
10
10 10
3
The difference is
sec.
10
2
1
7
4
(b) 3 + 3 + 3 + 3
5
2
10
5
4
5
7
8
=3 +3 +3 +3
10
10
10
10
24
4
2
= 12 = 14 = 14
10
10
5
35. (a) 3
35 64 40
5
=
⋅
=
=5
8 49
7
7
3
2
 1
 1 1 1 9 1 1 3
31.   +  2  ⋅ = + ⋅ = + = 1
 2
 4 3 4 4 3 4 4
18
72 4 72 1 18
2
3
14 ÷ 4 =
÷ =
⋅ =
=3
5 1
5 4 5
5
5
1
1
2
 2
 1 2 4 5 2
32.   +  2  ⋅ = + ⋅
 3
 2 9 9 2 9
4 5 9
= + = =1
9 9 9
The average is 3
1
1
3
1
1
36. (a) 2 + 3 + 2 + 4 + 1 + 2 + 3
2
4
4
4
2
1
1
1
3 1
= 2 + 3 + 3 + 2 +1 + 4 + 2
2
2
4
4
4
5
1
1
2
= 5 + 6 + 6 = 6 + 7 + 6 = 19
4
4
4
2
1
The total hours is 19 hr.
4

7 
13 
33.  5 − 1  ÷  3− 
8   16 

 40 15   48 13 
= −  ÷ − 
 8 8   16 16 
25 35
=
÷
8 16
=
5
2
1
7
3
sec.
5
11
1
77 7 77 1 11
3
(b) 19 ÷ 7 =
÷ =
⋅ = =2
4
4 1
4 7 4
4
25 16 10
3
⋅
=
=1
8 35 7
7
1
The average per day is 2
111
3
hr.
4
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
2110
1
1
1
3
37. (a) 11 + 10 + 7 + 9 + 4 + 8
4
2
2
4
1
2
2
3
= 11 + 10 + 7 + 9 + 4 + 8
4
4
4
4
8
= 49
4
= 51
The total weight loss is 51 lb.
42.
1
= 14,770
He must pay $14,770 in federal income
tax.
1
61 4 61 1 61
13
43. 15 ÷ 4 = ÷ = ⋅ =
=3
4
4 1 4 4 16
16
13
Each piece is 3 ft.
16
51
3
1
(b) 51 ÷ 6 =
=8 =8
6
6
2
1
The average is 8 lb.
2
9
3 27 4
44. 27 ÷ =
⋅ = 36
4
1 3
1
3
5
3
2
1
(c) 11 − 4 = 10 − 4 = 6 = 6
4
4
4
4
4
2
1
The difference is 6 lb.
2
1
36 bags can be filled.
 1 1 1
 3 4 2
45. 3−  + +  = 3−  + + 
 4 3 6
 12 12 12 
9
3
1
= 3− = 2 = 2
12
12
4
1
2 lb of cheese was eaten.
4
1
1
3
1
2
3
38. (a) 4 + 2 + 3 + 8 = 4 + 2 + 3 + 8
4
2
4
4
4
4
6
2
1
= 17 = 18 = 18
4
4
2
1
The total distance is 18 mi.
2
1 1 12 3 4
5
− = − − =
4 3 12 12 12 12
5
of the candy bar left.
There is
12
1
37 4 37 1 37
5
(b) 18 ÷ 4 =
÷ =
⋅ =
=4
2
2 1 2 4 8
8
5
The average is 4 mi.
8
46. 1 −
1 65 13 65 4 260
47. 65 ÷ 3 =
÷ = ⋅ =
= 20
4 1
4
1 13 13
20 loaves can be made.
5
3
5
6
13
6
7
39. 15 − 11 = 15 − 11 = 14 − 11 = 3
8
4
8
8
8
8
8
7
The stock dropped $3 .
8
6
1 894 149 894 4
÷
=
⋅
= 24
48. 894 ÷ 37 =
4
1
4
1 149
3
1
3
2
5
1
40. 12 + 1 = 12 + 1 = 13 = 14
4
2
4
4
4
4
1
The closing price was $14 .
4
41.
7
7 52750
⋅
⋅ ( 60,000 − 7250) =
1
25
25
1
His hourly rate is $24 per hour.
1 3
2 3
5
1
49. 6 + = 6 + = 6 = 7
2 4
4 4
4
4
1
The new rate is 7 points.
4
1
1 80, 250 80, 250
⋅ 80, 250 = ⋅
=
= 26,750
3
3
1
3
George will receive $26,750.
112
Section 3.5
50. 1
Order of Operations and Applications of Fractions and Mixed Numbers
9 21
30
5
1
+
=1 = 2 = 2
25 25
25
25
5
United Kingdom consumes 2
 3
 91 53 
2
55. 30 ⋅  22 − 17  = 30 ⋅  − 
3
 4
 4 3
 273 212 
−
= 30 ⋅ 
12 
 12
1
kg per
5
capita.
 61 1830
= 30 ⋅   =
12
 12 
1
= 152
2
 1 1
 5 5
 10 5 
51. 3⋅  2 + 1  = 3⋅  +  = 3⋅  + 
 2 4
 2 4
 4 4
3 15 45
1
= ⋅ =
= 11
1 4
4
4
1
Stephanie will need 11 yd for the
4
dresses.
Joan saves 152
1
gal.
2
1
3
2
3
6
3
3
56. 36 − 5 = 36 − 5 = 35 − 5 = 30
2
4
4
4
4
4
4
3
30 lb of cookies were sold.
4
 1 1
 2  5 10
52. 5⋅  1 + 1  = 5⋅  2  = ⋅
 4 4
 4 1 4
50
2
1
=
= 12 = 12
4
4
2
1
Grace spends 12 hr each week traveling
2
to and from work.

1
1 
2
2
57.  2 + 9 + 7 + 9 + 2 − 14 = 30 − 14
2
2 
3
3

3
2
= 29 − 14
3
3
1
= 15
3
1
She needs 15 ft more.
3
1  1 1   1  13  3 2   13 
53. 6 −  +   6  = −  +   
2  2 3  2  2  6 6   2 
13 5 13
= − ⋅
2 6 2
13 65 78 65
= −
=
−
2 12 12 12
13
1
=
=1
12
12
1
Wilma has 1 lb left.
12
1
3
5
3
5
3
1
58. 2 + 3 + 7 + 8 + 7 + 3 + 2
4
8
8
4
8
8
4
2
3
5
6
5
3
2
= 2 +3 +7 +8 +7 +3 +2
8
8
8
8
8
8
8
26
2
1
= 32 = 32 + 3 = 35
8
8
4
1
The distance is 35 yd.
4
 2 1
 8 3
11
54. 1−  +  = 1−  +  = 1−
12
 3 4
 12 12 
12 11 1
= − =
12 12 12
1
Jeremy has
of the lawn left to mow.
12
4
 1  8 25
59. 8 ⋅  12  = ⋅
= 100
 2 1 2
1
The perimeter is 100 in.
2
8
 2  5  4 11 11 242
60. 4 ⋅  3 1  = ⋅ ⋅ =
= 26
9
9
 3  6  1 3 6
3
The total area of the 4 shutters is 26
113
8 2
ft .
9
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
1
1
1
2
3
3
61. 35 + 20 + 20 = 35 + 20 + 20
3
2
2
6
6
6
8
2
1
= 75 = 76 = 76
6
6
3
1
Matt needs 76 ft of gutter.
3
62.
1
1
3
(b) 5 + 16 + 13 + 16 + 14
2
5
10
1 1 3
+ +
2 5 10
5 2
3
10
= 64 + + + = 64 = 65
10 10 10
10
The perimeter is 65 m.
= 5 + 16 + 13 + 16 + 14 +
1
17 425
1
 1
= 212
(50)  8  = 25 ⋅ =
2
2
2
2
 2
1 2
The area is 212 ft .
2
66. (a)
1
 7  1  2 287 57
63. 2 ⋅  35 14  = ⋅
⋅
 8  4  1 8 4
( )
16,359
16
7
= 1022
16
=
1
3
2
3
5
1
(b) 12 + 9 = 12 + 9 = 21 = 22
2
4
4
4
4
4
1
22 ft of lights are needed.
4
7 2
ft .
16
67. Door and front plus back:
 1   1  1  1   1  
2 ⋅  6   4  +  6   1  
 2   3  2  2   2  
 13 13 1 13 3 
= 2⋅ ⋅ + ⋅ ⋅ 
 2 3 2 2 2
47
 7  1  37 31 1147
= 22
64.  3  6  = ⋅ =
50
50
 10  5  10 5
47
The area is 22
cm 2 .
50
 169 39 
 676 117 
+  = 2⋅
+
= 2⋅
8
24 
 6
 24
 1
1  1   1
65. (a)  5   13  + (16)  13 
2  2   5
 5
1
2 793 793 2
= ⋅
=
m
12
1 24
33
=
( )
135 ft 2 of paint is needed.
4
The area of the whole roof is 1022
 3
1  1
7  10 +  9  10

2  2
 4
1 15 10 39 10
= ⋅ ⋅ + ⋅
2 2 1 4 1
150 390 540
=
+
=
= 135
4
4
4
1 11 66 16 66
⋅ ⋅
+ ⋅
2 2 5
1 5
12
1
Sides:
363 1056 363 2112
=
+
=
+
10
5
10
10
2475
1
=
= 247
10
2
1 2
The area is 247 m .
2
 1
13 20 13 260 2
2 ⋅ (10)  4  = 20 ⋅ =
⋅ =
m
3
1 3
3
 3
793 260 793 1040
+
=
+
12
3
12
12
1833
3 2
=
= 152 m
12
4
Total area =
114
Chapter 3
Chapter 3
Review Exercises
2
Section 3.1
21 5 13 21 5 4 21 10 11
− ÷ = − ⋅ = − =
15.
13 2 4 13 2 13 13 13 13
1. 5 books + 3 books = 8 books
1
2. 12 cm + 6 cm = 18 cm
3
1 8 1
= ⋅ =
8 7 7
5. Fractions with the same denominators are
considered like fractions.
1
17.
4 2
6. For example: like fractions: , ;
7 7
1 3
unlike fractions: ,
9 16
3
(b)
2
25 19 12 56 28
+ + =
=
in.
10 10 10 10 5
28 4 24 12
− =
=
in.
10 10 10 5
Section 3.2
2
4 6 10 2
+ =
=
15 15 15 3
19. (a) 7, 14, 21, 28
(b) 13, 26, 39, 52
(c) 22, 44, 66, 88
5
1
6 1
+
=
=
12 12 12 2
20. 6 and 8 have many common multiples
including 24, 48, and 56. Of all the
common multiples, 24 is the least.
3
10.
2 7 9
+ = =1
9 9 9
11.
15 6 9
− =
7 7 7
21. (a) 1, 2, 4, 5, 10, 20, 25, 50, 100
(b) 1, 5, 13, 65
(c) 1, 2, 5, 7, 10, 14, 35, 70
5
22. (a) 5 25
21 6 15
12.
− = =3
5 5 5
2 50
3 3 3
9
3 12 3
⋅ + = + = =
8 2 16 16 16 16 4
2 100
100 = 2 ⋅ 2 ⋅ 5 ⋅ 5
2
14.
11 13 11 13 48
+ + + =
= 12 in. or 1 ft
4 4 4 4
4
18. (a)
5 4 9 3
7.
+ = =
6 6 6 2
13.
3
1
4. 13 CDs − 2 CDs = 11 CDs
9.
3
 7
2  8  5  8  1 8
16.  −  ⋅ =   ⋅ =   ⋅
 10 10  7  10  7  2  7
3. 25 mi − 13 mi = 12 mi
8.
Review Exercises
4 4
4 16 20
+  = +
=
9 3
9 9
9
13
(b) 5 65
65 = 5 ⋅ 13
115
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
7
(c) 5 35
33.
2 70
70 = 2 ⋅ 5 ⋅ 7
23. 30: 30, 60, 90, 120, 150, 180
25: 25, 50, 75, 100, 125, 150
LCM: 150
24.
34.
2) 22 144
72
2)11
36
2)11
2)11 18
9
3)11
11
3
3)
1
11) 11
1
1
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 11 = 1584
35.
25. 105 = 3 ⋅ 5 ⋅ 7
28 = 2 ⋅ 2 ⋅ 7
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 ⋅ 7 = 420
27. The LCM of 3 and 4 is 12. They will meet
on the 12th day.
29.
30.
31.
32.
5 5 ⋅ 3 15
=
=
6 6 ⋅ 3 18
7 147
=
10 210
72 144
=
105 210
8 112
=
15 210
27 162
=
35 210
112 < 144 < 147 < 162, so
8 72 7 27
,
, ,
15 105 10 35
Section 3.3
26. 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
32 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 = 96
28.
5 5 ⋅ 3 15
=
=
6 6 ⋅ 3 18
7 7 ⋅ 2 14
=
=
9 9 ⋅ 2 18
15 14
5 7
> , so >
18 18
6 9
5
5 ⋅ 3 15
=
=
16 16 ⋅ 3 48
9 9 ⋅ 7 63
=
=
5 5 ⋅ 7 35
7
7 ⋅5 35
=
=
12 12 ⋅5 60
36.
1 7
3 14 17
+ =
+
=
8 12 24 24 24
37.
9
61
90
61
29
−
=
−
=
10 100 100 100 100
38.
11 2 11 10 1
− =
−
=
25 5 25 25 25
39.
3
5
3 10 13 1
+ =
+
=
=
26 13 26 26 26 2
40.
25
25 22 47
+2=
+
=
11
11 11 11
41. 4 −
17 17 ⋅10 170
=
=
15 15⋅10 150
7 14
=
12 24
11 14
11 7
< , so
<
24 24
24 12
116
37 80 37 43
=
−
=
20 20 20 20
42.
4 0 4
4
− = −0=
15 3 15
15
43.
0
1
1
1
+
=0+
=
17 34
34 34
Chapter 3
44.
7
33
70
33
37
−
=
−
=
100 1000 1000 1000 1000
45.
2 5 1 16
75 40
51 17
+ − =
+
−
=
=
15 8 3 120 120 120 120 40
50. (a)
Review Exercises
3 7 3 7 9 14 9 14
+ + + = + + +
2 3 2 3 6 6 6 6
46 23
2
=
=
or 7 yd
6
3
3
1
7 3 7
1
(b)
⋅ = or 3 yd 2
3 2 2
2
11 4 3 11 8 21 24 12
46.
− + = − +
=
=
14 7 2 14 14 14 14 7
1
 2 1  15 4
47.  +  ÷ −
 5 40  8 25
 16 1  15 4
= +  ÷ −
 40 40  8 25
Section 3.4
51.
1
8
4 17 4
=
⋅ −
=
−
40 15 25 75 25
17
Ź 5
=
17 12 5
1
−
=
=
75 75 75 15
2
48.
52.
2
1 20  11 5 
1
20  11 1 
⋅ −  + =
⋅ −  +
7 7  15 15 
7
7  15 3 
2
=
20  6 
1
⋅  +
7  15 
7
2
53.
20  2 
1
=
⋅  +
7  5
7
4
=
20 4 1
⋅
+
7 25 7
5
16 1 16 5
+ =
+
35 7 35 35
21 3
=
=
35 5
54.
=
49. (a)
(b)
25 63 13 25 63 52
+ + =
+ +
16 16 4 16 16 16
140 35
3
=
=
or 8 m
16
4
4
55.
1  63  5  315
59 2
or 2
m
   =
2  16  4  128
128
117
8
56
= 9
9
63
2
18
+1 = +1
7
63
11
74
10 = 11
63
63
9
1
8
= 10
2
16
15
15
+3
= +3
16
16
7
23
13 = 14
16
16
10
5
5
29
=
=
7
6
24
24
24
7
14
14
−4
= −4
= −4
12
24
24
5
15
2 =2
8
24
7
1
2
14
=
=
5
4
6
12
12
3
3
1
−3
= −3
= −3
12
12
4
11
1
12
5
3
9
=
5
8
24
8
1
−2
= −2
24
3
1
3
24
5
Chapter 3
56.
57.
58.
59.
60.
Fractions and Mixed Numbers: Addition and Subtraction
63. Estimate: 2 + 4 + 2 = 8
1
2
29
9
8
29
Exact: 2 + 4 + 1 = 2 + 4 + 1
4
9
36
36
36
36
46
10
5
=7
=8 =8
36
36
18
4
12
= 3
5
15
4
4
−1
= −1
15
15
8
2
15
3
64. Estimate: 5 + 2 + 4 = 11
Exact:
2
9
19
12
27
19
5 +1 + 3 = 5 +1 + 3
5 10
30
30
30
30
58
28
= 9 = 10
30
30
14
= 10
15
4
8
=
6
7
14
11
11
+5
= +5
14
14
5
19
11 = 12
14
14
6
3
6
3
=
8
16
13
13
+2
= +2
16
16
3
19
5 =6
16
16
3
65. Estimate: 65 − 15 = 50
1
9
5
36
Exact: 65 − 14 = 65 − 14
8
10
40
40
45
36
9
= 64 − 14
= 50
40
40
40
5
5
3
3
−2
5 = −2
5
2
3
5
6
=
66. Estimate: 44 − 21 = 23
13
23
65
69
Exact: 43 − 20 = 43 − 20
15
25
75
75
140
69
= 42
− 20
75
75
71
= 22
75
5
14
14
11
11
−4
14 = − 4
14
3
3
14
8
=
7
1
2
3
4
7
1
67. 4 + 3 = 4 + 3 = 7 = 8
2
3
6
6
6
6
1
Corry drove a total of 8 hr.
6
1 1
1
2
9
2 7
68. 2 − 1 = 2 − 1 = 1 − 1 =
8
4
8 8
8
8 8
7
Denise will have
acre left.
8
1
2
61.
42
42
=
8
16
13
13
+ 21
= + 21
16
16
15
63
16
Section 3.5
9
9
62.
38
= 38
10
10
3
6
+ 11
= + 11
5
10
5
1
15
49 = 50 = 50
10
2
10
7
8
5
1
1
9 2 6 49 16
⋅
69. 1 + 4 ⋅ 2 = +
5
10 7 5 10 7
=
118
6 56 62
2
+
=
= 12
5 5
5
5
Chapter 3
1
2 23 47 47
3
70. 5 − 23 ÷ 5 =
−
÷
2
9 4
2
9
4
 5  1
3
74.  1  ÷  11 − 10 
4
 16   8
1
23 47 9
23 9
=
−
⋅
=
−
2 47
4
4 2
1
=
23 18 5
1
− = =1
4 4 4
4
2
7
4
1
3
16,000
1
3
The appraised value is $144,000.
 1
1  10  2
1  10
72.  5 + 1  ⋅ 2 =  5 + 1  ⋅ 2
 8 16  11  16 16  11
 3  10
= 6  ⋅2
 16  11
3
1
 1
76.  2 + 7 + 2  ÷ 10
4
2
 4
3
2
 1
=  2 + 7 + 2  ÷ 10
4
4
 4
 6
 2
= 11  ÷ 10 = 12  ÷ 10
 4
 4
2
99 32
=
⋅
= 18
16 11
5
1
50 10 50 1
⋅
=
÷ =
4 10
4 1
2
 1  1
5  6  3
5
73.  1  ⋅  4 + 3  =   ⋅  4 + 3 
6  5  6
6
 5  2


36
8
= ⋅ 7 
25  6 
1
=
There are 1
2
36 50
=
⋅
= 12
25 6
1
21  9
6
=
÷  10 − 10 
16  8
8
9
9 160,000
⋅
= 144,000
(160,000) =
75.
1
10
10
1
6
2
21 64 28
1
=
⋅
=
=9
3
3
16 9
13 4
4
=
⋅
=
9 39 27
2
21  1
6
=
÷  11 − 10 
16  8
8
2
 4
3 13 39
= 1  ÷ 9 = ÷
4
4 9
 9
1
2
21  3 
21 9
=
÷  =
÷
16  8 
16 64
 1
2
6
3  1
3
71.  8 − 6  ÷ 9 =  8 − 6  ÷ 9
3
9
4  9
4
 9
 10
6
3
= 7 − 6  ÷ 9
9
4
 9
9
Review Exercises
1
119
5
1
or 1
4
4
1
lb of nuts in each bag.
4
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
Chapter 3
Test
1.
4 3 7
+ =
5 5 5
13.
2.
23 15 8 1
− =
=
16 16 16 2
3
5
6
5
11
3
14. 6 + 10 = 6 + 10 = 16 = 17
4
8
8
8
8
8
3. When subtracting like fractions, keep the
same denominator and subtract the
numerators. When multiplying fractions,
multiply the denominators as well as the
numerators.
15. 12 − 9
(b) 1, 2, 3, 4, 6, 8, 12, 24
1
3
7
4
9
7
17. 15 − 12 − 1 = 15 − 12 − 1
6
8 24
24
24
24
28
9
7
= 14 − 12 − 1
24
24
24
12
1
=1 =1
24
2
3
(c) 2 6
2 12
2 24
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3 = 23 ⋅ 3
1
5 2
1 37 8 49 37 49
18. 4 ⋅ 2 − 8 =
⋅ −
=
−
8 3
6 8 3 6
3
6
5. 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2
24 = 2 ⋅ 2 ⋅ 2 ⋅ 3
30 = 2 ⋅ 3 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 5 = 240
5 5 ⋅ 7 35
=
=
9 9 ⋅ 7 63
7.
11 11 ⋅ 3 33
=
=
21 21 ⋅ 3 63
8.
4 4 ⋅ 9 36
=
=
7 7 ⋅ 9 63
9. 33 , 35 , 36, so
10
11
10
1
= 11 − 9 = 2
11
11
11
11
1
1
2
3
6
8
16. 22 + 35 + 2 = 22 + 35 + 2
4
2
3
12
12
12
17
5
= 59 = 60
12
12
4. (a) 24, 48, 72, 96
6.
3 1
9 1 10 2
+ = + = =
5 15 15 15 15 3
1
=
74 49 25
1
−
=
or 4
6
6
6
6
1
1
2 10 5 17
19. 3 ÷ 2 + 5 =
÷ +
3
2
3 3 2 3
2
10 2 17 4 17
=
⋅ +
= +
3 5 3 3 3
1
=
11 5 4
, ,
21 9 7
10.
3 3
6
3
9
+ = + =
8 16 16 16 16
11.
7
7 6 1
−2= − =
3
3 3 3
12.
7 1 7 3
4 1
− = − = =
12 4 12 12 12 3
120
21
=7
3
Chapter 3
24. Area:
2
 2
 1
5  4  11 17 
÷
+
20.   ÷  1 + 2  =
6  25  10 6 
 5
 10
4  33 85 
=
÷
+
25  30 30 
4 118
=
÷
25 30
2
=
4
6
⋅
30
25 118
5
=
11
2
 2  7  22 57 627
=
= 25
m2
⋅
 4  5  =
25
25
 5  10  5 10
5
Perimeter:
 2
 7
 22 
 57 
2 4  + 25  = 2  + 2 
 5
 10 
 5
 10 
44 114 88 114
=
+
= +
5 10 10 10
202
2
1
=
= 20 = 20 m
10
10
5
12
295
59
 1
1  3  29 31 8
21.  7 − 5  ⋅1 =  −  ⋅
6 5  4
6 5
 4
 87 62  8
= − ⋅
 12 12  5
5
 3 1
25. 14,000 −  +  ⋅14,000
 28 7 
 3
4
= 14,000 −  +  ⋅14,000
 28 28 
2
500
25 8 10
1
=
⋅ =
or 3
3
3
12 5
3
Test
= 14,000 −
1
7
28
⋅
14,000
1
Ź1
22.
= 14,000 − 7 ⋅500
= 14,000 − 3500 = 10,500
Justin has $10,500 for cabinets.
2 1 2 3 6
1  =   = = 1
3 2 3 2 6
1 lb is needed.
50
26. 28
19
179 2000
23. 4 (2000) =
⋅
= 8950
40
1
40
1
The Ford Expedition can tow 8950 lb.
Chapters 1–3
Cumulative Review Exercises
1. Twenty-three million, four hundred
thousand, eight hundred six
2.
3.
4.
72
+ 24
96
6
41
3
41
9
32
2
− 24 = 28 − 24 = 4 = 4
48
16
48
48
48
3
2
The difference is 4 ft.
3
72
24
288
+ 1440
1728
×
3
5. 24 72
−72
0
1
72
− 24
48
121
Chapter 3
Fractions and Mixed Numbers: Addition and Subtraction
6. 54,923 rounds to 50,000
28,543 rounds to 30,000
50,000
×
30,000
1,500,000,000
5
17. 3 15
3 45
2 90
2 4 2
7. 4 ⋅ 4 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 5 ⋅ 8 ⋅ 8 = 4 ⋅ 5 ⋅ 8
2 180
2 360
8. 72 ÷ (42 − 10) ⋅ 3 = 72 ÷ (16 − 10) ⋅ 3
= 72 ÷ 6 ⋅ 3 = 12 ⋅ 3 = 36
360 = 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 = 23 ⋅ 32 ⋅ 5
9. 17, 19, 23, 29, 31
18.
7
10. 5 35
180 180 ⋅1 1
=
=
900 180 ⋅5 5
3
1
8
1
15 2 3
19.
⋅ =
16 5 8
2 70
70 = 2 ⋅ 5 ⋅ 7
11. Numerator: 21; denominator: 17
12.
13.
4
1
7
1
20 5 20 9 4
20.
÷ =
⋅ =
63 9 63 5 7
1
4
or
4
16
17
5
had pepperoni, 22 − 17 = 5, so
22
22
did not have pepperoni.
14. (a) Improper
(b) Proper
(c) Improper
21.
13 7 6 3
− = =
8 8 8 4
22.
3 5 25 33
+ +
=
16 16 16 16
23. 4 −
15. (a) 2 + 3 + 9 + 0 = 14, which is not
divisible by 3, so 2390 is not divisible
by 3.
(b) 1 + 2 + 4 + 5 = 12, so 1245 is divisible
by 3. The ones-place digit is 5, so
1245 is also divisible by 5.
(c) The ones-place digit is not 0 or 5, so
9321 is not divisible by 5.
18 20 18 2
=
− =
5
5
5 5
5
1
3 61 25 305
19
24. 6 ⋅ 2 =
⋅
=
or 13
10 11 10 11
22
22
2
2
3
7 13 17 13 10 26
9
25. 2 ÷ 1 = ÷ = ⋅
=
or 1
5 10 5 10 5 17 17
17
1
16. (a) Composite; 51 = 3 ⋅ 17
(b) Composite: 52 = 2 ⋅ 26
(c) Prime
122
Chapters 1–3
2
2
 1
3  5 5  33 11 5 5
26.  8 ÷ 2  ⋅ + =  ÷  ⋅ +
4  18 6  4 4  18 6
 4
3
1
29.
= 32 ⋅
1
5 5
+
18 6
1
=
1 1
1 21 3 63
7
or 7 m2
 5  (3) = ⋅ ⋅ =
2 4
2 4 1 8
8
4
1
8
5
3
30. (a) 8 − 6 = 8 − 6 = 2
5
2
10
10
10
The difference in the Richter scale
3
values is 2 .
10
1
4
9

(b)  7 + 6 + 8 + 4  ÷ 4

2
5
10 
5
8
9

= 7 + 6 + 8 + 4  ÷ 4

10
10
10 
2
 22 
=  25  ÷ 4 = 27 ÷ 4
 10 
10
272 4 272 1 272
=
÷ =
⋅ =
10 1 10 4 40
32
4
=6 =6
40
5
The average of the Richter scale
4
values is 6 .
5
2
 33 4  5 5
=
⋅  ⋅ +
 4 11  18 6
1
Cumulative Review Exercises
9 5 5
⋅
+
1 18 6
2
5 5 15 5
= + = +
2 6 6 6
20 10
1
=
=
or 3
6
3
3
4
22
22 28
⋅ 28 =
⋅
= 88
27.
7
7 1
1
The distance around is approximately 88
cm.
2
1
1
4
3
8
1
28. 1 + 1 + 1 = 1 + 1 + 1 = 3
3
2
6
6
6
6
6
2
1
= 4 = 4 yd
6
3
123
Chapter 4
Decimals
Chapter Opener Puzzle
Section 4.1
Decimal Notation and Rounding
Section 4.1 Practice Exercises
1. (a) decimal
17. Ten-thousandths
(b) tenths; hundredths; thousandths
18. Thousandths
2. Three-thousand, five
19. Thousandths
3. 102 = 100
20. Ones
3
4. 10 = 1000
21. Ones
5. 104 = 10,000
22. Millionths
6. 105 = 100,000
23. Nine-tenths
24. Seven-tenths
2
1
 1 
7.   =
100
 10 
25. Twenty-three hundredths
26. Nineteen hundredths
3
1
1
8.   =
 10  1000
27. Thirty-three thousandths
4
28. Fifty-one thousandths
1
 1 
9.   =
10,000
 10 
29. Four hundred seven ten-thousandths
5
1
 1 
10.   =
 10  100,000
30. Twenty ten-thousandths
11. Tenths
32. Four and twenty-six hundredths
12. Hundredths
33. Five and nine-tenths
13. Hundredths
34. Three and four-tenths
14. Tens
35. Fifty-two and three tenths
15. Tens
36. Twenty-one and five-tenths
16. Ten-thousandths
37. Six and two hundred nineteen thousandths
31. Three and twenty-four hundredths
124
Section 4.1
38. Seven and three hundred thirty-eight
thousandths
39. 8472.014
40. 60,025.0401
41. 700.07
Decimal Notation and Rounding
57. 8.4 =
84 42
=
10 5
58. 2.5 =
25 5
=
10 2
59. 3.14 =
314 157
=
100 50
60. 5.65 =
565 113
=
100 20
61. 23.5 =
235 47
=
10
2
62. 14.6 =
146 73
=
10
5
42. 9000.009
43. 2,469,000.506
44. 82,000,614.0097
45. 3.7 = 3
46. 1.9 = 1
7
10
9
10
47. 2.8 = 2
48. 4.2 = 4
49. 0.25 =
8
4
=2
10
5
2
1
=4
10
5
63. 11.91 =
1191
100
64. 21.33 =
2133
100
65. 34.2, 34.25, 34.29, 34.3
25 1
=
100 4
66. 12.4, 12.46, 12.49, 12.5
67. 0.042, 0.043,
50. 0.75 =
75 3
=
100 4
51. 0.55 =
55 11
=
100 20
52. 0.45 =
45
9
=
100 20
68. 0.0499, 0.04999, 0.05001, 0.4999,
69. 6.312 < 6.321
70. 8.503 < 8.530
71. 11.21 > 11.2099
812
203
53. 20.812 = 20
= 20
1000
250
54. 32.905 = 32
56. 4.0015 = 4
72. 10.51 > 10.5098
73. 0.762 > 0.76
905
181
= 32
1000
200
55. 15.0005 = 15
4
, 0.42, 0.43
10
74. 0.1291 > 0.129
5
1
= 15
10,000
2000
75. 51.72 < 51.721
76. 49.06 < 49.062
3
15
=4
2000
10,000
77. a, b
125
5
10
Chapter 4
Decimals
78. a, c, d
88. 4.359 ≈ 4.36
79. 0.3444, 0.3493, 0.3558, 0.3585, 0.3664
89. 9.0955 ≈ 9.096
80. 148.148, 148.295, 148.466, 149.333
90. 2.9592 ≈ 2.959
81. These numbers are equivalent, but they
represent different levels of accuracy.
91. 21.0239 ≈ 21.0
82. a
92. 16.804 ≈ 16.80
83. 7.1
93. 6.9995 = 7.000
84. 34.50
94. 21.9997 = 22.000
85. 49.943 ≈ 49.9
95. 0.0079499 = 0.0079
86. 12.7483 ≈ 12.7
96. 0.00084985 = 0.0008
87. 33.416 ≈ 33.42
97. 0.00362005 ≈ 0.0036 mph
Number
Hundreds
Tens
Tenths
Hundredths
Thousandths
98.
349.2395
300
350
349.2
349.24
349.240
99.
971.0948
1000
970
971.1
971.09
971.095
100.
79.0046
100
80
79.0
79.00
79.005
101.
21.9754
0
20
22.0
21.98
21.975
102. 0.279
Section 4.2
103. 0.972
Addition and Subtraction of Decimals
Section 4.2 Practice Exercises
1. b, c
10. 2.78999 ≈ 2.790
2. a, c
11. Expression
44.6
+ 18.6
63.2
Estimate
45
+ 19
64
12. Expression
28.2
+ 23.2
51.4
Estimate
28
+ 23
51
13. Expression
5.306
+ 3.645
8.951
Estimate
5
+4
9
3. b, c
4. c, d
5. 23.489 ≈ 23.5
6. 42.314 ≈ 42.31
7. 8.6025 ≈ 8.603
8. 0.981 ≈ 1.0
9. 2.82998 ≈ 2.8300
126
Section 4.2
14. Expression
3.451
+ 7.339
10.790
Estimate
3
+7
10
15. Expression
12.900
+ 3.091
15.991
Estimate
13
+3
16
16. Expression
4.125
+ 5.900
10.025
Estimate
4
+6
10
17.
78.9000
+ 0.9005
79.8005
18.
44.2000
+ 0.7802
44.9802
19.
23.0000
+ 8.0148
31.0148
20.
21.
22.
23.
24.
25.
7.9302
+ 34.0000
41.9302
34.0000
23.0032
+ 5.6000
62.6032
23.0000
8.0100
+ 1.0067
32.0167
68.394
+ 32.020
100.414
2.904
+ 34.229
37.133
103.94
+ 24.50
128.44
127
Addition and Subtraction of Decimals
26.
93.200
+ 43.336
136.536
27.
54.200
23.993
+ 3.870
82.063
28.
13.9001
72.4000
+ 34.1300
120.4301
29. Expression
35.36
− 21.12
14.24
Estimate
35
− 21
14
30. Expression
53.9
− 22.4
31.5
Estimate
54
− 22
32
31. Expression
7.24
− 3.56
3.68
Estimate
7
−4
3
32. Expression
23.3
− 20.8
2.5
Estimate
23
− 21
2
33. Expression
45.02
− 32.70
12.32
Estimate
45
− 33
12
34. Expression
66.15
− 42.90
23.25
Estimate
66
− 43
23
35.
14.500
− 8.823
5.677
36.
33.200
− 21.932
11.268
Chapter 4
Decimals
48.
4.00
− 0.42
3.58
3.005
+ 25.127
28.132
− 13.700
14.432
49.
103.400
− 45.050
58.350
− 0.982
57.368
23.370
− 21.900
1.470
+ 5.111
6.581
50.
98.500
− 23.210
75.290
− 0.144
75.146
0.780
− 0.028
0.752
+ 6.100
6.852
51.
8.962
+ 51.000
59.962
− 40.050
19.912
52.
11.957
+ 45.000
56.957
− 3.550
53.407
53.
5.3000
5.0300
+ 5.0030
15.3330
− 5.0003
10.3327
54.
2.6000
2.0600
+ 2.0060
6.6660
− 2.0006
4.6654
55.
5.8400
+ 5.0840
10.9240
− 5.0084
5.9156
37.
2.000
− 0.123
1.877
38.
39.
40.
41.
55.9000
− 34.2354
21.6646
42.
49.100
− 24.481
24.619
43.
18.003
− 3.238
14.765
44.
21.030
− 16.446
4.584
45.
183.010
− 23.452
159.558
46.
164.2300
− 44.3893
119.8407
47.
6.007
+ 12.740
18.747
− 3.400
15.347
128
Section 4.2
56.
85.3000
− 47.0092
38.2908
+ 4.0600
42.3508
57.
10.000
− 0.900
9.100
− 0.090
9.010
− 0.009
9.001
58.
(c) 11.0
1.7
+ 1.7
14.4
At 3.00 P.M. the level will be 14.4 in.
62. (a)
339.7
− 322.7
$17.0 million
441.2
(c)
339.7
328.5
+ 322.7
$1432.1 million (or $1.4321 billion)
63.
59. (a)
686.980
− 365.256
321.724 days
(b) 224.70
− 67.97
156.73 days
2.8125
− 2.1250
0.6875 lb
(b)
2.3750
2.6875
2.8125
2.1250
2.4375
+ 2.8125
15.2500 lb
441.2
− 328.5
$112.7 million
(b)
5.000
− 0.900
4.100
− 0.990
3.110
− 0.999
2.111
60. (a)
Addition and Subtraction of Decimals
245.62
52.48
193.14
− 72.44
120.70
− 108.34
12.36
+ 1084.90
1097.26
− 23.87
1073.39
+ 200.00
1273.39
The ending balance was $1273.39.
−
64. The bank subtracted $1500 on Jan 6
instead of $150.00.
61. (a)
5.9
7.6
9.3
− 4.2
− 5.9
− 7.6
1.7
1.7
1.7
The water is rising 1.7 in./hr.
(b) 9.3
+ 1.7
11.0
At 1 P.M. the level will be 11 in.
129
65.
4.20
− 2.85
1.35
1.35 million cells per microliter
66.
1299.99
− 998.95
301.04
It was marked down by $301.04.
Chapter 4
Decimals
67. 3 quarters and 1 dime:
3 × 1.75 = 5.25
5.25
+ 1.35
6.60 mm
2 nickels and 2 pennies:
2 × 1.95 = 3.9
2 × 1.55 = 3.1
71. x:
y:
3.62
− 1.03
2.59 ft
P = 4.875
3.620
1.600
2.590
2.075
2.590
1.200
+ 3.620
22.170 ft
3.9
+ 3.1
7.0 mm
The pile containing 2 nickels and 2
pennies is higher.
68.
1.95
− 1.75
0.20
The nickel is 0.2 mm thicker.
69. x:
72. x: 35.05
− 8.35
26.70 yd
27.3
− 18.4
8.9 in.
y:
y: 22.1
− 6.7
15.4 in.
P=
30.9
8.4
20.8
53.4
10.1
+ 61.8
185.4 cm
8.35
10.90
26.70
11.20
26.70
15.03
8.35
+ 37.13
144.36 yd
73.
7.25
4.30
9.75
+ 5.90
27.20 mi
74.
7.75
8.50
8.50
+ 7.75
32.50 m
30.9
− 10.1
20.8 cm
P=
10.90
11.20
+ 15.03
37.13 yd
P=
27.3
6.7
8.9
15.4
18.4
+ 22.1
98.8 in.
70. x: 61.8
− 8.4
53.4 cm
y:
4.875
− 1.2
3.675
− 1.6
2.075 ft
75. 6 + 0.5 + 0.5 = 7 mm
130
Section 4.2
76. 1.65 − (0.15 + 0.15) = 1.65 − 0.30
= 1.35 cm
80. 128.57 – 125.53 = 3.04
Between February and March, IBM
increased the most, by $3.04 per share.
77. 132.45 – 130.46 = 1.99
IBM decreased by $1.99 per share.
81. 83.45 – 80.67 = 2.78
Between January and February, FedEx
decreased the most, by $2.78 per share.
78. 90.01 – 83.45 = 6.56
FedEx increased by $6.56 per share.
82. 132.45 – 125.53 = 6.92
Between January and February, IBM
decreased the most, by $6.92 per share.
79. 92.17 – 85.81 = 6.36
Between March and April, FedEx
increased the most, by $6.36 per share.
Section 4.3
Addition and Subtraction of Decimals
Multiplication of Decimals
Section 4.3 Practice Exercises
1. front
12.
0.9
×30
27.0
13.
60
× 0.003
0.180
14.
40
× 0.005
0.200
15.
22.38
× 0.8
17.904
16.
31.67
× 0.4
12.668
17.
14
× 0.002
0.028
18.
0.25
× 40
10.00
2. <
3
3. 10 = 1000
3
4. 0.1 = 0.001
2
5. 0.1 = 0.01
6. 102 = 100
7.
0.8
× 0.5
0.40
8.
0.6
× 0.5
0.30
9. 0.9
×4
3.6
10. 0.2
×9
1.8
11.
19. 100
0.4
×20
8.0
20. 500
21. 30
131
Chapter 4
Decimals
22. 50
23. 0.07
24. 0.08
25. 0.2
26. 0.3
27. Exact
8.3
× 4.5
4 15
33 20
37.35
Estimate
8
×5
40
28. Exact
4.3
× 9.2
86
38 70
39.56
Estimate
4
×9
36
29. Exact
0.58
× 7.2
116
4 060
4.176
Estimate
0.6
×7
4.2
30. Exact
0.83
× 6.5
415
4 980
5.395
Estimate
0.8
×7
5.6
31. Exact
5.92
× 0.8
4.736
Estimate
6
× 0.8
4.8
32. Exact
9.14
× 0.6
5.484
Estimate
9
× 0.6
5.4
33. Exact
0.413
× 7
2.891
Estimate
0.4
×7
2.8
34. Exact
0.321
× 6
1.926
Estimate
0.3
×6
1.8
35. Exact
35.9
× 3.2
718
107 70
114.88
Estimate
40
× 3
120
36. Exact
41.7
× 6.1
417
250 20
254.37
Estimate
40
× 6
240
37. Exact
562
× 0.004
2.248
Estimate
600
× 0.004
2.400
38. Exact
984
× 0.009
8.856
Estimate
1000
× 0.009
9.000
39. Exact
0.0004
× 3.6
24
120
0.00144
Estimate
0.0004
×
4
0.0016
40. Exact
0.0008
× 6.5
40
480
0.00520
Estimate
0.0008
×
7
0.0056
41.
0.3
× 0.3
0.09
(0.3) 2 = 0.09, which is not equal to 0.9.
42.
0.8
× 0.8
0.64
(0.8) 2 = 0.64, which is not equal to 6.4.
132
Section 4.3
43.
0.06
× 0.06
0.036
51. (0.1)3 = (0.1)(0.1)(0.1) = 0.001
52.
2
(0.6) = 0.036
44.
45.
46.
47.
0.16
× 0.16
96
+ 160
0.0256
(0.16)2 = 0.0256
53.
2.5
× 2.5
125
+ 1250
6.25
(2.5) 2 = 6.25
50.
0.2
× 0.2
0.04
× 0.2
0.008
× 0.2
0.0016
(0.2) 4 = 0.0016
54.
1.1
× 1.1
11
+ 110
1.21
(1.1) 2 = 1.21
0.3
× 0.3
0.09
× 0.3
0.027
55. The decimal point will move to the right 2
places.
0.4
× 0.4
0.16
56. The decimal point will move to the right 4
places.
57. (a) 5.1 × 10 = 51
(b) 5.1 × 100 = 510
(c) 5.1 × 1000 = 5100
0.7
× 0.7
0.49
(d) 5.1 × 10,000 = 51,000
(0.7) 2 = 0.49
49.
0.2
× 0.2
0.04
× 0.2
0.008
(0.2)3 = 0.008
(0.4) 2 = 0.16
48.
Multiplication of Decimals
58. The decimal point will move to the left
3 places.
1.3
× 1.3
39
1 30
1.69
(1.3) 2 = 1.69
59. The decimal point will move to the left
1 place.
60. (a) 5.1 × 0.1 = 0.51
(b) 5.1 × 0.01 = 0.051
(c) 5.1 × 0.001 = 0.0051
2.4
× 2.4
96
4 80
5.76
(d) 5.1 × 0.0001 = 0.00051
61. 34.9 × 100 = 3490
62. 2.163 × 100 = 216.3
(2.4) 2 = 5.76
133
Chapter 4
Decimals
63. 96.59 × 1000 = 96,590
86. 4 thousand = 4 × 1000 = 4000
64. 18.22 × 1000 = 18,220
87. 20.549 billion = 20.549 ×1,000,000,000
= $20,549,000,000
65. 93.3 × 0.01 = 0.933
66. 80.2 × 0.01 = 0.802
88. 4.8 billion = 4.8 × 1,000,000,000
= 4,800,000,000 gal
67. 54.03 × 0.001 = 0.05403
89. (a)
68. 23.11 × 0.001 = 0.02311
69. 2.001 × 10 = 20.01
(b)
70. 5.932 × 10 = 59.32
71. 0.5 × 0.0001 = 0.00005
6.3
×32
12 6
189 0
201.6 lb
32
× 20
640 lb of CO 2
73. $3.24 = 324¢
90. $2.27 × 10 = $22.70
+ $1.59
$24.29
74. $21.56 = 2156¢
91.
72. 0.8 × 0.0001 = 0.00008
75. $0.37 = 37¢
12.95
× 20
259.00
5.95
× 10
59.50
1.29
× 60
77.40
3.95
× 10
39.50
4.99
× 20
99.80
259.00
59.50
77.40
+ 27.71
$423.61
76. $0.75 = 75¢
77. 347¢ = $3.47
78. 512¢ = $5.12
79. 2041¢ = $20.41
92.
80. 5712¢ = $57.12
81. (a) $1.499 ≈ $1
8.69
× 40
347.60
347.60
39.50
99.80
+ 29.21
$516.11
(b) $1.499 ≈ $1.50
82. (a) $20.599 ≈ $21
(b) $20.599 ≈ $20.60
93.
83. 2.6 million = 2.6 × 1,000,000
= 2,600,000
84. 34.7 million = 34.7 × 1,000,000
= 34,700,000 gal
85. 400 thousand = 400 × 1000 = 400,000
134
70.20
4
×
$280.80 for 4 General Tires
280.80
− 231.99
$48.81 can be saved.
Section 4.3
94. 24.99
36
× 4
×2
99.96
72
99.96
− 72.00
$27.96 can be saved.
98. A = l ⋅ w
A = (56)(31.5)
31.5
× 56
189 0
1575 0
1764.0
95. A = l ⋅ w
A = (0.023)(0.05)
= 1764 cm2
99. (a)
0.023
× 0.05
0.00115
0.3
× 0.3
0.09
(0.3) 2 = 0.09
2
= 0.00115 km
(b)
96. A = l ⋅ w
A = (6.7)(4.5)
6.7
× 4.5
3 35
26 80
30.15
= 30.15 yd 2
100. (a)
0.09 = 0.3
0.5
× 0.5
0.25
(0.5) 2 = 0.25
(b)
97. A = l ⋅ w
A = (15)(22.2)
22.2
× 15
111 0
222 0
333.0
= 333 yd 2
Section 4.4
0.25 = 0.5
101.
0.01 = (0.1)2 = 0.1
102.
0.04 = (0.2)2 = 0.2
103.
0.36 = (0.6)2 = 0.6
104.
0.49 = (0.7)2 = 0.7
Division of Decimals
Section 4.4 Practice Exercises
1. (a) repeating
5.
(b) terminating
2. 35.00
3. 5.28 × 1000 = 5280
4.
Multiplication of Decimals
11.8
× 0.32
236
3 540
3.776
6. 102.400
+ 1.239
103.639
8.003
− 2.200
5.803
135
Chapter 4
7.
8.
Decimals
16.82
− 14.80
2.02
21.1
15. 5 105.5
−10
05
−5
0 5
−5
0
5.28
− 0.001
0.00528
0.9
9. 9 8.1
−8 1
0
0.9
×9
8.1
0.8
10. 6 4.8
−4 8
0
0.8
×6
4.8
0.18
11. 6 1.08
−6
48
−48
0
0.52
12. 4 2.08
−2 0
08
−8
0
0.53
13. 8 4.24
−40
24
−24
0
0.23
14. 25 5.75
−5 0
75
−75
0
16. 7
31.6
221.2
−21
11
−7
42
−4 2
0
1.96
17. 5 9.80
−5
48
−4 5
30
−30
0
0.18
× 6
1.08
0.69
18. 3 2.07
−1 8
27
−27
0
0.52
× 4
2.08
0.53
× 8
4.24
0.035
19. 8 0.280
−24
40
−40
0
0.23
× 25
1 15
4 60
5.75
0.0675
20. 8 0.5400
−48
60
−56
40
−40
0
136
21.1
× 5
105.5
31.6
× 7
221.2
Section 4.4
16.84
21. 5 84.20
−5
34
−30
42
−4 0
20
−20
0
Division of Decimals
5.33...
27. 3 16.00 5.33... = 5.3
−15
10
−9
10
−9
1
5.77 ...
28. 9 52.00 5.77... = 5.7
−45
70
−63
70
−63
7
44.55
22. 2 89.10
−8
09
−8
11
−1 0
10
−10
0
3.166...
29. 6 19.000 3.166... = 3.16
−18
10
−6
40
−36
40
−36
4
0.12
23. 50 6.00
−50
10
−10
0
3.033...
30. 3 9.100 3.033... = 3.03
−9
0 10
−9
10
−9
1
0.075
24. 80 6.000
−560
400
−400
0
0.16
25. 25 4.00
−25
150
−150
0
2.1515...
31. 33 71.0000 2.1515... = 2.15
−66
50
−33
170
−165
50
−33
170
−165
5
0.2
26. 60 12.0
−120
0
137
Chapter 4
Decimals
3.8181...
32. 11 42.0000 3.8181... = 3.81
−33
90
−8 8
20
−11
90
−88
20
−11
9
40. 0.2 5.51
27.55
2 55.10
−4
15
−14
11
−1 0
10
−10
0
41. 0.3 62.5
33. 5.03 ÷ 0.01 = 503
208.33...
3 625.00 208.33... = 208.3
−6
025
−24
10
−9
10
−9
1
34. 3.2 ÷ 0.001 = 3200
35. 0.992 ÷ 0.1 = 9.92
36. 123.4 ÷ 0.01 = 12,340
37. 1.02 57.12
56
102 5712
−510
612
−612
0
42. 1.05 22.4
21.33...
105 2240.00 21.33... = 21.3
−210
140
−105
35 0
−31 5
3 50
−3 15
35
38. 2.23 95.89
43
223 9589
−892
669
−669
0
39. 0.8 2.38
43. 0.13 6.305
2.975
8 23.800
−16
78
−7 2
60
−56
40
−40
0
48.5
13 630.5
−52
110
−104
65
−6 5
0
138
Section 4.4
Division of Decimals
61. 1.8 = 1.8888...
(a) 1.9
(b) 1.89
(c) 1.889
44. 0.25 42.9
171.6
25 4290.0
−25
179
−175
40
−25
15 0
−15 0
0
62. 4.7 = 4.7777...
(a) 4.8
(b) 4.78
(c) 4.778
63. 3.62 = 3.6262...
(a) 3.6
(b) 3.63
(c) 3.626
45. 1.1 ÷ 0.001 = 1100
46. 4.44 ÷ 0.01 = 444
47. 420.6 ÷ 0.01 = 42,060
64. 9.38 = 9.3838...
(a) 9.4
(b) 9.38
(c) 9.384
48. 0.31 ÷ 0.1 = 3.1
49. The decimal point will move to the left 2
places.
0.257
65. 7 1.800
−1 4
40
−35
50
−49
1
0.257 ≈ 0.26
50. The decimal point will move to the left 5
places.
51. 3.923 ÷ 100 = 0.03923
52. 5.32 ÷ 100 = 0.0532
53. 98.02 ÷ 10 = 9.802
54. 11.033 ÷ 10 = 1.1033
0.26
× 7
1.82
66. 2.1 75.3
55. 0.027 ÷ 100 = 0.00027
35.857
21 753.000
−63
123
−105
18.0
−16 8
1 20
−1 05
150
−147
3
35.857 ≈ 35.86
56. 0.665 ÷ 100 = 0.00665
57. 1.02 ÷ 1000 = 0.00102
58. 8.1 ÷ 1000 = 0.0081
59. 2.4 = 2.4444...
(a) 2.4
(b) 2.44
(c) 2.444
60. 5.2 = 5.2222...
(a) 5.2
(b) 5.22
(c) 5.222
139
35.86
2.1
3 586
71 720
75.306
×
Chapter 4
Decimals
67. 3.7 54.9
14.83
37 549.00
−37
179
−148
31 0
−29 6
1 40
−1 11
29
14.83 ≈ 14.8
4.49
68. 21 94.30
−84
10 3
−8 4
1 90
−1 89
1
4.49 ≈ 4.5
70.
2.46 27.88
11.3333
246 2788.0000
−246
328
−246
82 0
−73 8
8 20
−7 38
820
−738
820
−738
82
14.8
× 3.7
1036
44 40
54.76
4.5
× 21
45
90 0
94.5
11.333
2.46
67998
4 53320
22 66600
27.87918
×
11.3333 ≈ 11.333
71. 0.9 32.1
69. 0.24 4.96
20.6666
24 496.0000
−48
16 0
−14 4
1 60
−1 44
160
−144
160
−144
16
35.666
9 321.000
−27
51
−45
60
−5 4
60
−54
60
−54
6
20.667
× 0.24
82668
4 13340
4.96008
35.666 ≈ 35.67
20.6666 ≈ 20.667
140
35.67
× 0.9
32.103
Section 4.4
72. 0.6 81.4
135.666
6 814.000
−6
21
−18
34
−30
40
−36
40
−36
40
−36
4
78. 965.24 40,540.08
42
96524 4054008
−386096
193048
−193048
0
Brooke needs to pay for 42 months which
equals 3.5 years.
79. $560 − $50 = $510
135.67
× 0.6
81.402
42.5
12 510.0
−48
30
−24
60
−6 0
0
The monthly payment is $42.50.
135.666 ≈ 135.67
73. 2.13 237.1
111.31
213 23710.00
−213
241
−213
280
−213
670
−639
310
−213
97
Division of Decimals
×
111.3
2.13
3339
11130
222600
237.069
64.6
80. 650 42,000.0
−39 00
3 000
−2 600
400 0
−390 0
10 0
On average, 65 balls per match are used.
12.5
81. (a) 800 10,000.0
−8 00
2 000
−1 600
400 0
−400 0
0
13 bulbs would be needed (rounded
up to the nearest whole unit).
(b) 0.75
× 13
225
750
$9.75
(c) The energy efficient fluorescent bulb
would be more cost effective.
111.31 ≈ 111.3
74. Unreasonable; 32 miles per gallon
75. Unreasonable; $960
76. Unreasonable; $340.00
77. Unreasonable; $140,000
141
Chapter 4
Decimals
33.3
82. (a) 24 800.0
−72
80
−72
80
−7 2
8
After 33 days.
416.6
(b) 24 10,000.0
−96
40
−24
160
−144
160
−144
16
85. 5.5 12
2.18
55 120.00
−11 0
100
−55
45 0
−440
10
2.2 mph
86. 2.5 33.2
13.28
25 332.00
−25
82
−7 5
70
−50
200
−200
0
After 417 days (or 1 year, 52 days)
0.3420
8399
2873.0000
83.
−2519 7
353 30
−335 96
17 340
−16 798
5420
Babe Ruth’s batting average was 0.342.
13.3 mph
87. 47.265
88. 22.45
0.3663
84. 11, 434 4189.0000
−3430 2
758 80
−68604
72 760
−68 604
4 1560
−3 4302
7258
Ty Cobb’s batting average was 0.366.
142
89.
8.6
× 12.4
3 44
17 20
86 00
106.64
b, d
90.
5.3
× 15.8
4 24
26 50
53 00
83.74
a, b
91.
(2749.13)(418.2) = 1,149, 686.166
92.
(139.241)(24.5) = 3411.4045
93.
(43.75 )2 = 1914.0625
94.
(9.3)5 = 69, 568.83693
Section 4.4
Division of Decimals
100. 42, 475,000 ÷ 155,959
= 272 people per square mile
95. 21.5 2056.75 = 95.6627907
96. 14.2 4167.8 = 293.5070423
101. (a) 584,000,000 ÷ 365
= 1,600,000 mi per day
97. Answers will vary.
(b) 1, 600, 000 ÷ 24 = 66, 666.6 mph
98. Answers will vary.
9410
≈ 0.37
25369
(b) Yes, the claim is accurate. The
decimal, 0.37 is more than 0.3 , which
is equal to 13 .
102. When we say that 1 year is 365 days, we
are ignoring the 0.256 day each year. In 4
years, that amount is 4 × 0.256 = 1.024 ,
which is another whole day. This is why
we add one more day to the calendar every
4 years.
99. (a)
Problem Recognition Exercises: Operations on Decimals
1. (a) 123.04 + 100 = 223.04
4. (a)
(b) 123.04 ×100 = 12, 304
(c) 123.04 − 100 = 23.04
(b)
(d) 123.04 ÷ 100 = 1.2304
(e) 123.04 + 0.01 = 123.05
(f) 123.04 × 0.01 = 1.2304
632.4600
+ 98.0034
730.4634
98.0034
+ 632.4600
730.4634
32.9
× 1.6
1974
3290
52.64
1.6
(b)
× 32.9
144
320
4800
52.64
5. (a)
(g) 123.04 ÷ 0.01 = 12, 304
(h) 123.04 − 0.01 = 123.03
2. (a) 5078.3 + 1000 = 6078.3
(b) 5078.3 ×1000 = 5, 078, 300
(c) 5078.3 − 1000 = 4, 078.3
(d) 5078.3 ÷ 1000 = 5.0783
(e) 5078.3 + 0.001 = 5078.301
6. (a)
(f) 5078.3 × 0.001 = 5.0783
(g) 5078.3 ÷ 0.001 = 5, 078, 300
(h) 5078.3 − 0.001 = 5078.299
(b)
3. (a)
4.800
+ 2.391
7.191
(b)
2.391
+ 4.800
7.191
143
74.23
× 0.8
59.384
0.8
× 74.23
24
160
3200
56000
59.384
Chapter 4
Decimals
7. (a) 21.6
×4
86.4
(b)
8. (a)
(b)
12.
21.6
× 0.25
1080
4320
5.400
13. 0.07 280
4000
7 28000
−28
0000
−000
0
92.5
×2
185.0
14. 6400 ÷ 0.001 = 6, 400, 000
92.5
× 0.5
46.25
200,000
15. 490 98,000,000
−98 0
000000
−00000
0
9. (a) 5.6 448
80
56 4480
−448
00
−0
0
(b)
2700
16. 2000 5, 400,000
−4 000
1 400 0
−1 400 0
000
−000
0
5.6
× 80
448.0
10. (a) 9.2 496.8
54
92 4968
−460
368
−368
0
(b)
20
× 0.05
1.00
54
× 9.2
108
4860
496.8
11. 0.125
×8
1.000
144
17.
4500
× 300, 000
1, 350, 000, 000
18.
340
× 5000
1, 700, 000
19.
83.4000
− 78.9999
4.4001
20.
124.7000
− 47.9999
76.7001
Section 4.5
Section 4.5
Fractions as Decimals
Fractions as Decimals
Section 4.5 Practice Exercises
1.
9
= 0.9
10
17.
316 316 ⋅ 2 632
=
=
;0.632
500 500 ⋅ 2 1000
2.
39
= 0.39
100
18.
19
19 ⋅ 2
38
=
=
;0.038
500 500 ⋅ 2 1000
3.
141
= 0.141
1000
4.
71
= 0.0071
10,000
5. 0.6 =
0.875
19. 8 7.000
−6 4
60
−56
40
−40
0
6 3
=
10 5
6. 0.0016 =
7. 0.35 =
16
1
=
10,000 625
0.25
20. 64 16.00
−12 8
3 20
−3 20
0
35
7
=
100 20
8. 0.125 =
125 1
=
1000 8
3.2
21. 5 16.0
−15
10
−1 0
0
9. 4.25 = 4.2525 ≈ 4.25
10. 0.37 = 0.3737L ≈ 0.374
11.
2 2⋅2 4
=
= = 0.4
5 5 ⋅ 2 10
12.
4 4⋅2 8
=
= ;0.8
5 5 ⋅ 2 10
13.
49 49 ⋅ 2 98
=
=
;0.98
50 50 ⋅ 2 100
14.
3
3⋅ 2
6
=
=
;0.06
50 50 ⋅ 2 100
15.
7
7⋅4
28
=
=
;0.28
25 25 ⋅ 4 100
16.
4
4⋅4
16
=
=
;0.16
25 25 ⋅ 4 100
2.72
22. 25 68.00
−50
180
−175
50
−50
0
0.25
23. 12 3.00
−2 4
60
−60
0
145
7
= 0.875
8
16
= 0.25
64
16
= 3.2
5
68
= 2.72
25
5
3
= 5 + 0.25 = 5.25
12
Chapter 4
Decimals
0.0625
24. 16 1.0000
−96
40
−32
80
−80
0
0.2
25. 5 1.0
−1 0
0
0.625
26. 8 5.000
−4 8
20
−16
40
−40
0
0.75
27. 24 18.00
−16 8
1 20
−1 20
0
0.6
28. 40 24.0
−24 0
0
3.3125
29. 16 53.0000
−48
50
−4 8
20
−16
40
−32
80
−80
0
4
1.875
30. 56 105.000
−56
49 0
−44 8
4 20
−3 92
280
−280
0
1
= 4 + 0.0625 = 4.0625
16
1
1 = 1 + 0.2 = 1.2
5
31. 20
5
6 = 6 + 0.625 = 6.625
8
0.45
9.00
−8 0
1 00
−1 00
0
0.44
32. 25 11.00
−10 0
1 00
−1 00
0
18
= 0.75
24
0.88
25
22.00
33.
−20 0
200
−200
0
24
= 0.6
40
0.55
34. 20 11.00
−10 0
1 00
−1 00
0
53
= 3.3125
16
0.88...
35. 9 8.00
−7 2
80
−72
8
146
105
= 1.875
56
7
9
= 7 + 0.45 = 7.45
20
3
11
= 3 + 0.44 = 3.44
25
22
= 0.88
25
11
= 0.55
20
8
3 = 3.8
9
Section 4.5
0.77...
36. 9 7.00
−6 3
70
−63
7
0.466...
37. 15 7.000
−60
1 00
−90
100
−90
10
0.277
38. 18 5.000
−3 6
1 40
−1 26
140
−126
14
7
4 = 4.7
9
7
= 0.46
15
5
= 0.27
18
0.5277 ...
39. 36 19.0000
−18 0
1 00
−72
280
−252
280
−252
28
19
= 0.527
36
0.5833...
40. 12 7.0000
−6 0
1 00
−96
40
−36
40
−36
4
7
= 0.583
12
Fractions as Decimals
0.5454...
41. 11 6.0000
−5 5
50
−44
60
−55
50
−44
6
6
= 0.54
11
0.2424...
42. 33 8.0000
−6 6
1 40
−1 32
80
−66
140
−132
8
8
= 0.24
33
0.126126 ...
43. 111 14.000000
−11 1
2 90
−2 22
680
−666
140
−111
290
−222
680
−666
14
147
14
= 0.126
111
Chapter 4
Decimals
0.522522 ...
44. 111 58.000000
−55 5
2 50
−2 22
280
−222
580
−555
250
−222
280
−222
58
1.13636 ...
45. 22 25.00000
−22
30
−2 2
80
−66
140
−132
80
−66
140
−132
8
2.04545...
46. 22 45.00000
−44
10
−0
100
−88
120
−110
100
−88
120
−110
10
0.1428
47. 7 1.0000
−7
30
−28
20
−14
60
−56
4
58
= 0.522
111
0.2857
48. 7 2.0000
−1 4
60
−56
40
−35
50
−49
1
25
= 1.136
22
0.076
49. 13 1.000
−91
90
−78
12
0.692
50. 13 9.000
−7 8
1 20
−1 17
30
−26
4
45
= 2.045
22
0.93
51. 16 15.00
−14 4
60
−48
12
148
1
= 0.143
7
2
= 0.286
7
1
= 0.08
13
9
= 0.69
13
15
≈ 0.9
16
Section 4.5
0.27
52. 11 3.00
−2 2
80
−77
3
3
≈ 0.3
11
(c)
4
= 0.4
9
(d)
5
= 0.5
9
Fractions as Decimals
If we memorize that
0.714
53. 7 5.000
−4 9
10
−7
30
−28
2
0.125
54. 8 1.000
−8
20
−16
40
−40
0
1.19
55. 21 25.00
−21
40
−2 1
1 90
−1 89
1
1.38
56. 13 18.00
−13
50
−3 9
1 10
−1 04
6
57. (a)
1
= 0. 1
9
(b)
2
= 0.2
9
5
≈ 0.71
7
1
= 0. 1, then
9
2
1
= 2 ⋅ = 2 ⋅ 0. 1 = 0.2, and so on.
9
9
1
≈ 0.13
8
58. (a)
1
= 0.3
3
(b)
2
= 0.6
3
If we memorize that
1
= 0.3, then
3
2
1
= 2 ⋅ = 2 ⋅ 0.3 = 0.6.
3
3
59. (a) 0.45 =
45
9
=
100 20
5 13
(b) 1 =
8 8
25
≈ 1.2
21
18
≈ 1.4
13
(c) 0.7 =
= 1.625
7
9
0.4545...
(d) 11 5.0000
−4 4
60
−55
50
−44
60
−55
5
149
1.625
8 13.000
−8
50
−4 8
20
−16
40
−40
0
5
= 0.45
11
Chapter 4
60. (a)
Decimals
2
= 0.6
3
(b) 1.6 =
(c)
152 152 ⋅ 4 608
=
=
= 6.08
25
25 ⋅ 4 100
2
9
61. (a) 0.3 =
1
3
(b) 2.125 =
(c) 22
2125 17
1
=
or 2
1000 8
8
0.86363...
19.00000
−17 6
1 40
−1 32
80
−66
140
−132
80
−66
14
3
10
18
72
Stride Rite: 15 = 15
= 15.72
25
100
10
1
Intel: 28.10 = 28
= 28
100
10
3
15
Burger King: 24
= 24
= 24.15
20
100
64. Dell: 26.3 = 26
19
= 0.863
22
75 3
=
100 4
0.6363...
(b) 11 7.0000
−6 6
40
−33
70
−66
40
−33
7
8
17
(c) 1.8 = 1 or
9
9
65. 0.2 =
1
5
66. 1.5 =
3
2
67. 0.2 < 0.2
42 42 ⋅ 4 168
=
=
= 1.68
25 25 ⋅ 4 100
62. (a) 0.75 =
74 74 ⋅ 4 296
=
=
= 2.96
25 25 ⋅ 4 100
63. McGraw-Hill:
6925 277
1
69.25 =
=
or 69
100
4
4
4495 899
19
Walgreen: 44.95 =
=
or 44
100
20
20
1
Home Depot: 38 = 38.50
2
General Electric:
11
11⋅ 4
44
37 = 37
= 37 +
= 37.44
25
25⋅ 4
100
16 8
3
= or 1
10 5
5
(d) 0.2 =
(d)
(d)
7
= 0.63
11
68.
3
= 0.6 < 0.6
5
69.
1
= 0.3 > 0.3
3
70.
2
= 0.6 > 0.66
3
1
71. 4 = 4.25 < 4.25
4
72. 2.12 < 2.12
73. 0.5 =
150
5
9
Section 4.5
74.
7
3
= 1 = 1.75
4
4
75. 0.27 < 0.27 =
Fractions as Decimals
79.
3
11
80.
76. 6.43 > 6.43
81. 0.9 =
77.
9
=1
9
82. 1.9 = 1 + 0.9 = 1 + 1 = 2
83. 6.9 = 6 + 0.9 = 6 + 1 = 7
78.
84. 15.9 = 15 + 0.9 = 15 + 1 = 16
Section 4.6
Order of Operations and Applications of Decimals
Section 4.6 Practice Exercises
1. (a)
85 5 17
÷ =
100 5 20
(b) 5
6. 1.2 56.7
47.25
12 567.00
−48
87
−84
3.0
−2 4
60
−60
0
4.6
23
−20
30
−30
0
2
5
1
3
 24   35  10
2. 
=

 7   36  3
3.
4.
5.
5
1
55 11 55 4 5
7.
÷ =
⋅
=
16 4 16 11 4
34.1
× 9.2
6 82
306 90
313.72
4
8.
9.
790.90
+ 23.91
814.81
34 5 34 ⋅ 3 5 102 5 107
+
=
+
=
+
=
9 27 9 ⋅ 3 27 27 27 27
1
9 7 9 ⋅ 2 7 18 7 11
− =
− = − =
4 8 4⋅2 8 8 8 8
13.00
− 6.04
6.96
10. 1. Perform all operations inside
parentheses first.
151
Chapter 4
Decimals
2.5
× 2.5
125
500
6.25
2. Simplify expressions containing
exponents.
3. Perform multiplication or division in
the order that they appear from left to
right.
= 16.25 − 6.25
4. Perform addition or subtraction in the
order that they appear from left to
right.
= 10
(
14. 11.38 − 10.42 − 7.52
11. (3.7 − 1.2) 2
3.7
− 1.2
2.5
= 11.38 − 8.41
= 6.25
12. (6.8 − 4.7) 2
= 2.97
6.8
−4.7
2.1
= (2.1) 2
2.1
× 2.1
21
4 20
4.41
= 4.41
)2
10.42
−7.52
2.90
2.9
× 2.9
261
580
8.41
2.5
× 2.5
1 25
5 00
6.25
(
)2
= (2.9)2
= (2.5) 2
13. 16.25 − 18.2 − 15.7
16.25
− 6.25
10.00
18.2
−15.7
2.5
= (2.5) 2
152
11.38
− 8.41
2.97
Section 4.6
Order of Operations and Applications of Decimals
19. 6.8 ÷ 2 ÷ 1.7
2
15. 12.46 − 3.05 − 0.8
0.8
× 0.8
0.64
= 12.46 − 3.05 − 0.64
12.46
− 3.05
9.41
= 9.41 − 0.64
9.41
− 0.64
8.77
= 8.77
3.4
2 6.8
−6
08
−8
0
= 3.4 ÷ 1.7
1.7 3.4
2
17 34
−34
0
=2
2
16. 15.06 − 1.92 − 0.4
0.4
× 0.4
0.16
= 15.06 − 1.92 − 0.16
15.06
− 1.92
13.14
= 13.14 − 0.16
13.14
− 0.16
12.98
= 12.98
17. 63.75 − 9.5(4)
20. 8.4 ÷ 2 ÷ 2.1
2
4.2
8.4
−8
04
−4
0
= 4.2 ÷ 2.1
2.1 4.2
2
21 42
−42
0
9.5
× 4
38.0
=2
21. 2.2 + [9.34 + (1.2) 2 ]
= 63.75 − 38
1.2
× 1.2
24
1 20
1.44
63.75
− 38.00
25.75
= 25.75
= 2.2 + (9.34 + 1.44)
= 2.2 + 9.34 + 1.44
18. 6.84 + (3.6)(9)
3.6
× 9
32.4
2.20
9.34
+ 1.44
12.98
= 6.84 + 32.4
6.84
+ 32.40
39.24
= 12.98
= 39.24
153
Chapter 4
Decimals
24. 6[(3.1)(4) − 8.1]
22. (3.1) 2 − (4.2 ÷ 2.1)
2.1 4.2
(3.1)(4)
2
21 42
−42
0
3.1
× 4
12.4
= 6[12.4 − 8.1]
= (3.1) 2 − 2
0 12
3.1
× 3.1
31
930
9.61
1 2.4
− 8.1
4.3
= 6(4.3)
= 9.61 − 2
1
4.3
× 6
25.8
= 7.61
23. 16.04 ÷ [(2.2) 2 − 0.83)]
= 25.8
2.2
× 2.2
44
440
4.84
25. 42.82 − 3(4.8 − 1.6) 2
4.8
− 1.6
3.2
= 16.04 ÷ [4.84 − 0.83)]
= 42.82 − 3(3.2) 2
4.84
− 0.83
4.01
3.2
× 3.2
64
960
10.24
= 16.04 ÷ 4.01
4.01 16.04
= 42.82 − 3(10.24)
4
401 1604
−1604
0
1
10.24
× 3
30.72
=4
= 42.82 − 30.72
42.82
− 30.72
12.10
= 12.1
154
Section 4.6
Order of Operations and Applications of Decimals
26. 14.28 ÷ [(1.1) 2 + 5.79]
29. 20.04 ÷
1.1
× 1.1
11
110
1.21
4
= 20.04 ÷ 0.8
5
0.8 20.04
25.05
8 200.40
−16
40
−40
0 40
−40
0
= 14.28 ÷ [1.21 + 5.79]
11
1.21
+ 5.79
7.00
= 25.05
30. (78.2 − 60.2) ÷
= 14.28 ÷ 7
7
9
13
78.2
− 60.2
18.0
2.04
14.28
−14
2
−0
28
−28
0
= 18 ÷
9
13
2
18 13
=
⋅
1 9
1
= 2.04
= 26
7 1
 14 1 
31. 14.4 ×  −  = 14.4 ×  − 
 4 8
 8 8
 13 
= 14.4 ×  
8
4
3
1
27. 89.8 ÷ 1 = 89.8 ÷ = 89.8 ×
3
4
3
= 89.8 × 0.75
89.8
× 0.75
4490
62 860
67.350
= 67.35
18
144 13
=
×
10
8
1
234
10
= 23.4
=
3
8
5
28. 30.12 ÷ 1 = 30.12 ÷ = 30.12 ×
5
5
8
= 30.12 × 0.625
30.12
× 0.625
15060
60240
18 07200
18.82500
= 18.825
2
1  1
1 1
1
32. 6.5 + ×   = 6.5 + ×
= 6.5 +
8  5
8 25
200
= 6.5 + 0.005 = 6.505
155
Chapter 4
Decimals
52
1
5 23 5 23
33. 2.3 × =
× =
9 10 9 18
36. 1 × 6.24 ÷ 2.1 = 1 × 624 ÷ 21
12
12 100 10
2
1.277 ...
18 23.000
−18
50
−3 6
1 40
−1 26
140
−126
14
34. 4.6 ×
35. 6.5 ÷
1
1
52
10
52
26
=
⋅
=
=
100 21 210 105
23
≈ 1.28
18
10
0.247
105 26.000
−21 0
5 00
−4 20
800
−735
65
7 46 7 322 161
=
× =
=
30
6 10 6 60
5.366...
161
30 161.000
≈ 5.37
30
−150
11 0
−9 0
200
−180
200
−180
20
37. (42.81 − 30.01) ÷
26
≈ 0.25
105
9
2
42.81
− 30.01
12.80
1
9 128 2 128
⋅ =
= 12.8 ÷ =
2 10 9 45
5
2.844...
45 128.000
−90
38 0
−36 0
2 00
−1 80
200
−180
20
3
1 3 13 5 65
=6 ÷ = ⋅ =
5
2 5 2 3 6
10.833...
65
6 65.000
≈ 10.83
6
−6
05 0
−4 8
20
−18
20
−18
2
128
≈ 2.84
45
1
2
1 2 51 1
51
38.
× 5.1 × = ×
× =
7
10 7 10 10 350
5
0.145
350 51.000
−35 0
16 00
−14 00
2 000
−1 750
250
156
51
≈ 0.15
350
Section 4.6
Order of Operations and Applications of Decimals
1
4.433...
60 266.000
−240
260
−240
200
−1 80
200
−180
20
2
2 421 421
39.
× 4.21 = ×
=
9
9 100 450
50
0.9355...
450 421.0000
−405 0
16 00
−13 50
2 500
−2 250
2500
−2250
250
421
= 0.935
450
17
34
34
1 34 5 85
42.
× 2.5 =
×2 =
× =
11
11
2 11 2 11
1
301
7.7272 ...
11 85.0000
−77
80
−7 7
30
−22
80
−77
30
−22
8
22 602 23 6923
40. 6.02 ÷
=
=
×
23 100 22 1100
11
6.293636 ...
1100 6923.000000
−6600
323 0
−220 0
103 00
−99 00
4 000
−3 300
7000
−6600
4000
−3300
7000
−6600
400
266
1
20
3
266
= 4.43
60
6923
= 6.2936
1100
43. (a)
85
= 7.72
11
21,816.6
21,345.6
471.0
They drove 471 mi.
−
1
15 471 2 942
= 471 ÷ =
× =
2
2
1 15 15
62.8
15 942.0
−90
42
−30
12 0
−12 0
0
The average speed is 62.8 mph.
(b) 471 ÷ 7
6 532 5 266
41. 5.32 ÷ =
× =
5 100 6
60
157
Chapter 4
Decimals
44. (a)
43,984.8
43, 725.1
259.7
−
48.
She drove 259.7 miles.
(b) 9.8 259.7
98
26.5
2597
−196
637
−588
490
−490
0
49.
597
− 450
147 minutes over the included amount
147
× 0.40
58.80
20.00
− 15.23
4.77 change
Caren should get $4.77 in change.
39.95
+ 58.80
98.75
50.
Jorge will be charged $98.75.
46.
4.79
× 3
14.37 cost for 3 packages
14.37
+ 0.86
15.23 total cost
She got 26.5 miles per gallon.
45.
1
× 150,000.00 = 75,000.00
2
3
× 150,000.000 = 56, 250.00
8
75, 000.00
+ 56, 250.00
$131, 250.00 to daughter and stepson
150,000.00
− 131, 250.00
$18,750.00
The grandson will receive $18,750.00
28.42
+ 6.00
34.42
20 × 2 = 40
40.00
− 34.42
5.58
Mr. Timpson should receive $5.58 in
change.
185.95
+ 24.17
210.12 total nightly charges
210.12
×
5
1,050.6
51.
Radcliff’s bill is $1,050.60.
15
1
1 60
47.
× 60 = ×
= 15 g for breakfast
4
4
1
1
15.0
60.0
+ 20.7
− 35.7
35.7 g so far today
24.3 g left
She has 24.3 g left for dinner.
92
84
77
+ 62
315
78.75
4 315.00
−28
35
−32
30
−2 8
20
−20
0
Duncan’s average is 78.75.
158
Section 4.6
Order of Operations and Applications of Decimals
17.25
4 69.00
−4
29
−28
10
−8
20
−20
0
Owen’s average is 17.25.
58. (a) BMI =
19
52.
14
16
+ 20
69
53.
703(150)
2
(70)
h
105,450
=
≈ 21.5
4900
=
57. (a) BMI =
703(220)
=
(72)2
h2
154,660
=
≈ 29.8
5184
(b) Overweight
=
703(141)
3969
1
1
5 3 10 3 7
− = − =
2 4 4 4 4
3
= 1 = 1.75
4
=
 8 4
61. (0.8 + 0.4) × 0.39 =  +  × 0.39
 9 9
 12 
=   × 0.39
 9
4
4 × 0.39
= × 0.39 =
3
3
0.39
× 4
1.56
=
703(150)
4900
0.52
3 1.56
−1 5
06
−6
0
1.56
3
= 0.52
(b) Ideal
703W
2
5 2 3 5 9 3
60. 0.5 ÷ 0.2 − 0.75 = ÷ − = × −
9 9 4 9 2 4
55. Answers will vary.
=
703(141)
3 3
× + 3.375
9 10
9
+ 3.375
=
90
1
= + 3.375
10
= 0.1+ 3.375 = 3.475
4.45
5.6
54.
4 17.80
5.2
−16
3.3
18
+ 3.7
−1 6
17.8
20
−20
0
The average rainfall per month is 4.45 in.
2
=
59. 0.3 × 0.3+ 3.375 =
14.54
5 72.70
−5
22
−20
27
−25
20
−20
0
The average snowfall per month is 14.54
in.
703W
2
(63)
h
99,123
=
≈ 25.0
3969
(b) Overweight
6.6
18.1
18.8
16.8
+ 12.4
72.7
56. (a) BMI =
703W
62. (0.7 − 0.6) × 5.4 = 0. 1 × 5.4
703(220)
=
5184
6
1 54
6
= ×
=
= 0.6
9 10 10
1
159
Chapter 4
Decimals
237.5
278 66025.0
63. (a) 132.05
−556
× 50
1042
6602.50
−834
2085
−1946
139 0
−139 0
0
Deanna can buy 237 shares.
(b)
3352
× 275
16760
234640
670400
921,800
Approximately 921,800 homes could
be powered.
(d)
287, 409.60
− 120,000.00
167, 409.60
He will pay $167,409.60 in interest.
(b)
12
× 15
180
There are 180 months in 15 years.
(c)
849
× 180
152,820
Gwen will pay $152,820.00.
(d) 152,820.00
− 94,000.00
58,820.00
She will pay $58,820.00 in interest.
1246.1
79.8
× 275
232.5
128.0
62305
90.0
872270
209.0
2492200
120.0
342,677.5
120.0
59.8
80.5
126.5
1246.1
Approximately 342,678 additional
homes could be powered.
67.
65. (a) 145,000
− 25,000
120,000
Marty will have to finance $120,000.
(b)
798.36
×
360
287, 409.6
He will pay $287,409.60.
66. (a) 109,000
− 15,000
94,000
Gwen needs to finance $94,000.
(b) 27.80
× 0.5
13.90
$13.90 will be leftover.
64. (a)
(c)
68.
12
× 30
360
There are 360 months in 30 years.
160
1
(80, 460.60) = 26,820.20
3
80, 460.60
− 26,820.20
53,640.40
53,640.40 ÷ 4 = 13,410.10
Each person will get approximately
$13,410.10.
106.97
90.75
133.25
+ 110.15
441.12 ÷ 4 = 110.28
The average price is $110.28.
Chapter 4
Chapter 4
Review Exercises
19. a, b
Section 4.1
20. b, c
1. 32.16
The 3 is in the tens place, the 2 is in the
ones place, the 1 is in the tenths place, and
the 6 is in the hundredths place.
Section 4.2
2. 2.079
The 2 is in the ones place, the 0 is in the
tenths place, the 7 is in the hundredths
place, and the 9 is in the thousandths
place.
3. Five and seven-tenths
4. Ten and twenty-one hundredths
21.
45.030
+ 4.713
49.743
22.
239.30
+ 33.92
273.22
23.
34.89
− 29.44
5.45
24.
5.002
− 3.100
1.902
25.
221.00
− 23.04
197.96
26.
34.000
+ 4.993
38.993
27.
17.300
+ 3.109
20.409
− 12.600
7.809
28.
189.220
+ 13.100
202.320
− 120.055
82.265
5. Fifty-one and eight thousandths
6. One hundred nine and one-hundredth
7. 33,015.047
8. 100.01
9. 4.8 = 4
8
4
=4
10
5
10. 0.025 =
11. 1.3 =
25
1
=
1000 40
13
10
12. 6.75 =
Review Exercises
675 27
=
100 4
13. 15.032 > 15.03
14. 7.209 < 7.22
15. 4.3875, 4.3953, 4.4839, 4.5000, 4.5142
16. 89.92
29. x:
y: 47.10
53.4
− 48.9
− 42.03
4.5 in.
5.07 in.
P = 47.1 + 53.4 + 42.03 + 4.5 + 5.07 + 48.9
= 201 in.
17. 34.890
18. (a) The amount in the box is less than the
advertised amount.
(b) The amount rounds to 12.5 oz.
161
Chapter 4
30. (a)
(b)
Decimals
38. 92.01 × 0.01 = 0.9201
Day 2 : 12.495
− Day 1 : − 12.301
0.194
Between Days 1 and 2, the increase
was $0.194.
39. 104.22 × 0.01 = 1.0422
40. 28,100,000
41. 432,000
Day 2 : 12.495
− Day 3 : − 12.298
0.197
42. (a) For 8 batteries, buy 4 packages of two.
3.99
× 4
15.96
Eight batteries cost $15.96 on sale.
Day 3 : 12.298
− Day 4 : − 12.089
0.209
Between days 3 and 4, the decrease
was $0.209.
31.
(b)
1.40
1.00
0.09
+ 1.25
3.74
The total rainfall was 3.74 in.
43.
Section 4.3
32.
33.
34.
35.
17.99
− 15.96
2.03
A customer can save $2.03.
23
× 0.25
5.75
The call will cost $5.75.
44. A = l ⋅ w
= (40)(23.5)
3.9
× 2.1
39
7 80
8.19
23.5
× 40
940.0
= 940 ft 2
57.01
× 1.3
17 103
57 010
74.113
P = 2(40) + 2(23.5) = 80 + 47 = 127 ft
45. (a)
60.1
× 4.4
24 04
240 40
264.44
(b)
7.7
× 45
38 5
308 0
346.5
36. 85.49 × 1000 = 85,490
37. 1.0034 × 100 = 100.34
162
36.4
× 200
7280.0
7280 people
92.9
× 200
18,580.0
18,580 people
Chapter 4
Section 4.4
50. 0.7 18.9
46. 0.5 8.55
7
17.1
5 85.5
−5
35
−35
05
−5
0
27
189
−14
49
−49
0
51. 1.2 0.036
0.03
12 0.36
−36
0
47. 1.5 64.2
42.8
15 642.0
−60
42
−30
120
−120
0
52. 493.93 ÷ 100 = 4.9393
53. 90.234 ÷ 10 = 9.0234
54. 553.8 ÷ 0.001 = 553,800
55. 2.6 ÷ 0.01 = 260
56.
48. 0.06 0.248
4.133...
6 24.800
−24
08
−6
20
−18
20
−18
2
8.6
52.52
0.409
Tenths
8.7
52.5
0.4
Hundredths
8.67
52.53
0.41
Thousandths
8.667
52.525 0.409
Tenthousandths
8.6667 52.5253 0.4094
11.622 ...
57. 9 104.600
−9
14
−9
56
−5 4
20
−18
20
−18
2
4.133... = 4.13
49. 0.3 2.63
8.766...
3 26.300
−24
23
−2 1
20
−18
20
−18
2
Review Exercises
8.766... = 8.76
163
11.622... ≈ 11.62
Chapter 4
Decimals
11.966 ...
58. 6 71.800
−6
11
−6
58
−5 4
40
−36
40
−36
4
0.499
59. (a) 12 5.990
−48
119
−108
110
−108
2
0.572
(b) 4 2.290
−2 0
29
−28
10
−8
2
11.966... ≈ 11.97
Section 4.5
7
7⋅5
35
=
=
;0.35
20 20 ⋅ 5 100
62.
27
27 ⋅ 2
54
=
=
;0.054
500 500 ⋅ 2 1000
2
4
63. 2 = 2 = 2.4
5
10
64. 3
7
7 ⋅ 625
4375
=
=
= 0.4375
16 16 ⋅ 625 10,000
0.31818 ...
69. 22 7.00000
−6 6
40
−22
180
−176
40
−22
180
−176
4
(c) The 12-pack is the better buy.
61.
66.
7
= 0.583
12
1.5277 ...
68. 36 55.0000
−36
19 0
−18 0
1 00
−72
280
−252
280
−252
28
$0.57 per roll
3 3⋅ 2 6
=
= ;0.6
5 5 ⋅ 2 10
24
24 ⋅ 8 192
=
=
= 0.192
125 125 ⋅ 8 1000
0.5833...
67. 12 7.0000
−6 0
1 00
−96
40
−36
40
−36
4
$0.50 per roll
60.
65.
13
52
=3
= 3.52
25
100
164
55
= 1.527
36
4
7
= 4 + 0.318 = 4.318
22
Chapter 4
0.153846 ...
70. 13 2.000000
−1 3
70
−65
50
−39
110
−104
60
−52
80
−78
2
0.294
71. 17 5.000
−34
1 60
−1 53
70
−68
2
0.869
72. 23 20.000
−18 4
1 60
−1 38
220
−207
13
3.666
73. 3 11.000
−9
20
−1 8
20
−18
20
−18
2
Review Exercises
2.833
6
17.000
74.
−12
50
−4 8
20
−18
20
−18
2
2
= 0.153846
13
75. 0.2 =
2
9
76. 1.6 = 1 + 0.6 = 1 +
5
≈ 0.29
17
17
≈ 2.83
6
2
2
=1
3
3
77. 3.3 = 3 + 0.3 = 3 +
1
1
=3
3
3
78. 5.7 = 5 + 0.7 = 5 +
7
7
=5
9
9
1
2
= 13
= 13.02
50
100
50
1
Microsoft: 30.50 = 30
= 30
100
2
37
Citibank: 4.37 = 4
100
79. Ford: 13
20
≈ 0.87
23
1
80. 1 = 1.3 > 1.33
3
1 9
81. 2.25 = 2 =
4 4
11
≈ 3.67
3
82. 0.14 =
14
7
7 1
=
<
=
100 50 49 7
83. 0.28
84. 0.713
Section 4.6
5
1
1
1
3
1 3 15 2
85. 7.5 ÷ = 7 ÷ =
⋅ =5
2
2 2
2 3
165
Chapter 4
Decimals
86. 2(3.14)(20) = (6.28)(20)
89.
6.28
× 20
125.60
= 125.6
87. 3.14(5) 2 = 3.14(25)
3.14
× 25
15 70
62 80
78.50
90.
= 78.5
88.
1
1
(3.14)(2) 2 (6) = (3.14)(4)(6)
3
3
1
= (3.14)(24)
3
1
= (24)(3.14)
3
= 8(3.14)
3.14
× 8
25.12
= 25.12
Chapter 4
8.
(b) Hundredths place
2. Five hundred nine and twenty-four
thousandths
4. 0.4419, 0.4484, 0.4489, 0.4495
5. b is correct.
7.
34.09
− 12.80
21.29
28.1
× 4.5
14 05
112 40
126.45
5.08
9. 5 25.40
−25
0 40
−40
0
26
13 63
3. 1.26 = 1
=1 ;
100
50 50
49.002
+ 3.830
52.832
1
12.6
(51) = 17
3
5.4
14.2
51− 17 = 34
7.3
+ 11.5
51.0
Marvin must drive 34 mi more.
Test
1. (a) Tens place
6.
189.95
199.95
+ 219.95
609.85
− 519.95
89.90
$89.90 will be saved by buying the combo
package.
166
10.
4.00
− 2.78
1.22
11.
12.0300
+ 0.1943
12.2243
Chapter 4
(b) A = (13)(11) − 67.5
13
× 11
13
130
143
= 143 − 67.5
143.0
− 67.5
75.5
12. 0.33 39.82
120.66...
33 3982.00
−33
68
−66
220
−19 8
2 20
−1 98
22
13.
120.66... = 120.6
= 75.5 in.2
(c) A = (18.2)(16.5) − (75.5 + 67.5)
18.2
× 16.5
9 10
109 20
182 00
300.30
= 300.30 − (75.5 + 67.5)
75.5
+ 67.5
143.0
= 300.30 − 143.0
300.30
− 143.00
157.30
42.7
× 10.3
12 81
427 00
439.81
14. 45.92 × 0.1 = 4.592
15. 579.23 × 100 = 57,923
16. 80.12 ÷ 0.01 = 8012
17. 2.931 ÷ 1000 = 0.002931
18. (a) 161.0
− 99.6
61.4°F
(b)
= 157.3 in.2
21. Cost of Purchase:
36.625
× 200
7325.000
77.0
− 75.6
1.4°F
19. (a) 50,500,000 votes
Receipt from Sale:
52.16
× 200
10432.00
(b) 51,000,000 votes
(c) The difference is approximately
500,000 in favor of Al Gore.
Net Profit:
10424.00
− 7329.25
3094.75
He made $3094.75.
20. (a) A = (7.5)(9)
7.5
× 9
67.5
= 67.5 in.2
167
7325.00
+ 4.25
7329.25
10432.00
− 8.00
10424.00
Test
Chapter 4
Decimals
37.499
24
899.990
22. 1099.99
−72
− 200.00
179
899.99
−168
119
−9 6
239
−216
230
−216
14
She will pay approximately $37.50 per
month.
4.8
23. 23 110.4
−92
18 4
−18 4
0
2
2
7
3
7  21 3 
7  18 
27.  5.25 −  =  −  =  
3
4
3  4 4
3 4 
2
2
7  9
7 81
=   = ×
3  2
3 4
27
7  81  189
= 
= 47.25
=
3 4 
4
1
28. (a)
5.2
26 135.2
−130
52
−5 2
0
4.6
5.9
0
8.4
2.5
12.8
+ 4.6
38.8
He ran 38.8 mi.
5.54
(b) 7 38.80
−35
38
−3 5
30
−28
2
The average is 5.5 mi per day.
4.8 + 5.2 = 10.0
He will use 10 gal of gas.
6
24
= 38
= 38.24
25
100
23
38
= 38.23
100
30
3
37.30 = 37
= 37
100
10
21
38.21 = 38
100
24. 38
25.
1

26. (8.7)  1.6 −  = (8.7)(1.6 − 0.5)
2

= (8.7)(1.1)
8.7
× 1.1
87
8 70
9.57
= 9.57
168
Chapters 1–4
Chapters 1–4
Cumulative Review Exercises
1. (17 + 12) − (8 − 3) ⋅3 = (29) − (5) ⋅3
= 29 − 15 = 14
2
 7
 7  7  49
11.   =    =
 10 
 10  10  100
2. 4039
3.
4.
5.
4
3902
34
+ 904
4840
2
13.
4990
− 1118
3872
23, 444
103
70 332
2 344 400
2, 414,732 ≈ 2,415,000
15.
8
8 1 8 2
÷4= ⋅ =
=
3
3 4 12 3
2
30,000
(15,000) =
= 6000
5
5
15,000
− 6 000
9 000
There is $9000 left.
2
8 1 5  2
8 1 5 4
16.
+ ÷ −  =
+ ÷ −
25 5 6  5 
25 5 6 25
8 1 6 4
=
+ ⋅ −
25 5 5 25
8
6
4
=
+ −
25 25 25
10 2
=
=
25 5
7. To check a division problem, multiply the
whole-number part of the quotient and the
divisor. Then add the remainder to get the
dividend.
20 × 225 + 30 = 4530
8. Wal-mart: 217,799
Sears: − 35,843
181,956
The difference between sales for WalMart and Sears is $181,956 million.
17.
7
27
70
27
97
+
=
+
=
10 100 100 100 100
18.
5
5 33 38
+3= +
=
11
11 11 11
19. 5 −
1 6
6
⋅ =
5 11 55
3
1
 0 3 0 5 0
14.   ÷   = ⋅ = = 0
 5   5  5 3 15
×
 6
10. 
 15
1
 32   8  32 11 4
⋅
= =2
12.   ÷   =
 22   11  22 8 2
20
6. 225 4530
−450
30
dividend: 4530
divisor: 225
whole-number part of the quotient: 20
remainder: 30
9.
Cumulative Review Exercises
20.
2
  10  12 4
=
=

  7  21 7
169
2 35 2 33
=
− =
7 7 7 7
12 9 24 9 15 3
− =
− = =
5 10 10 10 10 2
Chapter 4
Decimals
5 3 15 2
21. A = l ⋅ w = ⋅ =
ft
8 8 64
 5
 3  10 6
P = 2l + 2 w = 2   + 2   = +
 8
 8 8 8
16
=
= 2 ft
8
22.
23.
27.
2
1 1
9
+1 +1 +1
5
4 5
10
90
40
25
20
=
+1
+1
+1
100 100 100 100
175
7
=3
=3
100
4
7
19 1 19
3
3 ÷4= ⋅ =
=1
4
4 4 16
16
3
The average is 1 km.
16
28. 1.7 79.02
46.482
17 790.200
−68
110
−102
82
−6 8
1 40
−1 36
40
−34
6
50.90
+ 123.23
174.13
24. 700.80
− 32.01
668.79
25.
4
4
(3.14)(9)2 = (3.14)(81)
3
3
4
= (81)(3.14) = (108)(3.14)
3
3.14
× 1 08
25 12
314 00
339.12
= 339.12
29. (a)
301.1
× 0.25
15 055
60 220
75.275
26. 3.2 51.2
16
32 512
−32
192
−192
0
(b)
46.482 ≈ 46.48
0.004
× 938.12
8
40
3200
12000
3 60000
3.75248
938.12
× 0.004
3.75248
(c) Commutative property of
multiplication
170
Chapters 1–4
30. 19.875 = 19
15
875
7
= 19
1000
8
15
= 15.9375
16
0.9375
16 15.0000
−14 4
60
−48
120
−112
80
−80
0
3
1
14 = 14 + 3⋅ = 14 + 3(0.125)
8
8
= 14 + 0.375 = 14.375
5
1
7.5 = 7 = 7
10
2
171
Cumulative Review Exercises
Chapter 5
Ratio and Proportion
Chapter Opener Puzzle
a. 0.8 + 12.12 + 7.33 = 20.25
b. 60.75 ÷ 0.3 = 202.5
c. 1.7625 − 1.56 = 0.2025
d. 8.1 × 0.25 = 2.025
Section 5.1
Ratios
Section 5.1 Practice Exercises
1. ratio
(c) Total = 3 + 2 = 5
3
5
2. The recipe calls for twice as much flour as
sugar. For example, if 3 cups of sugar is
used, then 6 cups of flour must be used.
3. 5 : 6 and
5
6
4. 3 : 7 and
3
7
5. 11 to 4 and
6. 8 to 13 and
9
11
11
(b)
9
(c) Total = 11 + 9 = 20
11
20
10. (a)
11
4
11. (a)
8
13
21
52
(b) Cars that are not silver = 52 − 21 = 31
21
31
7. 1 : 2 and 1 to 2
8. 1 : 8 and 1 to 8
12. (a)
3
2
2
(b)
3
9. (a)
10
21
(b) Houses without a pool = 21 − 10 = 11
10
11
172
Section 5.1
13.
4 yr 2
=
6 yr 3
29.
10 lb 5
14.
=
14 lb 7
15.
30.
5 mi 1
=
25 mi 5
16 54
9
18 10
1 14
1 83
=
=
5
4
11
8
84
5
189
10
4
2
1
9
84 189 84 10
8
⋅
=
=
÷
=
5 10
5 189 9
2
5 11 5 8 10
= ÷ = ⋅ =
4 8 4 11 11
1
16.
20 ft 5
=
12 ft 3
31.
16.80 16.80 × 100 1680 7
=
=
=
2.40
2.40 × 100
240 1
17.
8m 4
=
2m 1
32.
18.50 18.50 × 100 1850 5
=
=
=
3.70
3.70 × 100
370 1
18.
14 oz 2
=
7 oz 1
33.
33 cm 11
19.
=
15 cm 5
20.
21.
22.
75¢ 3
=
100¢ 4
23.
18 in. 1
=
36 in. 2
24.
3 cups 1
=
9 cups 3
25.
3.6 3.6 × 10 36 3
=
=
=
2.4 2.4 × 10 24 2
1
1 1 1
÷4= ⋅ =
2
2 4 8
1
4
1 12
35.
10.25 10.25 × 100 1025 5
=
=
=
8.2 × 100
820 4
8.2
36.
11.55 11.55 × 100 1155 7
=
=
=
6.6 × 100
660 4
6.6
=
1
4
3
2
=
1 3 1 2 2 1
÷ = ⋅ =
=
4 2 4 3 12 6
37. Increase = 90 – 66 = 24
24 4
=
66 11
38. Increase = 4344 − 2100 = 2244
2244 187
=
2100 175
1
ft = 12 + 4 = 16 in.
3
6 3
=
16 8
39. (a) 1
26.
10.15 10.15 × 100 1015 5
=
=
=
8.12
8.12 × 100
812 4
27.
8
8 8 28 8 3 24 6
= 28 = ÷
= ⋅
=
=
1
1
3
1
28
28
7
93
3
28.
4
=
34.
21 days 7
=
30 days 10
$60 6
=
$50 5
1
2
1
ft
2
1
1
2 = 2 =1÷4 =1⋅3=3
1 13 43 2 3 2 4 8
(b) 6 in. =
3
24 24 24 40 24 3
9
= 40 =
÷
=
⋅
=
1
1
3
1 40 5
13 3
3
5
173
Ratios
Chapter 5
Ratio and Proportion
3
week
7
50.
9 3
=
21 7
3
3 1 3
÷2= ⋅ =
7
7 2 14
51.
0.3 0.3 × 10 3 1
=
=
=
1.2 1.2 × 10 12 4
52.
3.5 3.5 × 10 35 5
=
=
=
4.2 4.2 × 10 42 6
40. (a) 3 days =
3
7
2
=
(b) 2 weeks = 14 days
3
14
53. 8 + 5 = 13 units
400,000
4
1
=
=
41.
4,400,000 44 11
42.
33,000,000 3
=
22,000,000 2
43. Increase = 4, 400, 000 − 400, 000
= 4, 000, 000
4,000,000 10
=
400,000
1
44. Increase = 33,000,000 − 22,000,000
= 11,000,000
11,000,000 1
=
22,000,000 2
54. (a)
3
2
(b)
5
3
(c)
8
5
(d)
13
8
55. (a)
3
= 1.5
2
(b)
5
= 1.6
3
45.
60 15
=
128 32
(c)
8
= 1.6
5
46.
111 37
=
159 53
(d)
13
= 1.625
8
47.
60 20
=
183 61
48.
49.
Yes, they are approaching 1.618.
56. Answers will vary.
136 34
=
172 43
1 12
2 14
=
3
2
9
4
=
57. Answers will vary.
3 9 3 4 12 2
÷ = ⋅ =
=
2 4 2 9 18 3
174
Section 5.2
Section 5.2
Rates and Unit Cost
Rates and Unit Cost
Section 5.2 Practice Exercises
1. (a) rate
(b) unit
2.
2
3
3
3. 3 : 5 and
5
4. 4 to 1 and
5.
36¢ 4
=
27¢ 3
6.
6 34
8 14
ft
ft
=
4
1
27
4
33
4
9
=
1
27 33 27 4
9
÷ =
⋅
=
4
4
4 33 11
1
9.
16.
2 in. 1 in.
=
6 hr 3 hr
17.
130 calories 65 calories
=
8 crackers
4 crackers
18.
14 plants 7 plants
=
22 ft
11 ft
19.
$30
$15
=
4 trays 2 trays
20.
50 students 25 students
=
4 advisors
2 advisors
22. a, b, c
23.
$28.40 28.4 × 10 284 71
=
=
=
$20.80 20.8 × 10 208 52
24.
$32
5 ft 2
10.
44 ft
5 sec
11.
234 mi 117 mi
=
4 hr
2 hr
12.
13 pages 1 page
=
26 sec
2 sec
21. a, c, d
11
1.08 mi 1.08 × 100 108 9
7.
=
=
=
2.04 mi 2.04 × 100 204 17
8.
15.
14 blooms 7 blooms
=
6 plants
3 plants
13.
$58 $29
=
8 hr 4 hr
14.
336 words 112 words
=
15 min
5 min
175
452 mi 113 mi
=
or 113 mi/day
4 days 1 day
$6,000,000
= $75,000 per year
80 years
75,000
80 6,000,000
−5 60
400
−400
0
25.
480 km
= 96 km/hr
5 hr
26.
1120 mi
= 280 mi/hr
4 hr
Chapter 5
27.
28.
29.
30.
31.
Ratio and Proportion
$660
= $55 per payment
12 payments
55
12 660
−60
60
−60
0
32.
7396 wins
= 172 wins per year
43 years
172
43 7396
−43
309
−301
86
−86
0
33.
$2.76
= $0.69 per lb
4 lb
0.69
4 2.76
− 24
36
−
36

0
$13.08
= $1.09 per tile
12 tiles
1.09
12 13.08
−12
1 08
−1 08
0
34.
$1,792,000
= $256,000 per person
7 people
256,000
7 1,792,000
−1 4
39
−35
42
−42
0
176
123 lb
≈ 11.18 lb per member
11 members
11.1818...
11 123.0000
−11
13
−11
20
−11
90
−88
20
−11
90
−88
2
500 m
≈ 14.3 m/sec
35 sec
14.28
35 500.00
−35
150
−140
100
−70
300
−280
20
500 m
≈ 13.2 m/sec
38 sec
13.15
38 500.00
−38
120
−114
60
−38
220
−190
30
35.
$10.95
= 0.219 per oz
50 oz
36.
$3.49
≈ $0.291 per oz
12 oz
Section 5.2
37.
$1.99
= $0.995 per liter
2 liters
38.
$221.00
= $55.25 per chair
4 chairs
39.
$210
= $52.50 per tire
4 tires
40.
$64.80
= $21.60 per shirt
3 shirts
41.
42.
65
= 3.25 g/fl oz
20
47
Mello Yello:
= 3.92 g/fl oz
12
24
Ginger Ale:
= 3 g/fl oz
8
Mello Yello has the greatest amount per
fluid oz.
47. Coca-Cola:
48. Coca-Cola:
$32.50
≈ $5.417 per bodysuit
6 bodysuits
170
= 14.2 cal/fl oz
12
90
Ginger Ale:
= 11.25 cal/fl oz
8
Ginger Ale has the least number of
calories per fluid oz.
$9.84
= $1.23 per battery
8 batteries
$8.35
= $0.334/oz
25 oz
$5.01
(b)
= $0.334/oz
15 oz
(c) Both sizes cost the same amount per
ounce.
49.
13,000,000
100
= 130,000 platelets per microliter;
Since the patient’s platelet count is above
20,000 per microliter, the patient does not
have a life-threatening condition.
$2.99
= $0.299/lb
10 lb
$8.97
(b)
= $0.299/lb
30 lb
(c) Both sizes cost the same amount per
pound.
44. (a)
46.
240
= 12 cal/fl oz
20
Mello Yello:
43. (a)
45.
Rates and Unit Cost
$1.85
≈ $0.123 per ounce
15 oz
$1.39
≈ $0.164 per ounce
8.5 oz
The larger can is $0.123 per ounce. The
smaller can is $0.164 per ounce. The
larger can is the better buy.
50.
4, 260,000
= 355,000 vehicles per year
12
51.
344,000
= 43,000 prisoners per year
8
22,000,000
= 2, 200,000 per year
10
10, 200,000
(b)
= 2,040,000 per year
5
(c) Mexico
52. (a)
$5.65
≈ $0.014 per napkin
400 napkins
$1.95
= $0.0195 per napkin
100 napkins
The package of 400 sells for $0.014 per
napkin. The package of 100 sells for
$0.020 per napkin. The larger package is
the better buy.
$18.24
= $0.76 per month
24
$22.80
(b)
= $1.90 per month
12
(c) IBM
53. (a)
177
Chapter 5
Ratio and Proportion
54. cheetah:
120
≈ 29 m/sec
4.1
antelope:
58.
50
≈ 24 m/sec
2.1
The cheetah is faster.
$8.69
≈ $0.181 per oz
48 oz
$6.15
≈ $0.280 per oz
22 oz
$4.59
= $0.255 per oz
18 oz
The best buy is the 48-oz jar.
55. (a)
328 wins
≈ 9.9 wins/yr
33 yr
59. (a)
(b)
250 wins
≈ 8.6 wins/yr
29 yr
(b)
$2.00
≈ $0.167 per oz
12 oz
(c)
$3.19
3.19
=
≈ $0.322 per oz
3 × 3.3 oz 9.9
(c) Shula
56. (a)
328 wins
≈ 2.1 wins/loss
156 losses
The best buy is the 12-oz can.
250 wins
(b)
≈ 1.5 wins/loss
162 losses
60. (a)
(c) Shula
(b)
$9.59 $9.59
=
≈ $0.38 per ounce
57. (a)
6 ( 4.25) 25.5
$2.99
2.99
=
≈ $0.062 per oz
6 × 8 oz
48
$3.33
3.33
=
≈ $0.023 per oz
12 × 12 oz 144
The case of 12 twelve-oz cans for $3.33 is
the better buy.
$6.39 $6.39
=
≈ $0.18 per ounce
(b)
8 ( 4.5)
36
(c)
$3.61
$3.61
=
≈ $0.401 per oz
2 × 4.5 oz 9 oz
$2.55 $2.55
=
≈ $0.19 per ounce
3( 4.5) 13.5
The best buy is Dial.
Section 5.3
Proportions
Section 5.3 Practice Exercises
1. (a) equation
4.
3 teachers
1 teacher
=
45 students 15 students
5.
6 apples 3 apples
=
2 pies
1 pie
6.
6 days 3
=
2 days 1
(b) proportion
$4.29
2.
≈ $0.268 / oz .
16 oz
3 ft
1
3.
=
45 ft 15
178
Section 5.3
7.
264 mi 22 mi
=
36 gal 3 gal
9 ? 8
=
10 9
22.
?
8.
$264 22
=
$36
3
9.
4
5
=
16 20
10.
3
4
=
18 24
11.
25 10
=
15 6
?
(16)(3) = (24)(2)
48 = 48
Yes
24.
35 20
=
14 8
13.
2 4
=
3 6
2 26
=
1 13
15.
30 12
=
25 10
16.
21.
2 12
?
=
3 23
15
22
?
 1
 2
=
2
(22)
 
 3  (15)
 2
 3
11
5
5 22 ? 11 15
= ⋅
⋅
3 1
2 1
24 8
=
18 6
1
1
55 = 55
Yes
26.
$115
$460
18.
=
1 week 4 weeks
20.
24
4
n
30
5
(4)(30) n (5)(24)
120 = 120
Yes
25.
$6.25 $187.50
17.
=
1 hr
30 hr
19.
16 ? 2
=
24 3
23.
12.
14.
(9)(9) = (10)(8)
81 ≠ 80
No
1 in. 5 in.
=
7 mi 35 mi
1 34
?
=
7
12
3
?
 3
1  (12) = (3)(7)
 4
3
7 12 ?
⋅
= 21
4 1
16 flowers 32 flowers
=
5 plants
10 plants
1
21 = 21
5 ? 4
=
18 16
Yes
?
(5)(16) = (18)(4)
80 ≠ 72
No
179
Proportions
Chapter 5
Ratio and Proportion
2 ? 10
=
3.2 16
27.
?
4.7 ? 23.5
=
7
35
?
(4.7)(35) = (7)(23.5)
164.5 = 164.5
Yes
33.
48 ? 24
=
18
9
29.
34.
35 ? 5
=
14 2
?
?
31.
1 12
?
7.1 ? 35.5
=
2.4
10
(7.1)(10) = (2.4)(35.5)
71 ≠ 85.2
No
(35)(2) = (14)(5)
70 = 70
Yes
=
6.3 ? 12.6
=
9
16
?
?
2 83
5
2 12
(6.3)(16) = (9)(12.6)
100.8 ≠ 113.4
No
(48)(9) = (18)(24)
432 = 432
Yes
30.
5
6
?
=
 2  1  ?  5 
1  2  =   (5)
 3  2   6 
5 5 ? 5 5
⋅ = ⋅
3 2 6 1
25 25
=
6
6
Yes
(2)(16) = (3.2)(10)
32 = 32
Yes
28.
1 23
32.
35. Divide by 2
9 12
36. Divide by 3
6
37. Divide by 5
?
 3
 1  1 
 2  (6) = 1  9 
 8
 2  2 
38. Divide by 7
39. Divide by 8
3
19 6 ? 3 19
⋅ = ⋅
8 1 2 2
40. Divide by 25
4
41. Divide by 0.6
57 57
=
4
4
42. Divide by 0.4
Yes
43.
x 1
= ; x=5
40 8
5 ? 1
=
40 8
?
(5)(8) = 40
40 = 40
Yes
180
Section 5.3
44.
14 12
= ; x = 21
x 18
14 ? 12
=
21 18
49.
?
(14)(18) = (21)(12)
252 = 252
Yes
12.4 8.2
=
; y = 20
45.
31
y
12.4 ? 8.2
=
31
20
50.
?
(12.4)(20) = (31)(8.2)
248 ≠ 254.2
No
46.
51.
4.2
z
; z = 15.2
=
9.8 36.4
4.2 ? 15.2
=
9.8 36.4
?
(4.2)(36.4) = (9.8)(15.2)
152.88 ≠ 148.96
No
47.
48.
12 3
=
16 x
12 x = (16)(3)
12 x = 48
12 x 48
=
12 12
x=4
Check:
52.
12 ? 3
=
16 4
?
(12)(4) = (16)(3)
48 = 48 
53.
20 ? 5
20 5
=
=
Check:
28 7
28 x
?
20 x = (28)(5)
(20)(7)
(28)(5)
=
20 x = 140

1
40
140
=
20 x 140
=
20
20
x=7
54.
181
9 x
=
21 7
(9)(7) = 21x
63 = 21x
63 21x
=
21 21
3= x
15 3
=
10 x
15 x = (10)(3)
15 x = 30
15 x 30
=
15 15
x=2
Check:
Proportions
9 ? 3
=
21 7
?
(9)(7) = (21)(3)
63 = 63 
Check:
15 ? 3
=
10 2
?
(15)(2) = (10)(3)
30 = 30 
75 ? 25
p 25
=
=
Check:
12
4
12 4
?
4 p = (12)(25)
(75)(4) = (12)(25)
4 p = 300
300 = 300 
4 p 300
=
4
4
p = 75
10 ? 30
p 30
=
=
Check:
8 24
8 24
?
24 p = (8)(30)
(10)(24)
(8)(30)
=
24 p = 240
240
240 
=
24 p 240
=
24
24
p = 10
6 4
=
n 8
(6)(8) = 4n
48 = 4n
48 4n
=
4
4
12 = n
49 14
=
n 18
(49)(18) = 14n
882 = 14n
882 14n
=
14
14
63 = n
Check:
6 ? 4
=
12 8
?
(6)(8) = (12)(4)
48 = 48 
Chapter 5
Ratio and Proportion
49 ? 14
=
63 18
Check:
59.
?
(49)(18) = (63)(14)
882 = 882 
55.
2 t
=
3 18
(2)(18) = 3t
36 = 3t
36 3t
=
3
3
12 = t
1
?
17 x = 51
17 x 51
=
17 17
x=3
(2)(18) = (3)(12)
36 = 36 
34 ? 2
=
51 3
Check:
?
 1
(17)(3) = (12)  4 
 4
?
(34)(3) = (51)(2)
102 = 102 
?
17
4
1
51 = 51 
1
60.
26 5 5
=
30 x
 1
26 x =  5  ( 30 )
 5
26 30
26 x = ⋅
5 1
6
26 30
26 x = ⋅
5 1
?
25
9
=
100 36
1
?
26 x = 156
26 x 156
=
26
26
x=6
(25)(36) = (100)(9)
900 = 900 
58.
3
51 = 12 ⋅
25 9
=
100 y
25 y = (100)(9)
25 y = 900
25 y 900
=
25
25
y = 36
Check:
1
17 ? 4 4
=
12
3
Check:
34 2
=
56.
51 t
34t = (51)(2)
34t = 102
34t 102
=
34 34
t =3
57.
3
17 12
17 x = ⋅
4 1
2 ? 12
=
3 18
Check:
1
17 4 4
=
x
12
 1
17 x =  4  (12 )
 4
65 26
65 ? 26
=
=
Check:
15 y
15
6
?
65 y = (15)(26)
(65)(6) = (15)(26)
65 y = 390
390 = 390 
65 y 390
=
65
65
y=6
Check:
1
26 ? 5 5
=
30
6
?
 1
(26)(6) = (30)  5 
 5
?
6
30 26
⋅
156 =
1 5
156 = 156 
182
1
Section 5.3
61.
m 5
=
12 8
8m = (12)(5)
8m = 60
8m 60
=
8
8
15
1
or 7 or 7.5
m=
2
2
7.5 ? 5
=
Check:
12 8
64.
( )
?
( 4.75) (16) = (8) ( 9.5 )
?
(7.5)(8) = (12)(5)
60 = 60 
76 = 76 
65.
16 21
=
62.
12 a
16a = (12)(21)
16a = 252
16a 252
=
16
16
63
3
or 15 or 15.75
a=
4
4
16 ? 21
=
Check:
12 15.75
?
?
66.
3.125 18.75
=
k
5
3.125 k = (5) 18.75
3.125k = 93.75
3.125k 93.75
=
3.125 3.125
k = 30
3.125 ? 18.75
Check:
=
5
30
(
)
(
0.5 1.8
=
h
9
(0.5)(9) = 1.8h
4.5 = 1.8h
4.5 1.8h
=
1.8 1.8
2.5 = h
0.5 ? 1.8
=
Check:
2.5
9
(0.5)(9) = (2.5)(1.8)
4.5 = 4.5 
(16)(15.75) = (12)(21)
252 = 252 
63.
4.75 9.5
=
k
8
4.75k = (8) 9.5
4.75k = 76
4.75k
76
=
4.75 4.75
k = 16
4.75 ? 9.5
Check:
=
8
16
)
2.6 1.3
=
h 0.5
(2.6)(0.5) = 1.3h
1.3 = 1.3h
1.3 1.3h
=
1.3 1.3
1= h
2.6 ? 1.3
=
Check:
1
0.5
?
(2.6)(0.5) = (1)(1.3)
1.3 = 1.3 
?
( 3.125) (30) = (5) (18.75)
93.75 = 93.75 
183
Proportions
Chapter 5
67.
Ratio and Proportion
3
8
x
6.75 72
3
6.75 x =   ( 72 )
8
=
69.
3 72
⋅
8 1
1
6.75 x = 27
6.75 x
27
=
6.75 6.75
x=4
Check:
1
10
=
3
8
?
6.75
?
 3
  ( 72 ) = ( 4 )( 6.75)
8
70.
27 = 27 
=
1
12 y =   (120 )
4
30
1 120
⋅
4 1
1
3
=
Check:
1
1
2
t
1
6 ? 2
=
1
1
3
12 y = 30
12 y 30
=
12 12
y = 2.5
Check:
6
 1  1 
6t =    
 3  2 
1
6t =
6
1
1 1
⋅6t = ⋅
6
6 6
1
t=
36
120
y
12 y =
12
1
4
?
=
80
1 ? 1 1
( 4 )   =  
 
 80   10  2 
1
1

=
20 20
1
1
4
1
4 ? 2
= 1
1
10
4
72
=
9
12
z
Check:
?
3
⋅ 72 = 27
8
68.
1
2
 1   1
4z =    
 10   2 
1
4z =
20
1
1 1
⋅ 4z = ⋅
4
4 20
1
z=
80
9
6.75 x =
4
36
1 ? 1 1
( 6 )   =  
 
 36   3  2 
1 1
= 
6 6
120
2.5
?
(12 )( 2.5 ) = 
1
 (120 )
4
30 = 30 
184
Problem Recognition Exercises: Operations on Fractions versus Solving Proportions
Problem Recognition Exercises: Operations on Fractions versus
Solving Proportions
1. (a) Proportion;
x 15
=
4 8
8 x = (4)(15)
8 x = 60
8 x 60
=
8
8
15
x=
2
(b) Product of fractions;
1
(b) Product of fractions;
5
48 16
5. (a) Proportion;
=
3
p
(48)(3) = 16 p
144 = 16 p
144 16 p
=
16
16
p=9
(b) Product of fractions;
1 15 15
⋅ =
4 8 32
1
2. (a) Product of fractions;
16
2 3
3
⋅
=
5 10 25
2
48 16 32
⋅
=
= 32
1
8 3
5
(b) Proportion;
1
1
6. (a) Product of fractions;
2 y
=
5 10
(2)(10) = 5 y
20 = 5 y
20 5 y
=
5
5
y=4
2
4
10 28 8
⋅
= =8
1
7 5
1
1
(b) Proportion;
1
3. (a) Product of fractions;
2 3
3
⋅
=
7 14 49
7
(b) Proportion;
4. (a) Proportion;
3 6
6
⋅
=
5 15 25
2 n
=
7 14
(2)(14) = 7n
28 = 7n
28 7 n
=
7
7
n=4
7. (a)
3 6
=
7 z
3 z = (7)(6)
3 z = 42
3 z 42
=
3
3
z = 14
1
m 6
=
5 15
15m = (5)(6)
15m = 30
15m 30
=
15 15
m=2
(b)
185
5
3 6
3 35 5
÷
= ⋅
=
7 35 7 6
2
1
(c)
10 28
=
7
t
10t = (7)(28)
10t = 196
10t 196
=
10 10
98
t=
5
2
3 6 3 ⋅ 5 6 15 6 15 + 6
+
=
+
=
+
=
7 35 7 ⋅ 5 35 35 35
35
21 3
=
=
35 5
Chapter 5
Ratio and Proportion
3 6
18
⋅ =
7 35 245
(d)
(c)
1
4 20 4 3
3
8. (a) ÷
= ⋅
=
5 3
5 20 25
7
14 10 14 7
49
(d)
÷
=
=
⋅
5
7
5 10 25
5
4 20
=
v 3
(4)(3) = 20v
12 = 20v
12 20v
=
20 20
3
v=
5
(b)
5
11 66
10. (a)
=
3
y
11 y = (3)(66)
11 y = 198
11 y 198
=
11
11
y = 18
11 66 11 6 11 6 ⋅ 3
+
= + = +
(b)
3 11 3 1 3 1 ⋅ 3
11 18 11 + 18 29
= + =
=
3 3
3
3
4
4 20 4 20 16
(c) ×
=
= ⋅
5 3
3
5 3
1
4 20 4 ⋅ 3 20 ⋅ 5 12 100
=
+
= +
(d) +
5 3 5 ⋅ 3 3 ⋅ 5 15 15
12 + 100
112
=
=−
15
15
2
1
11 66 11 11 11
(c)
÷
=
=
⋅
3 11 3 66 18
2
14 10 4
9. (a)
⋅
= =4
1
5 7
1
(b)
1
(d)
1
14 x
=
5 7
(14)(7) = 5 x
98 = 5 x
98 5 x
=
5
5
98
x=
5
Section 5.4
14 10 14 ⋅ 7 10 ⋅ 5 98 50
− =
−
=
−
5 7
5 ⋅ 7 7 ⋅ 5 35 35
98 − 50
48
=
=−
35
35
6
22
11 66 11 66 22
×
=
⋅
=
= 22
3 11 3 11
1
1
1
Applications of Proportions and Similar Figures
Section 5.4 Practice Exercises
1. (a) similar
2.
(b) proportional
4 ? 12
=
7 21
?
(4)(21) = (7)(12)
84 = 84
4 12
=
7 21
186
Section 5.4
3.
Applications of Proportions and Similar Figures
3 ? 15
=
13 65
8.
?
(3)(65) = (13)(15)
195 = 195
3 15
=
13 65
4.
2 ? 21
=
5 55
9.
?
(2)(55) = (5)(21)
110 ≠ 105
2 21
≠
5 55
5.
12 ? 35
=
7 19
7.
3 12
=
2 13
k
4
 1
 1
 3  (4) =  2  k
 2
 3
7 4 7
⋅ = k
2 1 3
28 7
= k
2 3
4
?
3 28 3 7
⋅
= ⋅ k
7 2
7 3
(12)(19) = (7)(35)
228 ≠ 245
12 35
≠
7 19
6.
p 1
=
9 6
6 p = (9)(1)
6p = 9
6p 9
=
6 6
3
1
p = or 1 or 1.5
2
2
1
12
=k
2
6=k
2 3
=
7 x
2 x = (7)(3)
2 x = 21
2 x 21
=
2
2
21
1
x=
or 10 or 10.5
2
2
10.
4 n
=
3 5
(4)(5) = 3n
20 = 3n
20 3n
=
3
3
20
2
n=
or 6 or 6.6
3
3
11.
12.
187
2.4 m
=
3
5
(2.4)(5) = 3m
12 = 3m
12 3m
=
3
3
4=m
3
7
=
2.1 y
3 y = (2.1)(7)
3 y = 14.7
3 y 14.7
=
3
3
y = 4.9
1.2 3
=
4 a
1.2a = (4)(3)
1.2a = 12
1.2a 12
=
1.2 1.2
a = 10
Chapter 5
Ratio and Proportion
13. Let x represent the number of miles.
244 mi x mi
=
4 gal 10 gal
(244)(10) = 4 x
2440 = 4 x
2440 4 x
=
4
4
610 = x
Pam can drive 610 mi on 10 gal of gas.
8 cm 7 cm
=
91 mi x mi
8 x = (91)(7)
8 x = 637
8 x 637
=
8
8
x = 79.625
The actual distance is about 80 mi.
18. Let x represent the distance from Dallas to
Little Rock.
3.5 in 4.75 in
=
x mi
210 mi
3.5x = (210)(4.75)
3.5x = 997.5
3.5x 997.5
=
3.5
3.5
x = 285
The distance is 285 mi.
14. Let x represent the number of beats per
minute (60 seconds).
13 beats
x
=
10 sec
60 sec
(13)(60) = 10 x
780 = 10 x
780 10 x
=
10
10
78 = x
This is 78 beats/min.
19. Let x represent the number of male
students.
6200 female 31
=
19
x male
(6200)(19) = 31x
117,800 = 31x
117,800 31x
=
31
31
3800 = x
There are 3800 male students.
15. Let x represent the amount of crushed rock
needed.
x kg 3.25
=
24 kg
1
x = (24)(3.25)
x = 78
78 kg of crushed rock will be required.
16. Let x represent the number of pounds of
garbage.
2 adults 50 adults
=
63.4 lb
x lb
2 x = (63.4)(50)
2 x = 3170
2 x 3170
=
2
2
x = 1585
They will produce 1585 lb of garbage.
20. Let x represent the opponent’s votes.
7230 votes 6
=
x votes
5
(7230)(5) = 6 x
36,150 = 6 x
36,150 6 x
=
6
6
6025 = x
The opponent received 6025 votes.
17. Let x represent the distance from
Sacramento to Modesto.
188
Section 5.4
Applications of Proportions and Similar Figures
21. Let x represent the number of heads.
x heads 1
=
630 flips 2
2 x = (630)(1)
2 x = 630
2 x 630
=
2
2
x = 315
Heads would come up about 315 times.
25. Let x represent the number of Euros.
38.0 Euros x Euros
=
$50
$900
(38.0)(900) = 50 x
34, 200 = 50 x
34, 200 50 x
=
50
50
684 = x
Pierre can buy 684 Euros.
26. Let x represent Canadian dollars.
$103 Canadian
$ x Canadian
=
$100 American $235 American
(103)(235) = 100 x
24, 205 = 100 x
24, 205 100 x
=
100
100
242.05 = x
Erik can buy $242.05 Canadian.
22. Let x represent the number of times a 4
comes up.
x times 1
=
366 rolls 6
6 x = (366)(1)
6 x = 366
6 x 366
=
6
6
x = 61
The number 4 should come up about
61 times.
27. Let x represent the number of falls.
x falls
3 falls
=
60 visits 4 visits
4 x = (60)(3)
4 x = 180
4 x 180
=
4
4
x = 45
45 visits would be a result of falls.
23. Let x represent the number of earned runs.
x runs
42 runs
=
9 innings 126 innings
126 x = (9)(42)
126 x = 378
126 x 378
=
126 126
x=3
There would be approximately 3 earned
runs for a 9-inning game.
28. Let x represent the number of smokers.
x smokers
24 smokers
=
850 Americans 100 Americans
100 x = (850)(24)
100 x = 20, 400
100 x 20, 400
=
100
100
x = 204
204 smokers would be expected.
24. Let x represent yards gained.
34 passes 22 passes
=
357 yd
x yd
34 x = (357)(22)
34 x = 7854
34 x 7854
=
34
34
x = 231
231 yards would be gained.
189
Chapter 5
Ratio and Proportion
29. Let x represent amount of water needed.
1 cup sugar
1 cup sugar
= 6
4 cups water x cups water
1
x = ( 4)  
6
4
  1 
x =   
 1  6 
4 2
x= =
6 3
6 crackers
14 crackers
=
225 mg sodium x mg sodium
6 x = ( 225)(14)
6 x = 3,150
= 3,150
6
x = 525
6x
6
There are 525 mg of sodium in 14
crackers.
(b)
2
cup of water should be added.
3
525 + 225 = 750 mg
2,300 − 750 = 1,550 mg
30. Let x represent amount of depreciation.
$3000
x
= 1
5 years 3 2 years
1,550 mg of sodium is allowed from other
foods.
( )
3,000
5x = ( 1 ) ( )
5 x = ( 3,000)
3 12
7
2
33. Let x represent the memory.
21,000
30 frames/sec 12 frames/sec
=
2.45 megabytes x megabytes
30 x = ( 2.45)(12)
30 x = 29.4
30 x = 29.4
30
30
x = 0.98
5x = 2
5 x = 10,500
5x
5
10,500
= 5
x = 2,100
The car will depreciate $2,100.
31. (a)Let x represent the number of emails.
0.98 megabytes or 980 kilobytes would be
used.
14 students 30 students
=
91 emails
x emails
14 x = ( 91)( 30)
14 x = 2,730
14 x
14
34. Let x represent the earnings.
$312 earnings
x earnings
=
$800 investment $1100 investment
(320)(1100) = 800 x
343, 200 = 800 x
2,730
= 14
x = 195
343,200
800
x
= 800
800
429 = x
He would expect to receive 195 emails.
(b)
Sodium in 14 crackers + sodium in 6
crackers = sodium in 20 crackers
195 emails × 3 minutes per email
= 585 minutes
$1100 would have earned $429.
585minutes ÷ 60 minutes/hour
= 9.75 hours
He would expect to spend 585 minutes or
9.75 hours answering emails.
32. (a)Let x represent a sodium.
190
Section 5.4
Applications of Proportions and Similar Figures
35. Let x represent the height of the model.
40. Let x represent the number of manatees.
150 manatees 3 manatees
=
x manatees
40 manatees
(150)(40) = 3 x
6000 = 3 x
6000 3x
=
3
3
2000 = x
There are approximately 2000 manatees in
Florida.
x inches 1 inch
=
1454 feet 50 feet
50 x = (1454)(1)
50 x = 1454
50 x = 1454
50
50
x = 29.08
The model would be 29.08 inches tall.
37. Let x represent the amount of acid.
41. Let x represent the number of bison.
200 bison
6 bison
=
x bison
120 bison
(200)(120) = 6 x
24,000 = 6 x
24,000 6 x
=
6
6
4000 = x
There are approximately 4000 bison in the
park.
3 parts acid
x mL
=
5 parts water 420 mL
(3)(420) = 5 x
1260 = 5 x
1260 = 5 x
5
5
252 = x
The chemist needs 252 mL of acid.
42. Let x represent the number of deer.
20 deer
x deer
=
2
1 mi
492 mi 2
(20)(492) = 1x
9840 = x
There are about 9840 deer in Cass County.
38. Let x represent the amount of blue paint.
5 parts white 4 quarts white
=
7 parts blue
x quarts blue
5 x = (7)(4)
5 x = 28
5 x = 28
5
5
x = 5.6
43.
Mitch needs 5.6 quarts of blue paint.
39. Let x represent the number of fish.
21 fish 75 fish
=
100 fish x fish
21x = (100)(75)
21x = 7500
21x 7500
=
21
21
x ≈ 357
There are approximately 357 bass in the
lake.
44.
191
16 12
=
32 x
16 x = (32)(12)
16 x = 384
16 x 384
=
16
16
x = 24 cm
16 18
=
32 y
16 y = (32)(18)
16 y = 576
16 y 576
=
16
16
y = 36 cm
2 2.3
=
5
x
2 x = (5)(2.3)
2 x = 11.5
2 x 11.5
=
2
2
x = 5.75 ft
2 2
=
5 y
2 y = (5)(2)
2 y = 10
2 y 10
=
2
2
y = 5 ft
Chapter 5
45.
46.
47.
Ratio and Proportion
x 6
=
1.5 9
9 x = (1.5)(6)
9x = 9
9x 9
=
9 9
x = 1 yd
6 7
=
9 y
6 y = (9)(7)
6 y = 63
6 y 63
=
6
6
y = 10.5 yd
x 9
=
4 3
3 x = (4)(9)
3 x = 36
3x 36
=
3
3
x = 12 m
9 15
=
3 y
9 y = (3)(15)
9 y = 45
9 y 45
=
9
9
y=5m
49.
 1
 2  h = (3)(10)
 2
5
h = 30
2
2 5
2 30
⋅ h= ⋅
5 2
5 1
60
h=
5
h = 12
The flagpole is 12 ft high.
50. Let x represent the length of the shadow.
15 90
=
x
4
15 x = (4)(90)
15 x = 360
15 x 360
=
15
15
x = 24
The shadow will be 24 ft.
x 10
=
10 6 23
(6 23 ) x = (10)(10)
20
x = 100
3
20 3
100 3
⋅ x=
⋅
3 20
1 20
300
x=
20
x = 15 cm
51. Let x represent the height of the platform.
1.6 x
=
1 1.5
(1.6)(1.5) = 1x
2.4 = x
The platform is 2.4 m tall.
15 10
=
y
6
15 y = (6)(10)
15 y = 60
15 y 60
=
15 15
y = 4 in.
48.
x 19.2
=
25 20
20 x = (25)(19.2)
20 x = 480
20 x 480
=
20
20
x = 24 yd
h 10
=
3 2 12
52. Let x represent the length of the shadow.
32 22
=
18 x
32 x = (18)(22)
32 x = 396
32 x 396
=
32
32
x = 12.375
The shadow will be 12.375 ft.
25 7
=
20 y
25 y = (20)(7)
25 y = 140
25 y 140
=
25
25
y = 5.6 ft
53.
192
4 7
=
10 x
4 x = (10)(7)
4 x = 70
4 x 70
=
4
4
x = 17.5 in.
Section 5.4
54.
55.
56.
6 3
=
8 x
6 x = (8)(3)
6 x = 24
6 x 24
=
6
6
x=4m
6 4
=
8 y
6 y = (8)(4)
6 y = 32
6 y 32
=
6
6
1
y=5 m
3
1.6 4.8
=
x
2
1.6 x = (2)(4.8)
1.6 x = 9.6
1.6 x 9.6
=
1.6 1.6
x = 6 ft
2 10
=
1.6 y
2 y = (1.6)(10)
2 y = 16
2 y 16
=
2
2
y = 8 ft
4 9
=
24 x
4 x = (24)(9)
4 x = 216
4 x 216
=
4
4
x = 54 cm
24 30
=
4
y
24 y = (4)(30)
24 y = 120
24 y 120
=
24
24
y = 5 cm
8 in. 6 in.
=
28 ft x ft
8 x = (28)(6)
8 x = 168
8 x 168
=
8
8
x = 21 ft
8 in. 6 in.
=
28 ft y ft
8 y = (28)(6)
8 y = 168
8 y 168
=
8
8
y = 21 ft
Applications of Proportions and Similar Figures
59. Let x represent the number of crimes.
4743 crimes
x crimes
=
100,000 people 3,500,000 people
(4743)(3,500,000) = 100,000 x
16,600,500,000 = 100,000 x
16,600,500,000 100,000 x
=
100,000
100,000
166,005 = x
There were approximately 166,005 crimes
committed.
60. Let x represent the height of the
Washington Monument.
3.25 328
=
5.5
x
3.25x = (5.5)(328)
3.25x = 1804
3.25x 1804
=
3.25 3.25
x ≈ 555
The Washington Monument is
approximately 555 ft tall.
57.
8 in. 15.2 in.
=
z ft
28 ft
8 z = (28)(15.2)
8 z = 425.6
8 z 425.6
=
8
8
z = 53.2 ft
1
58. x = (3 yd) = 1 yd
3
1
4
1
y = (4 yd) = = 1 yd
3
3
3
193
Chapter 5
Ratio and Proportion
61. Let x represent the number of women with
breast cancer.
110 cases
x cases
=
100,000 women 14,000,000 women
(110)(14,000,000) = 100,000 x
1,540,000,000 = 100,000 x
1,540,000,000 100,000 x
=
100,000
100,000
15, 400 = x
Approximately 15,400 women would be
expected to have breast cancer.
Chapter 5
62. Let x represent the number of men with
prostate disease.
118 cases
x cases
=
1000 men 2,500,000 men
(118)(2,500,000) = 1000 x
295,000,000 = 1000 x
295,000,000 1000 x
=
1000
1000
295,000 = x
Approximately 295,000 men would be
expected to have prostate disease.
Review Exercises
Section 5.1
7.
52 cards 4
=
13 cards 1
8.
$21 7
=
$15 5
3. 8 : 7 and 8 to 7
9.
80 ft 2
=
200 ft 5
4. (a)
2
3
10.
(b)
3
2
1. 5 to 4 and
2. 3 : 1 and
5
4
3
1
11.
(c) Total children = 3 + 2 = 5
3
5
12.
4
5. (a)
5
(b)
5
4
(c) Total bottles = 4 + 5 = 9
5
9
6. (a)
7 days 1
=
28 days 4
1 12 hr
1
3
hr
2 yd
3
2 16 yd
=
=
3
2
1
3
=
2
3
13
6
3 1 3 3 9
÷ = ⋅ =
2 3 2 1 2
2
2 13 2 6
4
= ÷ = ⋅ =
3 6 3 13 13
1
13.
$2.56 2.56 × 100 256 4
=
=
=
$1.92 1.92 × 100 192 3
14.
42.5 mi 42.5 × 100 4250 170
=
=
=
3.25 mi 3.25 × 100 325
13
15. (a) 1200 + 320 = 1520
This year’s enrollment is 1520
students.
12
52
(b) Non-face cards = 52 − 12 = 40
12
40
(b)
194
320 students
4
=
1520 students 19
Chapter 5
16.
3.8 m 3.8 × 10 38 19
=
=
=
2.4 m 2.4 × 10 24 12
17.
12 1
=
60 5
30. (a)
(b)
18. Total smokers = 12 + 60 = 72
Total personnel = 32 + 115 = 147
72 24
=
147 49
20 hot dogs 4 hot dogs
=
45 min
9 min
20.
4 mi
2 mi
=
34 min 17 min
21.
130,000 tons 650 tons
=
1800 ft
9 ft
22.
9460 crimes
473 crimes
=
100,000 people 5000 people
1 1 hr
3
=
$4.19
= $0.175/oz
24 oz
(b)
$6.99
= $0.159/oz
44 oz
The 44-oz jar is the better buy.
32. $5.99 − $2.00 = $3.99
$3.99
≈ $0.499 per oz
8 oz
0.4987
8 3.9900
−3 2
79
− 72
70
− 64
60
− 56
4
44 mi
4
3
11
hr
= 44 ⋅
33.
3
mi/hr = 33 mi/hr or mph
4
1
25.
14°
= 4° per hour
3.5 hr
26.
2700 times
= 90 times/sec
30 sec
27.
66 min
= 11 min/lawn
6 lawns
28.
$5.99
= $0.599 per oz
10 oz
29.
$20.00
= $6.667 per towel
3 towels
$12.59
= $0.280/oz
32 oz
31. (a)
23. All unit rates have a denominator of 1, and
reduced rates may not.
24. 44 mi
$8.39
= $0.262/oz
32 oz
The 32-oz bottle is the better buy.
Section 5.2
19.
Review Exercises
195
$7.45
≈ $1.24 per roll
6 rolls
1.241
6 7.450
−6
14
−12
25
−24
10
−6
4
$1.24 − 99¢ = $1.24 − $0.99 = $0.25
The difference is about $0.25 or 25¢ per
roll.
Chapter 5
34.
Ratio and Proportion
15.06 in.
= 0.6275 in./hr
24 hr
0.6275
24 15.0600
−144
66
−48
180
−168
120
−120
0
42.
64 ? 8
=
81 9
43.
?
(64)(9) = (81)(8)
576 ≠ 648
No
3 12 ? 7
=
7 14
44.
?
(3 12 ) (14) = (7)(7)
35. (a) 250,000 −130,000 = 120,000
There was an increase of 120,000
hybrid vehicles.
(b)
2 in. 6 in.
=
5 mi 15 mi
7
7 14 ?
⋅
= 49
2 1
120,000
= 10,000
12
There will be 10,000 additional hybrid
vehicles per month.
1
49 = 49
Yes
36. (a) 449 − 386 = 63
There was an increase of 63 lb.
5.2 ? 15.6
=
3
9
45.
?
(b)
(5.2)(9) = (3)(15.6)
46.8 = 46.8
Yes
63
= 3.5
18
Americans increased the amount of
vegetables in their diet by 3.5 lb per
year.
?
Section 5.3
37.
38.
39.
40.
6 ? 6.3
=
10 10.3
46.
(6)(10.3) = (10)(6.3)
61.8 ≠ 63
No
16 12
=
14 10 12
2 18 ? 3 52
=
4 34 7 53
47.
8
6
=
20 15
 1  3  ?  3  2 
 2  7  =  4  3 
 8  5   4  5 
5 10
=
3 6
19
17 38 ? 19 17
⋅
= ⋅
4 5
8 5
4 20
=
3 15
4
323 323
=
20
20
$11 $88
41.
=
1 hr 8 hr
Yes
196
Chapter 5
48.
49.
5 12 ? 6 12
=
6
7
?
 1
 1
 5  (7) = (6)  6 
2
 
 2
?
11 7 6 13
⋅ = ⋅
2 1 1 2
77 78
≠
2
2
No
53.
1 76
=
13
21
b
 6
 1  (21) = 13b
 7
3
13 21
= 13b
⋅
7 1
1
39 = 13b
39 13b
=
13 13
3=b
4.25 ? 5.25
=
8
10
?
(4.25)(10) = (8)(5.25)
42.5 ≠ 42
No
50.
54.
 1
 1
 9  p =  6  (3)
 2
 3
12.4 ? 3.1
=
9.2
2.3
1
?
19
19 3
p= ⋅
2
3 1
(12.4)(2.3) = (9.2)(3.1)
28.52 = 28.52
Yes
51.
52.
p
3
= 1
1
63 92
1
19
p = 19
2
2 19
2
⋅ p = ⋅ 19
19 2
19
p=2
100 25
=
x
16
100 x = (16)(25)
100 x = 400
100 x 400
=
100 100
x=4
55.
y 45
=
6 10
10 y = (6)(45)
10 y = 270
10 y 270
=
10
10
y = 27
56.
197
2.5 5
=
6.8 h
2.5h = (6.8)(5)
2.5h = 34
2.5h 34
=
2.5 2.5
h = 13.6
0.3 k
=
1.2 3.6
(0.3)(3.6) = 1.2k
1.08 = 1.2k
1.08 1.2k
=
1.2
1.2
0.9 = k
Review Exercises
Chapter 5
Ratio and Proportion
Section 5.4
61.
57. Let x represent the number of human
years.
1 dog year
12 dog years
=
7 human years x human years
1x = (7)(12)
x = 84
The human equivalent is 84 years.
62.
58. Let x represent the number of yen.
9500 yen x yen
=
$100
$450
(9500)(450) = 100 x
4, 275,000 = 100 x
4, 275,000 100 x
=
100
100
42,750 = x
Lavu can buy 42,750 yen.
x
40
=
13.5 54
54 x = (13.5)(40)
54 x = 540
54 x 540
=
54
54
x = 10 in.
y 54
=
46 40
40 y = (46)(54)
40 y = 2484
40 y 2484
=
40
40
y = 62.1 in.
h 6
=
1 34
3
h = (1)(6)
4
3
h=6
4
4 3
4
⋅ h = ⋅6
3 4
3
24
h=
3
h=8
The building is 8 m high.
59. Let x represent the population of Alabama.
59,800 births
13
=
x people
1000
(59,800)(1000) = 13x
59,800,000 = 13x
59,800,000 13x
=
13
13
4,600,000 = x
Alabama had approximately 4,600,000
people.
63.
64.
60. Let x represent the amount of tax.
$25.00 item $145.00 item
=
$1.20 tax
$x tax
25 x = (1.2)(145)
25 x = 174
25 x 174
=
25
25
x = 6.96
The tax would be $6.96.
198
x 2
=
4 5
5 x = (4)(2)
5x = 8
5x 8
=
5 5
x = 1.6 yd
2
y
=
4.5 5
5 y = (4.5)(2)
5y = 9
5y 9
=
5 5
y = 1.8 yd
x 23.4
=
12
26
26 x = (12)(23.4)
26 x = 280.8
26 x 280.8
=
26
26
x = 10.8 cm
16
y
=
27 14.4
14.4 y = (27)(16)
14.4 y = 432
14.4 y 432
=
14.4 14.4
y = 30 cm
Chapter 5
Chapter 5
Test
1. 25 to 521, 25 : 521,
25
521
11.
17
23
(b) Non-sailboats = 23 − 17 = 6
17
6
2. (a)
3.
4
2
=
30 15
4.
22 11
=
12 6
5.
65 cm 5
=
104 cm 8
12.
13.
72
9
=
1000 125
(c) The poverty ratio was greater in New
Mexico.
=
1
2
3
2
1
min
2
1 3 1 2 1
÷ = ⋅ =
2 2 2 3 3
104.8 oz
≈ 2.29 oz/lb
45.8 lb
45.8 104.8
$6.72
= $0.22 per oz
30 oz
1
15.
1
(b) 1 min = 90 sec
2
30 1
=
90 3
8.
= 21.45 g/cm3
14. 2 packs = 6 rings
$3.00
= $0.50 per ring
6 rings
1
=
100 cm3
0.224
30 6.720
−6 0
72
− 60
120
− 120
0
(b)
7. (a) 30 sec =
2145 g
2.288
458 1048.000
−916
132 0
−91 6
40 40
−36 64
3 760
−3 664
96
168
21
6. (a)
=
1000 125
1
2
1 12
Test
255 mi 85 mi
=
2 hr
6 hr
$10.99
≈ $0.044/caplet
250 caplets
$4.29
= $0.179/capsule
24 capsules
The generic pain reliever is the better buy.
16. They form equal ratios or rates.
20 lb
10 lb
9.
=
6 weeks 3 weeks
17.
42 28
=
15 10
4g
1g
10.
=
8 cookies 2 cookies
18.
20 pages 30 pages
=
18 min
12 min
199
Chapter 5
19.
Ratio and Proportion
25. Let x represent the time to download the
file.
1.6 MB 4.8 MB
=
2.5 min
x min
1.6 x = (2.5)(4.8)
1.6 x = 12
1.6 x 12
=
1.6 1.6
x = 7.5
It will take 7.5 min.
$15 $75
=
1 hr 5 hr
105 ? 21
=
55 10
20.
?
(105)(10) = (55)(21)
1050 ≠ 1155
No
21.
22.
23.
25 45
=
p 63
(25)(63) = 45 p
1575 = 45 p
1575 45 p
=
45
45
35 = p
26. Let x represent the number of hours spent
on homework.
x hr
7.5 hr
=
3 cr hr 12 cr hr
(7.5)(12) = 3x
720 = 3x
90 3x
=
3
3
30 = x
Cherise spends 30 hr each week on
homework outside of class.
32 20
=
20 x
32 x = (20)(20)
32 x = 400
32 x 400
=
32
32
x = 12.5
27. Let x represent the number of goldfish.
8 3
=
x 10
(8)(10) = 3 x
80 = 3 x
80 3 x
=
3
3
27 ≈ x
There are approximately 27 fish in her
pond.
1
n 33
=
9 6
 1
6n = (9)  3 
 3
3
6n =
9 10
⋅
1 3
1
28.
6n = 30
6n 30
=
6
6
n=5
24.
y
7.2
=
14 16.8
16.8 y = (14)(7.2)
16.8 y = 100.8
16.8 y 100.8
=
16.8
16.8
y=6
200
x 1
=
6 4
4 x = (6)(1)
4x = 6
4x 6
=
4 4
3
1
x = or 1 mi
2
2
1 2
=
4 y
1 y = (2)(4)
y = 8 mi
Chapter 5
29.
x 24
=
10 15
15x = (10)(24)
15x = 240
15x 240
=
15
15
x = 16 cm
Chapters 1–5
Cumulative Review Exercises
1. Five hundred three thousand, forty-two
2.
7.
251
300
492 →
500
+ 631
+ 600
1400
8.
245 35 ⋅ 7 7
=
=
175 35 ⋅ 5 5
9.
13 3 39
⋅ =
2 7 14
Approximately 1400
3. 226 × 100,000 = 22,600,000
22
4. 16 355
−32
35
−32
3
2
3
 3  3  9
10.   =    =
5
 5  5  25
22 R 3
11.
22.1875
5. 16 355.0000
−32
35
−32
30
−16
140
−128
120
−112
80
−80
0
1
6 3
1
(6) = = = 1
4
4 2
2
1
1
6 −1 = 4
2
2
1
Bruce has 4 in. of sandwich left.
2
1
12.
 7 3 5  7 4  5
 8 ÷ 4  + 6 =  8 ⋅ 3  + 6
2
7 5 12
= + =
=2
6 6 6
6. 22 × (32 − 11) ÷ 14 = 22 × 21 ÷ 14
= 4 × 21 ÷ 14
= 84 ÷ 14
=6
201
13.
8
8 27 35
+3= +
=
9
9 9
9
14.
9 0 9
9
− = −0=
13 3 13
13
Test
Chapter 5
Ratio and Proportion
21. 43.923 × 100 = 4392.3
1
1 33 5
15. Longer walls: 8 − 2 =
−
4
2 4 2
33 10 23
=
−
=
ft
4
4
4
1
3 13 11
Wall with door: 4 − 2 = −
3
4 3 4
52 33 19
=
−
=
ft
12 12 12
23 23 19 69 69 19
+ + =
+
+
4
4 12 12 12 12
157
1
=
or 13
12
12
1
Emil needs 13
ft of wallpaper border.
12
16.
22. 237.9 ÷ 100 = 2.379
23.
3  1  3 329 987
11
=
or 61
 82  = ⋅
4 4 4 4
16
16
1
11 329 987
−
82 − 61 =
4
16
4
16
1316 987
=
−
16
16
329
9
=
or 20
16
16
11
9
It sold 61 acres, and 20
acres are
16
16
left.
24.
61
; 61 : 44
44
25.
1950 13
=
150
1
26.
27.
1, 200,000
9600 mi
2
= 125 people/mi 2
7.5 ? 9
=
10 12
?
(7.5)(12) = (10)(9)
90 = 90
Yes
(b)
18. One thousand four and seven hundred one
thousandths
31 ? 33
=
5
6
?
(31)(6) = (5)(33)
186 ≠ 165
No
23.880
+ 11.300
35.180
− 7.123
28.057
20. 4.36 =
7
840
=
6000 50
Approximately 7 out of 50 deaths are due
to cancer.
28. (a)
5 6 × 9 + 5 59
17. 6 =
=
9
9
9
There are 59 ninths.
19.
29.20
10.75
30.50
34.20
+ 26.25
130.90 cm
29.
436 109
=
100 25
202
13
5
=
11.7 x
13x = (11.7)(5)
13 x = 58.5
13x 58.5
=
13
13
x = 4.5
Chapters 1–5
30. Let x represent the distance Jim can travel.
150 mi x mi
=
6 gal
4 gal
(150)(4) = 6 x
600 = 6 x
600 6 x
=
6
6
100 = x
Jim can drive 100 mi on 4 gal.
203
Cumulative Review Exercises
Chapter 6
Percents
Chapter Opener Puzzle
Section 6.1
Percents and Their Fraction and Decimal Forms
Section 6.1 Practice Exercises
1. percent
2. 18% =
18
100
10.
5
= 5%
100
11.
70
= 70%
100
12.
26
= 26%
100
18 students earned an A.
3.
48
= 48%
100
1
(or ÷ 100).
100
Then reduce the fraction to lowest terms.
13. Replace the symbol % by ×
84
4.
= 84%
100
5.
50
= 50%
100
14. 50% = 50 ×
6.
10
= 10%
100
15. 3% = 3 ×
1
3
=
100 100
7.
25
= 25%
100
16. 7% = 7 ×
8.
75
= 75%
100
1
7
=
100 100
17. 84% = 84 ×
9.
$2
= 2%
$100
1
84 21
=
=
100 100 25
18. 32% = 32 ×
1
32
8
=
=
100 100 25
204
1
50 1
=
=
100 100 2
Section 6.1
Percents and Their Fraction and Decimal Forms
19. 25% = 25 ×
1
25 1
=
=
100 100 4
1
25 1
25
1
34. 6 % = ×
=
=
4
4 100 400 16
20. 20% = 20 ×
1
20 1
=
=
100 100 5
35. Replace the % symbol by × 0.01 (or ÷ 100).
36. 58% = 58 × 0.01 = 0.58
1
3.4
34
17
=
=
=
100 100 1000 500
21. 3.4% = 3.4 ×
37. 72% = 72 × 0.01 = 0.72
38. 15% = 15 × 0.01 = 0.15
1
5.2
52
13
22. 5.2% = 5.2 ×
=
=
=
100 100 1000 250
39. 66% = 66 × 0.01 = 0.66
23. 115% = 115 ×
1
115 23
3
=
=
or 1
100 100 20
20
40. 8.5% = 8.5 × 0.01 = 0.085
24. 150% = 150 ×
1
150 3
1
=
= or 1
100 100 2
2
42. 72.31% = 72.31 × 0.01 = 0.7231
41. 12.9% = 12.9 × 0.01 = 0.129
43. 41.05% = 41.05 × 0.01 = 0.4105
1
175 7
3
25. 175% = 175 ×
=
= or 1
100 100 4
4
26. 120% = 120 ×
27. 0.5% = 0.5 ×
44. 142% = 142 × 0.01 = 1.42
45. 201% = 201 × 0.01 = 2.01
1
120 6
1
=
= or 1
100 100 5
5
46. 0.55% = 0.55 × 0.01 = 0.0055
1
0.5
5
1
=
=
=
100 100 1000 200
47. 0.75% = 0.75 × 0.01 = 0.0075
2
4
48. 26 % = 26 × 0.01 = 26.4 × 0.01 = 0.264
5
10
1
0.2
2
1
28. 0.2% = 0.2 ×
=
=
=
100 100 1000 500
1
49. 16 % = 16.25 × 0.01 = 0.1625
4
1
0.25
25
=
=
29. 0.25% = 0.25 ×
100 100 10,000
1
=
400
30. 0.75% = 0.75 ×
=
50. 55
1
0.75
75
=
=
100 100 10,000
1
5
% = 55
× 0.01 = 55.05 × 0.01
20
100
= 0.5505
1
2
51. 62 % = 62 × 0.01 = 62.2 × 0.01 = 0.622
5
10
3
400
2
200 1
200 2
31. 66 % =
×
=
=
3
3 100 300 3
1
31 1
31
32. 5 % = ×
=
6
6 100 600
1
49 1
49
33. 24 % = ×
=
2
2 100 200
205
52.
1 50
=
= 50%
2 100
53.
1 25
=
= 25%
4 100
54.
3 75
=
= 75%
4 100
Chapter 6
Percents
55.
1
= 100%
1
70. Move the decimal point 2 places to the
left.
56.
9 225
=
= 225%
4 100
57.
3 150
=
= 150%
2 100
71. 7.6% = 7.6 × 0.01 = 0.076
1
76
1
7.6% = 7.6 ×
=
×
100 10 100
76
19
=
=
1000 250
72. 75% = 76 × 0.01 = 0.75
1
75 3
75% = 75 ×
=
=
100 100 4
2
200 1
200 2
58. 66 % =
×
=
=
3
3 100 300 3
c
59. 10% = 10 ×
1
10
1
=
=
100 100 10
73. 4.3% = 4.3 × 0.01 = 0.043
1
43 1
43
4.3% = 4.3 ×
= ×
=
100 10 100 1000
1
90
9
=
=
100 100 10
74. 5.8% = 5.8 × 0.01 = 0.058
1
58 1
58
29
5.8% = 5.8 ×
= ×
=
=
100 10 100 1000 500
1
75 3
=
=
100 100 4
75. 2% = 2 × 0.01 = 0.02
1
2
1
2% = 2 ×
=
=
100 100 50
1
25 1
=
=
100 100 4
76. 18.2% = 18.2 × 0.01 = 0.182
1
182
1
18.2% = 18.2 ×
=
×
100 10 100
182
91
=
=
1000 500
d
60. 90% = 90 ×
e
61. 75% = 75 ×
b
62. 25% = 25 ×
f
63. 150% = 150 ×
1
150 3
=
=
100 100 2
a
64. 30% = 30 × 0.01 = 0.30
e
77. 35% = 35 × 0.01 = 0.35
1
35
7
=
=
35% = 35 ×
100 100 20
1
65. 33 % = 33.3 × 0.01 = 0.3
3
d
78. 29% = 29 × 0.01 = 0.29
1
29
29% = 29 ×
=
100 100
66. 125% = 125 × 0.01 = 1.25
f
67. 50% = 50 × 0.01 = 0.50
b
68. 1% = 1 × 0.01 = 0.01
a
69. 80% = 80 × 0.01 = 0.80
c
206
Section 6.1
Percents and Their Fraction and Decimal Forms
79. 40% = 40 × 0.01 = 0.4
1
40 2
40% = 40 ×
=
=
100 100 5
42.5% = 42.5 × 0.01 = 0.425
1
425
1
42.5% = 42.5 ×
=
×
100 10 100
425 17
=
=
1000 40
42% = 42 × 0.01 = 0.42
1
42 21
42% = 42 ×
=
=
100 100 50
47.8% = 47.8 × 0.01 = 0.478
1
478
1
47.8% = 47.8 ×
=
×
100 10 100
478 239
=
=
1000 500
48.0% = 48.0 × 0.01 = 0.48
1
48 12
48.0% = 48.0 ×
=
=
100 100 25
48.8% = 48.8 × 0.01 = 0.488
1
488
1
48.8% = 48.8 ×
=
×
100 10 100
488
61
=
=
1000 125
59% = 59 × 0.01 = 0.59
1
59
59% = 59 ×
=
100 100
73% = 73 × 0.01 = 0.73
1
73
73% = 73 ×
=
100 100
80. 41.2% = 41.2 × 0.01 = 0.412
1
412
1
41.2% = 41.2 ×
=
×
100 10 100
412 103
=
=
1000 250
Section 6.2
Fractions and Decimals and Their Percent Forms
Section 6.2 Practice Exercises
1. (a) False; 5% = 5 ×
6. 80% = 80 × 0.01 = 0.8
1
5
1
=
=
100 100 20
(b) True; 10% = 10 ×
1
7. 6 % = 6.3 × 0.01 = 0.063
3
1
10
1
=
=
100 100 10
8. 143% = 143 × 0.01 = 1.43
(c) True;
200% = 200 ×
2. 60% = 60 ×
9. 0.3% = 0.3 × 0.01 = 0.003
1
200 2
=
= =2
100 100 1
10. 0.68 × 100% = 68%
1
60 3
=
=
100 100 5
11. 1.62 × 100% = 162%
12. 0.005 × 100% = 0.5%
1
130 13
3
3. 130% = 130 ×
=
=
or 1
100 100 10
10
13. 0.26 × 100% = 26%
14. When multiplying a decimal by 100, move
the decimal point 2 places to the right.
1
33 1
33
4. 16 % = ×
=
2
2 100 200
5. 0.5% = 0.5 ×
15.
1
5
1
5
1
= ×
=
=
100 10 100 1000 200
207
5
5 100
500
× 100% = ×
%=
% = 125%
4
4 1
4
Chapter 6
16.
Percents
5
2
2 100
200
× 100% = ×
%=
% = 40%
5
5 1
5
34.
1
1
77
77 100
17.
× 100% =
×
% = 77%
100
1
100
35.
1
1
113
113 100
18.
×
% = 113%
× 100% =
1
100
100
1
36.
19. 0.27 = 0.27 × 100% = 27%
20. 0.51 = 0.51 × 100% = 51%
21. 0.19 = 0.19 × 100% = 19%
37.
22. 0.33 = 0.33 × 100% = 33%
23. 1.75 = 1.75 × 100% = 175%
24. 2.8 = 2.8 × 100% = 280%
38.
25. 0.124 = 0.124 × 100% = 12.4%
26. 0.277 = 0.277 × 100% = 27.7%
27. 0.006 = 0.006 × 100% = 0.6%
39.
28. 0.0008 = 0.0008 × 100% = 0.08%
29. 1.014 = 1.014 × 100% = 101.4%
30. 2.203 = 2.203 × 100% = 220.3%
40.
1
31.
7
7
7
100
% = 35%
=
× 100% =
×
1
20 20
20
71
71
71 100
%
=
× 100% =
×
1
100 100
100
7 100
700
7 7
= × 100% = ×
%=
%
8
1
8
8 8
1
= 87.5 or 87 %
2
5 100
500
5 5
= × 100% = ×
%=
%
8
1
8
8 8
1
= 62.5% or 62 %
2
13 100
1300
13 13
= × 100% = ×
%=
%
16
1
16
16 16
1
= 81.25% or 81 %
4
11 100
1100
11 11
= × 100% = ×
%=
%
16
1
16
16 16
3
= 68.75% or 68 %
4
5 100
500
5 5
= × 100% = ×
%=
%
6
1
6
6 6
1
= 83.3% or 83 %
3
5
5 100
500
5
= × 100% = ×
%=
%
12
1
12
12 12
2
= 41.6% or 41 %
3
1
= 71%
41.
1
32. 89 = 89 × 100% = 89 × 100 %
1
100 100
100
1
= 89%
42.
5
33. 19 = 19 × 100% = 19 × 100 % = 95%
20 20
1
20
1
43.
208
4 100
400
4 4
= × 100% = ×
%=
%
9
1
9
9 9
4
= 44.4% or 44 %
9
1 100
100
1 1
= × 100% = ×
%=
%
9
1
9
9 9
1
= 11. 1% or 11 %
9
1 1
1 100
100
= × 100% = ×
%=
% = 25%
4 4
4 1
4
Section 6.2
Fractions and Decimals and Their Percent Forms
1
5 14
14 100
1400
54. 1 = × 100% = ×
%=
%
9 9
9
1
9
5
= 155.5% or 155 %
9
37
37
37 100
×
% = 37%
=
× 100% =
44.
100 100
1
100
1
45.
1
1
1 100
100
= × 100% = ×
%=
%
10 10
10
1
10
= 10%
2 5
5 100
500
55. 1 = × 100% = ×
%=
%
3 3
3 1
3
2
= 166.6% or 166 %
3
2
3
3
100
3
=
× 100% =
×
% = 6%
46.
1
50 50
50
56.
1
47.
2 100
200
2 2
= × 100% = ×
%=
%
3
1
3
3 3
2
= 66.6% or 66 %
3
57.
1 1
1 100
100
48. 8 = 8 × 100% = 8 × 1 % = 8 %
= 12.5%
25
7 100
700
7 7
= × 100% = ×
%=
%
6
1
6
6 6
2
= 116.6% or 116 %
3
3 3
= × 100%
7 7
3 100
= ×
%
7 1
300
=
%
7
≈ 42.9%
42.85
7 300.00
−28
20
−14
60
−5 6
40
−35
5
6 6
= × 100%
7 7
6 100
= ×
%
7 1
600
=
%
7
≈ 85.7%
85.71
7 600.00
−56
40
−35
50
−4 9
10
−7
3
1 1
= × 100%
13 13
1 100
= ×
%
13 1
100
=
%
13
≈ 7.7%
7.69
13 100.00
−91
90
−7 8
1 20
−1 17
3
3 7
7 100
% = 175%
49. 1 = × 100% = ×
4 4
4
1
1
58.
50
50.
7 7
7 100
% = 350%
= × 100% = ×
1
2 2
2
1
5
27 27
27 100
% = 135%
=
× 100% =
×
51.
1
20 20
20
1
1 17
17 100
1700
52. 2 =
× 100% =
×
%=
%
8 8
8
1
8
1
= 212.5% or 212 %
2
53.
59.
11 100
1100
11 11
= × 100% = ×
%=
%
9
9
1
9
9
2
= 122.2% or 122 %
9
209
Chapter 6
60.
61.
Percents
3 3
= × 100%
13 13
3 100
= ×
%
13 1
300
=
%
13
≈ 23.1%
23.07
13 300.00
−26
40
−39
1 00
−91
9
5 5
= × 100%
11 11
5 100
%
= ×
11 1
500
=
×
11
≈ 45.5%
45.45
11 500.00
−44
60
−55
50
−4 4
60
−55
5
8 8
= × 100%
62.
11 11
8 100
%
= ×
11 1
800
=
%
11
≈ 72.7%
63.
64.
13 13
= × 100%
15 15
13 100
= ×
%
15 1
1300
=
%
15
≈ 86.7%
1
1
= × 100%
15 15
1 100
= ×
%
15 1
100
=
%
15
≈ 6.7%
65. The fraction
1
= 0.5 and
2
1
% = 0.5% = 0.005.
2
66. The fraction
3
= 0.75 and
4
3
% = 0.75% = 0.0075.
4
67. 25% = 0.25 and 0.25% = 0.0025.
68. 10% = 0.1 and 0.10% = 0.1% = 0.001.
69. a, c
70. c, d
71. a, c
72. b, c
72.72
11 800.00
−77
30
−22
80
−7 7
30
−22
8
73. (a)
(b)
(c)
86.66
15 1300.00
−120
100
−90
10 0
−9 0
1 00
−90
10
(d)
(e)
1 25
=
= 0.25
4 100
1
100
× 100% =
% = 25%
4
4
92 23
0.92 =
=
100 25
0.92 × 100% = 92%
1
15
3
15% = 15 ×
=
=
100 100 20
15% = 15 × 0.01 = 0.15
16 8
3
1.6 = = or 1
10 5
5
1.6 × 100% = 160%
1
= 0.01
100
1
= 1%
100
(f) 0.5% = 0.5 ×
6.66
15 100.00
−90
10 0
−9 0
1 00
−90
10
1
5
1
5
=
×
=
100 10 100 1000
1
200
0.5% = 0.5 × 0.01 = 0.005
=
210
Section 6.2
Fractions and Decimals and Their Percent Forms
1
100
6
1
= ×
10 100
6
=
1000
3
=
500
0.6% = 0.6 × 0.01 = 0.006
2 4
= = 0.4
5 10
2 2
200
= × 100% =
% = 40%
5 5
5
2
2 = or 2
1
2 = 2 × 100% = 200%
1 5
= = 0.5
2 10
1 1
100
= × 100% =
% = 50%
2 2
2
12
3
0.12 =
=
100 25
0.12 = 0.12 × 100% = 12%
1
45
9
45% = 45 ×
=
=
100 100 20
45% = 45 × 0.01 = 0.45
74. (a) 0.6% = 0.6 ×
(b)
(c)
(d)
(e)
(f)
(e) 0.2% = 0.2 ×
1
2
1
2
= ×
=
100 10 100 1000
1
500
0.2% = 0.2 × 0.01 = 0.002
19 95
(f)
=
= 0.95
20 100
=
5
19 100
19 19
=
× 100% =
×
%
20 20
1
20
1
= 95%
3
13
or
10
10
1.3 = 1.3 × 100% = 130%
1
22 11
(b) 22% = 22 ×
=
=
100 100 50
22% = 22 × 0.01 = 0.22
3 75
(c)
=
= 0.75
4 100
3 3
300
= × 100% =
% = 75%
4 4
4
73
(d) 0.73 =
100
0.73 = 0.73 × 100% = 73%
76. (a) 1.3 = 1
2
200
(e) 22.2% = 22 % =
%
9
9
1
14
7
=
=
100 100 50
14% = 14 × 0.01 = 0.14
87
(b) 0.87 =
100
0.87 = 0.87 × 100% = 87%
1
(c) 1 = or 1
1
1 = 1 × 100% = 100%
1
(d) = 0.3
3
100
1 1
= × 100% =
%
3
3 3
1
= 33.3% or 33 %
3
75. (a) 14% = 14 ×
2
200
1
2
=
×
=
9
100 9
1
22.2% = 22.2 × 0.01 = 0.2
1
5
(f)
=
= 0.05
20 100
1
1
100
=
× 100% =
% = 5%
20 20
20
77. 1.4 = 1.4 × 100% = 140%
1.4 > 100%
78. 0.0087 = 0.0087 × 100% = 0.87%
0.0087 < 1%
211
Chapter 6
Percents
79. 0.052 = 0.052 × 100% = 5.2%
0.052 < 50%
Section 6.3
80. 25 = 25 × 100% = 2500%
25 > 25%
Percent Proportions and Applications
Section 6.3 Practice Exercises
1. (a) percent
15. Yes, because
(b) cross
50
= 50%.
100
50
= 50%.
100
2. 0.55 = 0.55 × 100% = 55%
16. No, because
3. 1.30 = 1.30 × 100% = 130%
17. No, because no denominator is 100.
4. 0.0006 × 100% = 0.06%
5.
6.
7.
18. Yes, because
3 100
300
3 3
= × 100% = ×
%=
%
8
1
8
8 8
1
= 37.5% or 37 %
2
3
20. 24
1
1 100
1
=
× 100% =
×
%
100
1
100 100
100
% = 1%
=
100
21. 45
1
125
1
125 5
8. 62 % =
×
=
=
2
2 100 200 8
22. 36
1
2
1
9. 2% = 2 ×
=
=
100 100 50
1
77
=
100 100
23. 32
11. 82% = 82 × 0.01 = 0.82
12. 0.3% = 0.3 × 0.01 = 0.003
13. 100% = 100 × 0.01 = 1
14. Yes, because
1
= 1 %.
100
2
3
19. Yes, because 4 = %.
100 4
5 5
5 100
500
= × 100% = ×
%=
%
2 2
2
1
2
= 250%
10. 77% = 77 ×
1 12
7
= 7%.
100
212
Section 6.3
Percent Proportions and Applications
35. Amount: 21,684
base: 20,850
p = 104
104 21,684
=
100 20,850
24. 3
36. Amount: 41,200
base: 40,000
p = 103
103 41, 200
=
100 40,000
25. Amount: 12
base: 20
p = 60
26. Amount: 100
base: 400
p = 25
37. Amount: x
base: 200
p = 54
54
x
=
100 200
100 x = (54)(200)
100 x = 10,800
100 x 10,800
=
100
100
x = 108 employees
27. Amount: 99
base: 200
p = 49.5
28. Amount: 45
base: 50
p = 90
29. Amount: 50
base: 40
p = 125
38. Amount: x
base: 412
p = 35
35
x
=
100 412
100 x = (35)(412)
100 x = 14, 420
100 x 14, 420
=
100
100
x = 144.2
30. Amount: 3.5
base: 2
p = 175
31. Amount: 12
base: 120
p = 10
10
12
=
100 120
32. Amount: 3
base: 20
p = 15
15
3
=
100 20
39. Amount: x
base: 40
1
p=
2
1
2
x
100 40
1
100 x =   (40)
2
100 x = 20
100 x 20
=
100 100
x = 0.2
33. Amount: 72
base: 90
p = 80
80 72
=
100 90
34. Amount: 21
base: 105
p = 20
20
21
=
100 105
213
=
Chapter 6
Percents
40. Amount: x
base: 900
p = 1.8
1.8
x
=
100 900
100 x = (1.8)(900)
100 x = 1620
100 x 1620
=
100
100
x = 16.2 g
115
x
=
100 19,000
100 x = (115)(19,000)
100 x = 2,185,000
100 x 2,185,000
=
100
100
x = 21,850
The sticker price is $21,850.
45. Let x represent the approximate number of
teens not wearing seat belts.
base: 304
p = 72
72
x
=
100 304
100 x = (72)(304)
100 x = 21,888
100 x 21,888
=
100
100
x ≈ 219
41. Amount: x
base: 500
p = 112
112
x
=
100 500
100 x = (112)(500)
100 x = 56,000
100 x 56,000
=
100
100
x = 560
Approximately 219 of the 304 teens were
not wearing seat belts.
42. Amount: x
base: 1050
p = 106
106
x
=
100 1050
100 x = (106)(1050)
100 x = 111,300
100 x 111,300
=
100
100
x = 1113
46. Let x represent the number of freshmen.
base: 42
p = 61.9
61.9 x
=
100 42
100 x = (61.9)(42)
100 x = 2599.8
100 x 2599.8
=
100
100
x = 25.998
There are approximately 26 freshmen.
43. Let x represent the amount Pedro pays in
taxes.
base: 72,000
p = 28
28
x
=
100 72,000
100 x = (28)(72,000)
100 x = 2,016,000
100 x 2,016,000
=
100
100
x = 20,160
Pedro pays $20,160 in taxes.
47. Amount: 18
base: x
p = 50
50 18
=
100 x
50 x = (100)(18)
50 x = 1800
50 x 1800
=
50
50
x = 36
44. Let x represent the sticker price.
base: 19,000
p = 115
214
Section 6.3
Percent Proportions and Applications
48. Amount: 44
base: x
p = 22
22 44
=
100 x
22 x = (44)(100)
22 x = 4400
22 x 4400
=
22
22
x = 200 ft
52. Amount: 9.5
base: x
p = 200
200 9.5
=
100
x
200 x = (100)(9.5)
200 x = 950
200 x 950
=
200 200
x = 4.75
49. Amount: 69
base: x
p = 30
30 69
=
100 x
30 x = (100)(69)
30 x = 6900
30 x 6900
=
30
30
x = 230 lb
53. Let x represent Albert’s monthly income.
amount: 120
p = 7.5
7.5 120
=
100
x
7.5 x = (100)(120)
7.5 x = 12,000
7.5 x 12,000
=
7.5
7.5
x = 1600
Albert makes $1600 per month.
50. Amount: 28
base: x
p = 70
70 28
=
100 x
70 x = (100)(28)
70 x = 2800
70 x 2800
=
70
70
x = 40
54. Let x represent the total distance.
amount: 56
p = 80
80 56
=
100 x
80 x = (100)(56)
80 x = 5600
80 x 5600
=
80
80
x = 70
The total distance is 70 mi.
51. Amount: 9
base: x
2
p=
3
2
3
=
55. Let x represent the total number of emails.
amount: 14
p = 40
40 14
=
100 x
40 x = (100)(14)
40 x = 1400
40 x 1400
=
40
40
x = 35
Amiee has a total of 35 e-mails.
9
x
100
2
x = (100)(9)
3
2
x = 900
3
3 2
3 900
⋅ x= ⋅
2 3
2 1
2700
x=
2
x = 1350
215
Chapter 6
Percents
56. Let x represent the population of
Charlotte, North Carolina.
amount: 32,000
p=5
5
32,000
=
100
x
5 x = (100)(32,000)
5 x = 3, 200,000
5 x 3, 200,000
=
5
5
x = 640,000
The population is approximately 640,000.
60. Amount: 4
base: 12
p unknown
4
p
=
100 12
12 p = (100)(4)
12 p = 400
12 p 400
=
12
12
p = 33 1
3
33 1 % of 12 letters is 4 letters.
3
57. Amount: 42
base: 120
p unknown
p
42
=
100 120
120 p = (100)(42)
120 p = 4200
120 p 4200
=
120
120
p = 35
35% of $120 is $42.
61. Amount: 280
base: 320
p unknown
p
280
=
100 320
320 p = (100)(280)
320 p = 28,000
320 p 28,000
=
320
320
p = 87.5
87.5% of 320 mi is 280 mi.
58. Amount: 112
base: 400
p unknown
p 112
=
100 400
400 p = (100)(112)
400 p = 11, 200
400 p 11, 200
=
400
400
p = 28
28% of 400 is 112.
62. Amount: 54
base: 48
p unknown
p
54
=
100 48
48 p = (100)(54)
48 p = 5400
48 p 5400
=
48
48
p = 112.5
112.5% of 48 is 54.
59. Amount: 84
base: 70
p unknown
p
84
=
100 70
70 p = (100)(84)
70 p = 8400
70 p 8400
=
70
70
p = 120
120% of 70 is 84.
216
Section 6.3
Percent Proportions and Applications
63. Let p represent the percent of the
questions answered correctly.
amount: 29
base: 40
p
29
=
100 40
40 p = (100)(29)
40 p = 2900
40 p 2900
=
40
40
p = 72.5
She answered 72.5% correctly.
67. Amount: 160
base: 600
p unknown
p 160
=
100 600
600 p = (100)(160)
600 p = 16,000
600 p 16,000
=
600
600
p = 26.6
Approximately 26.7% of the officers were
promoted.
64. Let p represent Jeff’s shooting percentage.
amount: 520
base: 1280
p
520
=
100 1280
1280 p = (100)(520)
1280 p = 52,000
1280 p 52,000
=
1280
1280
p = 40.625
Jeff’s shooting percentage was about 41%.
68. Amount: 440
base: 600
p unknown
p
440
=
100 600
600 p = (100)(440)
600 p = 44,000
600 p 44,000
=
600
600
p = 73.3
Approximately 73.3% of the officers were
not promoted.
65. Amount: 120
base: 600
p unknown
p 120
=
100 600
600 p = (100)(120)
600 p = 12,000
600 p 12,000
=
600
600
p = 20
20% of the officers are female.
69. Let x represent the amount of rain that fell
in August.
base: 56
p = 125
125 x
=
100 56
100 x = (125)(56)
100 x = 7000
100 x 7000
=
100
100
x = 70
70 mm of rain fell in August.
66. Amount: 480
base: 600
p unknown
p
480
=
100 600
600 p = (100)(480)
600 p = 48,000
600 p 48,000
=
600
600
p = 80
80% of the officers are male.
217
Chapter 6
Percents
70. Let x represent the number of people
influenced by gas prices.
base: 500
p = 38
38
x
=
100 500
100 x = (38)(500)
100 x = 19,000
100 x 19,000
=
100
100
x = 190
190 people are expected to be influenced.
p 136
=
100 373
373 p = (100)(136)
373 p = 13,600
373 p 13,600
=
373
373
p ≈ 36.5
Terry made approximately 36.5% of his
three-point shots.
74. Let x represent the completion percentage.
amount: 393
base: 571
p
393
=
100 571
571 p = (100)(393)
571 p = 39,300
571 p 39,300
=
571
571
p ≈ 68.8
Manning’s completion percentage is
approximately 68.8%.
71. Let x represent the number of freshmen
admitted.
amount: 209
p = 11
11 209
=
100
x
11x = (100)(209)
11x = 20,900
11x 20,900
=
11
11
x = 1900
Approximately 1900 freshmen were
admitted.
75. (a) Let x represent the number of fiveperson households that own a dog.
base: 200
p = 53
53
x
=
100 200
100 x = (53)(200)
100 x = 10,600
100 x 10,600
=
100
100
x = 106
106 five-person households own a
dog.
72. Let x represent the total area of the park.
amount: 3366
p = 99
99 3366
=
100
x
99 x = (100)(3366)
99 x = 336,600
99 x 336,600
=
99
99
x = 3400
The area of Yellowstone National Park is
3400 mi2 .
73. Let p represent the percent of shots made.
amount: 136
base: 373
218
Section 6.3
(b) Let x represent the number of threeperson households that own a dog.
base: 50
p = 46
46
x
=
100 50
100 x = (46)(50)
100 x = 2300
100 x 2300
=
100
100
x = 23
23 three-person households own a
dog.
Percent Proportions and Applications
38
x
=
100 182
100 x = (38)(182)
100 x = 6916
100 x 6916
=
100
100
x = 69.16
69 were Fords.
79. Let x represent the total vehicles sold.
amount: 27
p = 15
15 27
=
100 x
15 x = (100)(27)
15 x = 2700
15 x 2700
=
15
15
x = 180
There were 180 total vehicles.
76. Let p represent the percent of memory
used.
amount: 7.56
base: 74.4
p
7.56
=
100 74.4
74.4 p = (100)(7.56)
74.4 p = 756
74.4 p 756
=
74.4 74.4
p ≈ 10
10% of the memory is used.
80. Let x represent the total vehicles sold.
amount: 10
p=4
4 10
=
100 x
4 x = (100)(10)
4 x = 1000
4 x 1000
=
4
4
x = 250
There were 250 total vehicles.
77. Let x represent the number of Chevys.
base: 215
p = 34
34
x
=
100 215
100 x = (34)(215)
100 x = 7310
100 x 7310
=
100
100
x = 73.10
73 were Chevys.
78. Let x represent the number of Fords.
base: 182
p = 38
219
Chapter 6
Percents
81. Let x represent the amount spent on
clothes. Let y represent the amount spent
on dinner.
44
x
=
100 600
100 x = (44)(600)
100 x = 26, 400
100 x 26, 400
=
100
100
x = 264
52.00
− 12.48
39.52
1
y = (39.52) = 19.76
2
12.48
+ 19.76
32.24
Melissa spent $32.24.
83. Step 1: $57.65 ≈ $58
Step 2: 10% of 58 is 5.8.
Step 3: 2 × 5.8 = 11.6
A 20% tip is $11.60.
600
− 264
336 left after clothes
20
y
=
100 336
100 y = (20)(336)
100 y = 6720
100 y 6720
=
100
100
y = 67.20
84. Step 1: $18.79 ≈ $19
Step 2: 10% of 19 is 1.9.
Step 3: 2 × 1.9 = 3.8
A 20% tip is $3.80.
85. Step 1: $42 is already a whole dollar
amount.
Step 2: 10% of 42 is 4.2
1
Step 4: (4.2) = 2.1
2
2.1 + 4.2 = 6.3
A 15% tip is $6.30.
264
+ 67.20
331.20
Carson spent $331.20.
86. Step 1: $12 is already a whole dollar
amount.
Step 2: 10% of 12 is 1.2.
1
Step 4: (1.2) = 0.6
2
0.6 + 1.2 = 1.8
A 15% tip is $1.80.
82. Let x represent the amount spent on
makeup. Let y represent the amount spent
on lunch.
24
x
=
100 52
100 x = (24)(52)
100 x = 1248
100 x 1248
=
100
100
x = 12.48
Section 6.4
Percent Equations and Applications
Section 6.4 Practice Exercises
3. 3x = 27
3x 27
=
3
3
x=9
1. Divide both sides of the equation by 26 to
get x = 2.5.
2. Divide both sides of the equation by 6 to
get x = 9.
220
Section 6.4
Percent Equations and Applications
4. 12 x = 48
12 x 48
=
12 12
x=4
13. Let x represent the unknown amount.
x = (0.55%)(900)
x = (0.0055)(900)
x = 4.95
5. 0.15 x = 45
0.15 x
45
=
0.15 0.15
x = 300
14. Let x represent the unknown amount.
x = (0.4%)(75)
x = (0.004)(75)
x = 0.3
15. Let x represent the unknown amount.
x = (133%)(600)
x = (1.33)(600)
x = 798
6. 0.32 x = 60
0.32 x
60
=
0.32 0.32
x = 187.5
16. Let x represent the unknown amount.
x = (120%)(40.4)
x = (1.2)(40.4)
x = 48.48
7. 1.02 x = 841.5
1.02 x 841.5
=
1.02
1.02
x = 825
17. 50% equals one-half of the number. So
divide the number by 2.
8. 1.06 x = 90.1
1.06 x 90.1
=
1.06 1.06
x = 85
18. 10% equals one-tenth of the number. So
divide the number by 10.
19. 2 × 14 = 28
165 693
=
9.
100
x
165 x = (100)(693)
165 x = 69,300
165 x 69,300
=
165
165
x = 420
10.
16
x
=
100 60
(16)(60) = 100 x
960 = 100 x
960 100 x
=
100 100
x = 9.6
20.
3
× 80 = 60
4
21.
1
× 40 = 20
2
22.
1
× 32 = 3.2
10
23. Let x represent the amount of active
ingredient.
x = (6%)(64)
x = (0.06)(64)
x = 3.84
There are 3.84 oz of sodium hypochlorite
in household bleach.
11. Let x represent the unknown amount.
x = (35%)(700)
x = (0.35)(700)
x = 245
24. Let x represent the amount of alcohol.
x = (40%)(12.5)
x = (0.4)(12.5)
x=5
There are 5 L of alcohol in 12.5 L of
antifreeze.
12. Let x represent the unknown amount.
x = (12%)(625)
x = (0.12)(625)
x = 75
221
Chapter 6
Percents
25. Let x represent the number of completed
passes.
x = (60%)(8358)
x = (0.6)(8358) ≈ 5015
Marino completed approximately
5015 passes.
33. Let x represent the number tested.
47 = (0.04) x
47 0.04 x
=
0.04 0.04
1175 = x
There were 1175 subjects tested.
26. Let x represent the minimum number of
questions.
x = (80%)(60)
x = (0.8)(60)
x = 48
She must answer 48 questions correctly to
score 80%.
34. (a) Let x represent the total number of
pages.
8 = (0.80) x
8 0.8 x
=
0.8 0.8
10 = x
There are 10 pages in the paper.
27. Let x represent the base.
18 = (0.40) x
18 0.4 x
=
0.4 0.4
45 = x
(b) 10 − 8 = 2
Ted has 2 pages left to type.
35. Let x represent the total population.
61.6 = (0.22) x
61.6 0.22 x
=
0.22 0.22
280 = x
At that time, the population was about
280 million.
28. Let x represent the base.
72 = (0.30) x
72 0.3 x
=
0.3 0.3
240 = x
36. Let x represent the population last year.
245,300 = (1.10) x
245,300 1.1x
=
1.1
1.1
223,000 = x
The population was 223,000 last year.
29. Let x represent the base.
(0.92) x = 41.4
0.92 x 41.4
=
0.92 0.92
x = 45
30. Let x represent the base.
(0.84) x = 100.8
0.84 x 100.8
=
0.84
0.84
x = 120
37. 0.13 = 0.13 × 100% = 13%
31. Let x represent the base.
3.09 = (1.03) x
3.09 1.03x
=
1.03 1.03
3= x
40. 2.2 = 2.2 × 100% = 220%
38. 0.4 = 0.4 × 100% = 40%
39. 1.08 = 1.08 × 100% = 108%
41. 0.005 = 0.005 × 100% = 0.5%
42. 0.007 = 0.007 × 100% = 0.7%
43. 0.17 = 0.17 × 100% = 17%
32. Let x represent the base.
189 = (1.05) x
189 1.05 x
=
1.05 1.05
180 = x
44. 0.9 = 0.9 × 100% = 90%
222
Section 6.4
45. Let x represent the percent.
x ⋅ 480 = 120
480 x 120
=
480 480
x = 0.25
x = 0.25 × 100%
x = 25%
Percent Equations and Applications
51. Let x represent the percent.
x ⋅ 120 = 84
120 x 84
=
120 120
x = 0.7
x = 0.7 × 100%
x = 70%
70% of the hot dogs were sold.
46. Let x represent the percent.
180 = x ⋅ 2000
180 2000 x
=
2000 2000
0.09 = x
x = 0.09 × 100%
x = 9%
52. Let x represent the percent.
x ⋅ 2500 = 900
2500 x 900
=
2500 2500
x = 0.36
x = 0.36 × 100%
x = 36%
36% of the goal has been achieved.
47. Let x represent the percent.
666 = x ⋅ 740
666 740 x
=
740 740
0.9 = x
x = 0.9 × 100%
x = 90%
53. (a) 4 + 2 + 14 + 10 + 16 + 18 + 10 + 6 =
80
There are 80 total employees.
(b) Let x represent the percent.
x ⋅ 80 = 10
80 x 10
=
80 80
x = 0.125
x = 0.125 × 100%
x = 12.5%
12.5% missed 3 days of work.
48. Let x represent the percent.
x ⋅ 60 = 2.88
60 x 2.88
=
60
60
x = 0.048
x = 0.048 × 100%
x = 4.8%
(c) 2 + 14 + 10 + 16 + 18 = 60
Let x represent the percent.
x ⋅ 80 = 60
80 x 60
=
80 80
x = 0.75
x = 0.75 × 100%
x = 75%
75% missed 1 to 5 days of work.
49. Let x represent the percent.
x ⋅ 300 = 400
300 x 400
=
300 300
x = 1.333
x = 1.333 × 100%
x = 133.3%
50. Let x represent the percent.
28 = x ⋅ 24
28 24 x
=
24 24
1.167 ≈ x
x ≈ 1.167 × 100%
x ≈ 116.7%
54. (a) 16 + 18 + 10 + 6 = 50
Let x represent the percent.
x ⋅ 80 = 50
80 x 50
=
80 80
x = 0.625
x = 0.625 × 100%
x = 62.5%
62.5% missed at least 4 days of work.
223
Chapter 6
Percents
(b) Let x represent the percent.
x ⋅80 = 4
80 x 4
=
80 80
x = 0.05
x = 0.05 × 100%
x = 5%
5% did not miss any days.
60. Let x represent the percent.
360 = (510) x
360 510 x
=
510 510
0.706 ≈ x
0.706 × 100% = 70.6%
70.6% of the earth is covered by water.
61. Let x represent the total cost of the TV.
1440 = (0.60) x
1440 0.60 x
=
0.60 0.60
2400 = x
The total cost of the TV is $2400.
55. Let x represent the total number of
hospital stays.
6.3 = (0.18) x
6.3 0.18 x
=
0.18 0.18
35 = x
There were 35 million total hospital stays
that year.
62. Let x represent the number of females
homeschooled in 1999.
875,000 = (2.02) x
875,000 2.02 x
=
2.02
2.02
433,000 ≈ x
There were approximately 433 thousand
females homeschooled in 1999 for grades
K-12.
56. (a) Let x represent the amount of liquid in
the bottle.
4.8 = (0.10 )x
4.8 0.10 x
=
0.10 0.10
48 = x
(b) 48 − 4.8 = 43.2 oz
63. Let x represent the minutes of
commercials.
x = (0.26)(60)
x = 15.6
15.6 min of commercials would be
expected.
57. Let x represent the percent.
x ⋅ 87 = 11
87 x 11
=
87 87
x ≈ 0.126
0.126 × 100% = 12.6%
Approximately 12.6% of Florida’s
panthers live in Everglades National Park.
64. Let x represent the amount of water.
x = (0.65)(150)
x = 97.5
There is 97.5 lb of water in a 150-lb person.
58. Let x represent the number using online
travel sites.
x = (0.44)(400)
x = 176
176 people would be expected to use
online travel sites.
65. Let x represent the number that made over
$10 per hour.
x = (0.635)(10,000,000)
x = 6,350,000
6,350,000 people ages 25−34 made over
$10/hr.
59. Let x represent the number saving for their
children’s education.
x = (0.52)(800)
x = 416
416 parents would be expected to have
started saving for their children’s
education.
66. Let x represent the number that made over
$10 per hour.
x = (0.652)(6,600,000)
x = 4,303,200
4,303,200 people ages 55−64 made over
$10/hr.
224
Section 6.4
Percent Equations and Applications
69. (a) 220 − 20 = 200 beats per minute
67. Let x represent the total workers ages 16−24.
4,000,000 = (0.25) x
4,000,000 0.25 x
=
0.25
0.25
16,000,000 = x
There are a total of 16,000,000 workers in
the 16−24 age group.
(b) (0.60)(200) = 120
(0.85)(200) = 170
Between 120 and 170 beats per
minute.
70. (a) 220 − 42 = 178 beats per minute
68. Let x represent the total workers ages
45−54.
9,000,000 = (0.72) x
9,000,000 0.72 x
=
0.72
0.72
12,500,000 = x
There are a total of 12,500,000 workers in
the 45-54 age group.
(b) (0.60)(178) ≈ 107
(0.85)(178) ≈ 151
Between 107 and 151 beats per
minute
Problem Recognition Exercises: Percents
1. 0.10 (82 ) = 8.2
12. Let x represent the percent.
x
120
=
100 500
500 x = (100)(120)
500 x = 12,000
500 x 12,000
=
500
500
x = 24
24%
2. 0.05 (82 ) = 4.1
3. 0.20 (82 ) = 16.4
4. 0.50 (82 ) = 41
5. 2.00 (82 ) = 164
13. Let x represent the number.
x = (0.12)(40)
x = 4.8
6. 0.15 (82 ) = 12.3
7. Greater than, since 104% > 100%.
14. Let x represent the percent.
x ⋅ 180 = 27
180 x 27
=
180 180
x = 0.15
x = 0.15 × 100%
x = 15%
8. Less than, since 8% < 10% and 10% of 50
is 5.
9. Greater than, since 11% > 10% and 10%
of 90 is 9.
10. Greater than, since 52% > 50% and 50%
of 200 is 100.
11. Let x represent the base.
6 = (0.002) x
6
0.002 x
=
0.002 0.002
3000 = x
225
Chapter 6
Percents
23. Let x represent the amount.
x = (0.50)(50)
x = 25
15. Let x represent the base.
150 105
=
100
x
150 x = (100)(105)
150 x = 10,500
150 x 10,500
=
150
150
x = 70
24. Let x represent the amount.
x = (0.15)(900)
x = 135
25. Let x represent the base.
50 = (0.50) x
0.50 x
50
=
0.50 0.50
100 = x
16. Let x represent the amount.
x = (0.30)(120)
x = 36
17. Let x represent the amount.
x = (0.07)(90)
x = 6.3
26. Let x represent the base.
900 = (0.15) x
900 0.15x
=
0.15 0.15
6000 = x
18. Let x represent the base.
100 = (0.40) x
100 0.40 x
=
0.40 0.40
250 = x
27. Let x represent the percent.
x ⋅ 250 = 2
2
250 x
=
250 250
x = 0.008
x = 0.008 × 100%
x = 0.8%
19. Let x represent the percent.
x ⋅ 60 = 180
60 x 180
=
60
60
x=3
x = 3 × 100%
x = 300%
28. Let x represent the percent.
x ⋅ 60 = 75
60 x 75
=
60 60
x = 1.25
x = 1.25 × 100%
x = 125%
20. Let x represent the amount.
x = (0.005)(140)
x = 0.7
21. Let x represent the base.
75 = (0.001) x
0.001x
75
=
0.001 0.001
75,000 = x
29. Let x represent the amount.
x = (0.10)(26)
x = 2.6
30. Let x represent the base.
11 = (0.55) x
0.55x
11
=
0.55 0.55
20 = x
22. Let x represent the percent.
x ⋅ 72 = 27
72 x 27
=
72 72
x = 0.375
x = 0.375 × 100%
x = 37.5%
226
Problem Recognition Exercises: Percents
31.
Let x represent the percent.
x ⋅ 248 = 186
248 x 186
=
248 248
x = 0.75
x = 0.75 × 100%
x = 75%
33. Let x represent the percent.
x ⋅186 = 248
186 x 248
=
186 186
x = 1.333
x = 1.333× 100%
x = 133.3% or 133 1 %
3
32. Let x represent the amount.
x = (0.55)(11)
x = 6.05
34. Let x represent the percent.
x ⋅5 = 20
5x 20
=
5
5
x=4
x = 4 × 100%
x = 400%
Section 6.5 Applications Involving Sales Tax, Commission,
Discount, and Markup
Section 6.5 Practice Exercises
7. Let x represent the number.
52 = (0.002) x
52
0.002 x
=
0.002 0.002
26,000 = x
Sales   Sales   Cost of 
1. (a) 
=
 tax   tax rate  merchandise
(b) ( Commission ) = 

(c)
( Discount ) = 
Commission   Total 
  sales 
rate
8. Let x represent the amount.
x = (2.25)(36)
x = 81
Discount   Original
rate   price 
9. Let x represent the percent.
x
6
=
100 25
25 x = (100)(6)
25 x = 600
25 x 600
=
25
25
x = 24
24%
Markup  Original
(d) ( Markup) = 
 rate   price 
2. (a) 82% = 82 × 0.01 = 0.82
(b) 0.003 = 0.003 × 100% = 0.3%
3. 12
10. Let x represent the base.
18 = (0.75) x
18
0.75 x
=
0.75 0.75
24 = x
1
4. 33 %
3
5. 28
6. 200%
227
Chapter 6
Percents
11. Let x represent the amount.
x = (0.016)(550)
x = 8.8
21. Let x represent the amount of tax.
x = (0.05)(68.25)
x = 3.41
68.25 + 3.41 = 71.66
The total bill is $71.66.
12. Let x represent the percent.
x
32.2
=
100
28
28 x = (100)(32.2)
28 x = 3220
28 x 3220
=
28
28
x = 115
115%
22. Let x represent the amount of tax.
x = (0.045)(64)
x = 2.88
The tax is $2.88.
23. Let x represent the tax rate.
16.80 = x ⋅ $240.00
16.80 240.00 x
=
240.00 240.00
0.07 = x
The tax rate is 7%.
13. (0.05)($20.00) = $1.00
$20.00 + $1.00 = $21.00
14. (0.06)($56.00) = $3.36
$56.00 + $3.36 = $59.36
15.
16.
24. (a) $44.10 − $42.00 = $2.10
The tax is $2.10.
$0.50
= 0.04 or 4%
$12.50
$12.50 + $0.50 = $13.00
(b) Let x represent the tax rate.
2.10 = x ⋅ 42.00
2.10 42.00 x
=
42.00 42.00
0.05 = x
The tax rate is 5%.
$14.84
= 0.07 or 7%
$212.00
$212.00 + $14.84 = $226.84
25. Let x represent the price of the fruit
basket.
2.67 = (0.06) x
2.67 0.06 x
=
0.06 0.06
44.5 = x
The price is $44.50.
17. Let x represent the cost.
$2.75 = (0.025) x
$2.75 0.025x
=
0.025 0.025
$110.00 = x
$110.00 + $2.75 = $112.75
18. Let x represent the cost.
$18.00 = (0.03) x
$18.00 0.03x
=
0.03
0.03
$600.00 = x
$600.00 + $18.00 = $618.00
26. Let x represent the price of the groceries.
1.50 = (0.06) x
1.50 0.06 x
=
0.06 0.06
25 = x
The price is $25.00.
19. $58.30 − $55.00 = $3.30
$3.30
= 0.06 = 6%
$55.00
27. (0.05)($20,000.00)=$1000.00
28. (0.16)($540.00) = $86.40
20. $220.42 − $214.00 = $6.42
$6.42
= 0.03 or 3%
$214.00
29.
228
$10,000.00
= 0.08 or 8%
$125,000.00
Section 6.5
30.
Applications Involving Sales Tax, Commission, Discount, and Markup
42,000 = 0.03x
42,000 0.03x
=
0.03
0.03
1, 400,000 = x
Her sales were $1,400,000.
$24.00
= 0.03 or 3%
$800.00
31. Let x represent the total sales.
$540.00 = (0.10) x
$540.00 0.10 x
=
0.10
0.10
$5400.00 = x
38. Amount of commission
= $1090 − $100 = $990
Let x represent last weeks sales.
990 = 0.055 x
990
0.055 x
=
0.055 0.055
18,000 = x
She had $18,000 in sales.
32. Let x represent the total sales.
$159.00 = (0.15) x
$159.00 0.15 x
=
0.15
0.15
$1060.00 = x
39. $86,000 − $60,000 = $26,000.
Commission on $60,000:
(0.06)(60,000) = 3600
Commission on $26,000:
(0.085)(26,000) = 2210
3600 + 2210 = 5810
Kahir’s commissions totaled $5810.00.
33. Let x represent the amount of commission.
x = (0.07)($48,000)
x = $3360
Zach made $3360 in commission.
34. (a) $750 − $400 = $350
Marisa sold $350 over $400.
(b) Let x represent the amount of
commission.
x = (0.15)($350)
x = $52.50
She earned $52.50 in commission.
40. Discount = (0.20)($56.00) = $11.20
Sale price = $56.00 − $11.20 = $44.80
41. Discount = (0.15)($175.00) = $26.25
Sale price = $175.00 − $26.25 = $148.75
35. Let x represent the commission rate.
300 = x ⋅ 2000
300 2000 x
=
2000 2000
0.15 = x
Rodney’s commission rate is 15%.
42. Discount = $900.00 − $600.00 = $300.00
(rate)($900.00) = $300.00
$300.00
rate =
$900.00
1
1
= or 33 %
3
3
36. Let x represent the commission rate.
7600 = x ⋅ 95,000
7600
95,000 x
=
95,000 95,000
0.08 = x
His commission rate is 8%.
43. Discount = $900.00 − $630.00 = $270.00
(rate)($900.00) = $270.00
$270.00
rate =
$900.00
= 0.30 or 30%
37. Amount of commission
= $67,000 − $25,000
= $42,000
Let x represent total sales.
44. Original price = $76.50 + $8.50 = $85.00
(rate)($85.00) = $8.50
$8.50
rate =
$85.00
= 0.10 or 10%
229
Chapter 6
Percents
45. Original price = $77.00 + $33.00 =
$110.00
(rate)($110.00) = $33.00
$33.00
rate =
$110.00
= 0.30 or 30%
The discount is $80.70 and the sale price
is $188.30.
53. The set of dishes is not free. After the first
discount, the price was 50% or one-half of
$112, which is $56. Then the second
discount is 50% or one-half of $56, which
is $28.
46. (0.50)(original price) = $38.00
$38.00
original price =
= $76.00
0.50
Sale price = $76.00 − $38.00 = $38.00
54. Let x represent the discount amount.
x = (0.10)($109.99) = $11.00
Sale price = $109.99 − $11 = $98.99
The discount is $11.00, and the sale price
is $98.99.
47. (0.40)(original price) = $23.36
$23.36
original price =
0.40
= $58.40
Sale price = $58.40 − $23.36 = $35.04
55. Discount = $235 − $188 = $47
Let x represent the discount rate.
( x)($235) = $47
$47
x=
= 0.2
$235
The discount is $47.00, and the discount
rate is 20%.
48. Let x represent the amount of discount.
x = (0.15)($5.60) = $0.84
Sale price = $5.60 − $0.84 = $4.76
The discounted lunch bill is $4.76.
56. Discount = $70 − $50 = $20
Let x represent the discount rate.
( x)($70) = $20
$20
x=
≈ 0.286
$70
The discount is $20.00, and the rate is
approximately 28.6%.
49. (a) Let x represent the amount of
discount.
x = (0.10)($550) = $55
The discount is $55.
(b) Sale price = $550 − $55 = $495
The discounted yearly membership
will cost $495.
57. Markup = (0.05)($92.00) = $4.60
Retail price = $92.00 + $4.60 = $96.60
50. Discount = $60 − $45 = $15
Let x represent the discount rate.
( x)($60) = $15
$15
x=
= 0.25
$60
The discount rate is 25%.
58. Markup = (0.10)($25.00) = $2.50
Retail price = $25.00 + $2.50 = $27.50
59. Markup = $118.80 − $110.00 = $8.80
(rate)($110.00)=$8.80
$8.80
rate =
$110.00
= 0.08 or 8%
51. Discount = $229 − $183.20 = $45.80
Let x represent the discount rate.
( x)($229) = $45.80
$45.80
x=
= 0.2
$229
The discount rate is 20%.
60. Markup = $57.50 − $50.00 = $7.50
(rate)($50.00) = $7.50
$7.50
rate =
$50.00
= 0.15 or 15%
52. Let x represent the discount amount.
x = (0.30)($269) = $80.70
sale price = $269 − $80.70 = $188.30
230
Section 6.5
Applications Involving Sales Tax, Commission, Discount, and Markup
66. Let x represent the amount of markup.
x = (1.10)($84) = $92.40
Retail price = $84 + $92.40 = $176.40
The retail price is $176.40.
61. Original price = $422.50 − $97.50
= $325.00
(rate)($325.00) = $97.50
$97.50
rate =
$325.00
= 0.3 or 30%
67. Markup amount = $375 − $300 = $75
Let x represent the markup rate.
$75 = ( x)($300)
$75
=x
$300
0.25 = x
The markup rate is 25%.
62. Original price = $875.00 − $175.00
= $700.00
(rate)($700.00) = $175.00
$175.00
rate =
$700.00
= 0.25 or 25%
68. Markup amount = $69 − $60 = $9
Let x represent the markup rate.
$9 = ( x)($60)
$9
=x
$60
0.15 = x
The markup rate is 15%.
63. (0.20)(original price) = $9.00
$9.00
original price =
0.20
= $45.00
Retail Price = $45.00 + $9.00 = $54.00
64. (0.18)(original price) = $31.50
$31.50
original price =
0.18
= $175.00
69. Original price = $123.20 − $43.20 =
$80.00
Let x represent the markup rate.
$43.20 = ( x)($80.00)
$43.20
=x
$80.00
0.54 = x
The markup rate is 54%.
Retail Price = $175.00 + $31.50 = $206.50
65. (a) Let x represent the amount of markup.
x = (0.18)($150.00) = $27.00
The markup is $27.00.
70. Original price = $420 − $70 = $350
Let x represent the markup rate.
$70 = ( x)($350)
$70
=x
$350
0.2 = x
The markup rate is 20%.
(b) Retail price = $150.00 + $27.00
= $177.00
The retail price is $177.00.
(c) Tax = (0.07)($177.00) = $12.39
The total price is
$177.00 + $12.39 = $189.39.
Section 6.6
Percent Increase and Decrease
Section 6.6 Practice Exercises
1. (a)  Percent  =  Amount of increase  × 100%
 increase  Original amount 
(b)  Percent  =  Amount of decrease  × 100%
 decrease
 Original amount 
231
Chapter 6
Percents
6. Hourly earnings = ($12)(30) = $360
Commission = (0.03)($1290.00) = $38.70
Total earnings = $360 + $38.70 = $398.70
Sean’s salary is $398.70.
12
x
=
60 100
(12)(100) = 60 x
1200 = 60 x
1200 60 x
=
60
60
x = 20
20%
80
x
=
(b)
50 100
(80)(100) = 50 x
8000 = 50 x
8000 50 x
=
50
50
x = 160
160%
2. (a)
7. Multiply the decimal by 100% by moving
the decimal point 2 places to the right and
attaching the % sign.
8. 0.23 = 23%
9. 0.05 = 5%
10. 0.88 = 88%
11. 0.12 = 12%
12. (a) Subtract the original price from the
increased price.
3. (a) Tax = (0.05)($65) = $3.25
$65 + $3.25 = $68.25
The total price will be $68.25.
(b) Subtract the decreased price from the
original price.
13. (a) Increase
(b) Let x represent the discount amount.
x = (0.20)($65) = $13
Sale price = $65 − $13 = $52
Tax = (0.05)($52) = $2.60
$52 + $2.60 = $54.60
The total price with the 20% discount
would be $54.60.
(b) 59 − 48 = 11
14. (a) Increase
(b) 123 − 78 = 45
15. (a) Decrease
(b) 145 − 135 = 10
(c) $68.25 − $54.60 = $13.65
Damien will save $13.65.
16. (a) Decrease
4. (a) Tax = (0.06)($32.50) = $1.95
$32.50 + $1.95 = $34.45
The total price will be $34.45.
(b) 190 − 109 = 81
17. (a) Decrease
(b) 654 − 645 = 9
(b) Let x represent the discount amount.
x = (0.25)($32.50) ≈ $8.12
Sale price = $32.50 − $8.12 = $24.38
Tax = (0.06)($24.38) ≈ $1.46
$24.38 + $1.46 = $25.84
The total price with the 25% discount
would be $25.84.
18. (a) Increase
(b) 42 − 24 = 18
19. (a) Increase
(b) 79 − 67 = 12
(c) $34.45 − $25.84 = $8.61
Kira will save $8.61.
20. (a) Decrease
(b) 205 − 105 = 100
5. $425 − $200 = $225
Commission = (0.14)($225) = $31.50
Pablo’s commission is $31.50.
232
Section 6.6
Percent Increase and Decrease
29. Increase = 170 – 165 = 5
5
Percent increase =
× 100%
165
≈ 0.03 × 100%
≈ 3%
21. $60 − $30
$30
× 100% = 1 × 100% = 100%
$30
c
22. $30 − $20 = $10
$10
× 100% = 0.5 × 100% = 50%
$20
b
30. Increase = 45,000 – 42,000 = 3000
3000
Percent increase =
× 100%
42,000
≈ 0.07 × 100%
≈ 7%
23. Increase = 14 − 8 = 6
6
Percent increase = × 100%
8
= 0.75 × 100%
= 75%
31. $62 − $31 = $31
$31
× 100% = 0.5 × 100% = 50%
$62
a
24. Increase = 309,000 – 300.000 = 9000
900
× 100%
Percent increase =
300,000
= 0.03 × 100%
= 3%
32. $40 − $10 = $30
$30
× 100% = 0.75 × 100% = 75%
$40
d
25. Increase = 42,000 – 21,000 = 21,000
21,000
× 100%
Percent increase =
21,000
= 1× 100%
= 100%
33. Decrease = $12.60 − $11.97 = $0.63
$0.63
Percent decrease =
× 100%
$12.60
= 0.05 × 100%
= 5%
26. Increase = 20,700 – 20,200 = 500
500
Percent increase =
× 100%
20, 200
≈ 0.025 × 100%
≈ 2.5%
34. Decrease = 27 − 17 = 10
10
Percent decrease = × 100%
27
≈ 0.37 × 100%
= 37%
27. Increase = 5500 – 5000 = 500
500
Percent increase =
× 100%
5000
= 0.10 × 100%
= 10%
35. Decrease = 5 − 1.6 = 3.4
3.4
Percent decrease =
× 100%
5
= 0.68 × 100%
= 68%
28. Increase = 1000 – 800 = 200
200
Percent increase =
× 100%
800
= 0.25 × 100%
= 25%
36. Decrease = 160 − 140 = 20
20
Percent decrease =
× 100%
160
= 0.125 × 100%
= 12.5%
233
Chapter 6
Percents
37. Decrease = 79 − 59 = 20
20
Percent decrease =
× 100%
79
= 0.253 × 100%
= 25.3%
43. Change = 8.11 − 7.78 = 0.33
0.33
Percent decrease =
× 100%
8.11
≈ 0.041 × 100%
= 4.1%
38. Decrease = 279 − 249 = 30
30
Percent decrease =
× 100%
279
= 0.108 × 100%
= 10.8%
44. Change = 1.075 − 1.047 = 0.028
0.028
Percent decrease =
× 100%
1.075
≈ 0.026 × 100%
= 2.6%
39. Decrease = 12 − 10.2 = 1.8
1.8
Percent decrease =
× 100%
12
= 0.15 × 100%
= 15%
45. (a) Change = 15.07 − 8.92 = 6.35 million
An increase of 6.35 million people
6.35
× 100%
8.92
≈ 0.712 × 100%
= 71.2%
(b) Percent increase =
40. Decrease = 3000 − 1800 = 1200
1200
Percent decrease =
× 100%
3000
= 0.4 × 100%
= 40%
An increase of 71.2% in unemployment
(c) Change = 15.27 – 14.84 = 0.43 million
41. Change = 112.3 − 110.8 = 1.5
1.5
Percent increase =
× 100%
110.8
≈ 0.014 × 100%
= 1.4%
A decrease of 0.43 million people
42. Change = 62.6 − 61.4 = 1.2
1.2
Percent increase =
× 100%
61.4
≈ 0.019 × 100%
= 1.9%
A 2.8% decrease in unemployment
0.43
× 100%
15.27
≈ 0.028 × 100%
= 2.8%
(d) Percent decrease =
46. Change = 6320 − 5814 = 506 million
metric tons
506
Percent increase =
× 100%
5814
≈ 0.087 × 100%
= 8.7%
234
Section 6.7
Section 6.7
Simple and Compound Interest
Simple and Compound Interest
Section 6.7 Practice Exercises
= $240
$4000 + $240 = $4240
1. (a) Simple; principal
(b) I = Prt
9. I = Prt = ($5050)(0.06)(4) = $303(4)
= $1212
$5050 + $1212 = $6262
(c) Sompound
r

(d) A = P  1 + 

n
n −t
10. I = Prt = ($4800)(0.04)(3) = $192(3)
= $576
$4800 + $576 = $5376
2. Change = 1181 − 995 = 186
186
Percent increase =
× 100%
995
≈ 0.19 × 100%
= 19%
 1
11. I = Prt = ($12,000)(0.04)  4 
 2
= $480(4.5) = $2160
$12,000 + $2160 = $14,160
3. Change = 1099 − 987 = 112
112
Percent decrease =
× 100%
1099
≈ 0.10 × 100%
= 10%
 1
12. I = Prt = ($6230)(0.07)  6 
 3
 19 
= $436.10   ≈ $2761.97
 3
4. Change = 605 − 580 = 25
25
Percent decrease =
× 100%
605
≈ 0.04 × 100%
= 4%
$6230 + $2761.97 = $8991.97
()
13. I = Prt = ($10,500)(0.045) 4
()
= $472.50 4 = $1890
$10,500 + $1890 = $12,390
5. Change = 404 − 364 = 40
40
Percent increase =
× 100%
364
≈ 0.11× 100%
= 11%
14. I = Prt = ($9220)(0.08)(4)
= $737.60(4) = $2950.40
$9220 + $2950.40 = $12,170.40
15. (a) I = Prt = ($2500)(0.035)(4)
= $87.50(4) = $350
6. Change = 2034 − 1729 = 305
305
Percent increase =
× 100%
1729
≈ 0.18 × 100%
= 18%
(b) $2500 + $350 = $2850
16. (a) I = Prt = ($3400)(0.04)(5) = $136(5)
= $680
(b) $3400 + $680 = $4080
7. I = Prt = ($6000)(0.05)(3) = $300(3)
= $900
$6000 + $900 = $6900
17. (a) I = Prt = ($400)(0.08)(1.5) = $32(1.5)
= $48
8. I = Prt = ($4000)(0.03)(2) = $120(2)
(b) $400 + $48 = $448
235
Chapter 6
Percents
18. (a) I = Prt = ($1000)(0.08)(2.25)
= $80(2.25) = $180
22. I = Prt = ($750)(0.08)(0.5) = $60(0.5)
= $30
$750 + $30 = $780
(b) $1000 + $180 = $1180
23. There are 2(3) = 6 total compounding
periods.
19. I = Prt = ($10,300)(0.04)(5) = $412(5)
= $2060
$10,300 + $2060 = $12,360
24. There are 4(2) = 8 total compounding
periods.
20. I = Prt = ($20,000)(0.06)(10)
= $1200(10) = $12,000
$20,000 + $12,000 = $32,000
25. There are 12(2) = 24 total compounding
periods.
26. There are 12(1.5) = 18 total compounding
periods.
21. I = Prt = ($4500)(0.10)(2.5) = $450(2.5)
= $1125
$4500 + $1125 = $5625
27. (a) I = Prt = $500 (0.04 )(3) = $20 (3) = $60
$500 + $60 = $560
(b) Year
Interest
Total
1
($500)(0.04) = $20
$500 + $20 = $520
2
($520)(0.04) = $20.80
$520 + $20.80 = $540.80
3
($540.80)(0.04) = $21.63
$540.80 + $21.63 = $562.43
28. (a) I = Prt = $12, 000 (0.05 )(3) = $600 (3) = $1800
$12, 000 + $1800 = $13, 800
(b) Year
Interest
Total
1
($12,000)(0.05) = $600
$12,000 + $600 = $12,600
2
($12,600)(0.05) = $630
$12,600 + $630 = $13,230
3
($13,230)(0.05) = $661.50
$13,230 + $661.50 = $13,891.50
29. (a) I = Prt = ($8000)(0.04)(3) = $960
$8000 + $960 = $8960
I = Prt
Total in account
1
I = ($8000)(0.04)(1) = $320
$8000 + $320 = $8320
2
I = ($8320)(0.04)(1) = $332.80
$8320 + $332.80 = $8652.80
(b) Year
3
I = ($8652.80)(0.04)(1) = $346.11 $8652.80 + $346.11 = $8998.91
(c) $8998.91 − $8960 = $38.91
30. (a) I = Prt = ($12,000)(0.08)(1) = $960
$12,000 + $960 = $12,960
236
Section 6.7
(b)
Simple and Compound Interest
Compound
Period
I = Prt
Total in account
1st quarter
I = ($12,000)(0.08)(0.25) = $240
$12,000 + $240 = $12,240
2nd quarter
I = ($12,240)(0.08)(0.25) = $244.80
$12,240 + $244.80 = $12,484.80
3rd quarter
I = ($12,484.80)(0.08)(0.25) ≈ $249.70 $12,484.80 + $249.70 = $12,734.50
4th quarter
I = ($12,734.50)(0.08)(0.25) = $254.69 $12,734.50 + $254.69 = $12,989.19
(c) $12,989.19 − $12,960 = $29.19
31. A = total amount in the account;
P = principal;
r = annual interest rate;
n = number of compounding periods per
year;
t = time in years
 0.03 
36. A = $4000 1 +

2 

 0.06 
37. A = $10,000  1 +

4 

32. P = $1000
r = 0.08
n = 12
t=3
 0.04 
38. A = $9000  1 +
4 

1⋅5
 0.045 
33. A = $5000  1 +

1 

Chapter 6
4⋅1.5
≈ $9941.60
12⋅2
 0.08 
40. A = $9000  1 +

12 

≈ $14,725.49
≈ $6622.88
Review Exercises
2.
33
= 33%
100
3.
5 125
=
= 125%
4 100
4.
5. b, c
75
= 75%
100
6. c, d
7. 30% =
30
3
=
100 10
f
1
33.3 1
=
8. 33 % = 33.3% =
3
100 3
d
1 50
=
= 50%
2 100
237
≈ $16,019.47
≈ $10,555.99
Section 6.1
1.
≈ $10,934.43
4⋅2.5
 0.045 
39. A = $14,000  1 +

12 

≈ $6230.91
1⋅4
2⋅2
≈ $4373.77
12⋅3
 0.0525 
34. A = $12,000  1 +

1 

 0.05 
35. A = $6000 1 +

2 

2⋅3
Chapter 6
9. 33% =
Percents
33
100
1
505 1
=
×
100 100 100
101 1
101
=
×
=
20 100 2000
24. 5.05% = 5.05 ×
a
10. 50% =
50 1
=
100 2
Section 6.2
b
25.
17
17
=
× 100% = 17%
100 100
26.
22 22
=
× 100% = 44%
50 50
27.
4 4
= × 100% = 80%
5 5
13. 7.5% = 7.5 × 0.01 = 0.075
e
28.
7 7
= × 100% = 175%
4 4
14. 75% = 75 × 0.01 = 0.75
c
29. 0.12 = 0.12 × 100% = 12%
2
66.6 2
=
11. 66 % = 66.6% =
3
100 3
c
12. 60% =
60
6 3
= =
100 10 5
e
30. 1.1 = 1.1 × 100% = 110%
15. 50% = 50 × 0.01 = 0.5
f
31. 0.005 = 0.005 × 100% = 0.5%
16. 100% = 100 × 0.01 = 1
a
32. 0.4 = 0.4 × 100% = 40%
17. 0.25% = 0.25 × 0.01 = 0.0025
d
33.
14
= 0.875 × 100% = 87.5%
16
18. 25% = 25 × 0.01 = 0.25
b
34.
19
= 0.76 × 100% = 76%
25
19. 42% = 42 ×
35.
3
= 0.6 × 100% = 60%
5
1
20 1
20. 20% = 20 ×
=
=
100 100 5
20% = 20 × 0.01 = 0.20
36.
1
= 0.1 × 100% = 10%
10
1
42 21
=
=
100 100 50
42% = 42 × 0.01 = 0.42
37.
21. 6.15% = 6.15 × 0.01 = 0.0615
22. 5.29% = 5.29 × 0.01 = 0.0529
9
45
=
= 0.45
20 100
9
9
=
× 100% = 45%
20 20
38. 1 = 1
1 = 100%
1
915 1
=
×
100 100 100
183 1
183
=
×
=
20 100 2000
23. 9.15% = 9.15 ×
238
Chapter 6
51. Let x represent the amount.
x
12
=
50 100
100 x = (50)(12)
100 x 600
=
100 100
x=6
1
6
3
=
=
100 100 50
6% = 6 × 0.01 = 0.06
39. 6% = 6 ×
12 6
=
10 5
1.2 = 1.2 × 100% = 120%
40. 1.2 =
41.
52. Let x represent the amount.
3
x 54
=
64 100
 3
100 x = (64)  5 
 4
100 x 368
=
100 100
x = 3.68
9
= 0.009
1000
9
9
9
=
× 100% = % or 0.9%
1000 1000
10
1
75 3
=
=
100 100 4
75% = 75 × 0.01 = 0.75
42. 75% = 75 ×
53. Let x represent the percent.
x
11
=
100 88
88 x = (100)(11)
88 x 1100
=
88
88
x = 12.5
12.5%
Section 6.3
43. Amount: 67.50
base: 150
p = 45
44. Amount: 360
base: 3000
p = 12
54. Let x represent the percent.
x
8
=
100 2500
2500 x = (100)(8)
2500 x 800
=
2500 2500
x = 0.32
0.32%
45. Amount: 30.24
base: 144
p = 21
46. Amount: 31.8
base: 30
p = 106
47.
6 75
=
8 100
48.
27
15
=
180 100
49.
840 200
=
420 100
50.
6
0.3
=
2000 100
Review Exercises
55. Let x represent the base.
33 13 13
=
100 x
100
x = (100)(13)
3
100
x = 1300
3
3 100
3
⋅
⋅ 1300
x=
100 3
100
x = 39
239
Chapter 6
Percents
56. Let x represent the base.
120 24
=
100 x
120 x = (100)(24)
120 x 2400
=
120
120
x = 20
62. x = 0.29 ⋅ 404
x = 117.16
63. 18.90 = x ⋅ 63
18.90 63x
=
63
63
0.3 = x
x = 30%
57. Let x represent the number of no-shows.
x
4.2
=
260 100
100 x = (260)(4.2)
100 x 1092
=
100
100
x = 10.92
Approximately 11 people would be noshows.
64. x ⋅ 250 = 86
250 x 86
=
250 250
x = 0.344 or 34.4%
58. Let x represent the number surveyed.
58 493
=
x
100
58 x = (100)(493)
58 x 49,300
=
58
58
x = 850
850 people were surveyed.
65.
30 = 0.25 ⋅ x
30 0.25 x
=
0.25 0.25
120 = x
66.
26 = 1.30 ⋅ x
26 1.3x
=
1.3 1.3
20 = x
67. Let x represent the original price.
0.80 ⋅ x 54.40
=
0.80
0.80
x = 68
The original price is $68.00.
59. Let x represent the percent spent on rent.
x
720
=
100 1800
1800 x = (100)(720)
1800 x 72,000
=
1800
1800
x = 40
Victoria spends 40% on rent.
68. Let x represent the percent.
x ⋅ 600 330
=
600
600
x = 0.55
Veronica read 55% of the novel.
69. Let x represent the number of fat calories.
x = 0.30 ⋅ 2400
x = 720
Elaine can consume 720 fat calories.
60. Let x represent the total cars on the lot.
40 26
=
100 x
40 x = (100)(26)
40 x 2600
=
40
40
x = 65
There are 65 cars.
70. (a) 0.13 ⋅ 300,000,000 = 39,000,000
(b) 0.20 ⋅ 404,000,000 = 80,800,000
Section 6.5
Section 6.4
71. Tax = (0.06)(1279) = 76.74
The sales tax is $76.74.
61. 0.18 ⋅ 900 = x
162 = x
240
Chapter 6
Review Exercises
72. Let x represent the tax rate.
x ⋅ 685 47.95
=
685
685
x = 0.07
The sales tax rate is 7%.
80. Discount = (0.10)($1747) = $174.70
Sale price = $1747 − $174.70 − $50
= $1522.30
The discount is $174.70. The final price is
$1522.30.
73. (a) $97.47− $90.25 = $7.22
The tax is $7.22.
81. Markup = $208 − $160 = $48
Let x represent the markup rate.
48 x ⋅ 160
=
160
160
0.3 = x
The markup rate is 30%.
(b) Let x represent the tax rate.
x ⋅ 90.25 7.22
=
90.25
90.25
x = 0.08
The tax rate is 8%.
82. Markup = (0.18)($50) = $9
Retail price = $50 + $9 = $59
The baskets will sell for $59 each.
74. Sales tax = (0.06)(225) = 13.50
resort tax = (0.11)(225) = 24.75
Total for one night:
225 + 13.50 + 24.75 = $263.25
Total for 4 nights: $263.25 × 4 = $1053.
The total amount for 4 nights will be
$1053.00.
Section 6.6
83. (a) Increase
(b) 107.5 − 86 = 21.5
21.5
× 100%
86
= 0.25 × 100%
= 25%
75. Let x represent the rate.
11 x ⋅ 104
=
104
104
0.106 ≈ x
The commission rate was approximately
10.6%.
Percent increase =
84. (a) Decrease
(b) 410 − 82 = 328
76. Commission = 0.4 ⋅ 4075 = 163
Andre earned $163 in commission.
328
× 100%
410
= 0.8 × 100%
= 80%
Percent decrease =
77. 8 hr × $15 = $120
$420 − $200 = $220
0.05 ⋅ $220 = $11
$120 + $11 = $131
Sela will earn $131 that day.
85. Increase = 574 − 263 = 311
311
Percent increase =
× 100%
263
≈ 1.183 × 100%
= 118.3%
78. Let x represent the rate.
5600
x ⋅ 160,000
=
160,000
160,000
0.035 = x
The commission rate is 3.5%.
86. Decrease = 3.2 − 1.8 = 1.4
1.4
Percent decrease =
× 100%
3.2
≈ 0.4375 × 100%
= 43.75%
79. Discount = (0.30)($28.95) ≈ $8.69
Sale price = $28.95 − $8.69 = $20.26
The discount is $8.69. The sale price is
$20.26.
241
Chapter 6
Percents
90. I = Prt = ($7000)(0.04)(5) = $280(5)
= $1400
$7000 + $1400 = $8400
87. Increase = 285,000 − 128,000 = 157,000
157,000
Percent increase =
× 100%
128,000
= 1.23 × 100%
= 123%
91. I = Prt = ($2500)(0.05)(1.5) = $125(1.5)
= $187.50
$2500 + $187.50 = $2687.50
Jean-Luc will have to pay $2687.50.
88. Increase = 16,000 − 300 = 15,700
15,700
Percent increase =
× 100%
300
≈ 52.33 × 100%
= 5233%
92. I = Prt = ($800)(0.025)(2) = $20(2)
= $40
$800 + $40 = $840
Win’s brother will owe him $840.
Section 6.7
89. I = Prt = ($10, 200)(0.03)(4) = $306(4)
= $1224
$10,200 + $1224 = $11,424
93.
94.
Year
Interest
Total
1
($6000)(0.04) = $240
$6000 + $240 = $6240
2
($6240)(0.04) = $249.60
$6240 + $249.60 = $6489.60
3
($6489.60)(0.04) ≈ $259.58
$6489.60 + $259.58 = $6749.18
Period
Interest
Total
1
I = ($10,000)(0.03)(0.5) = $150
$10,000 + $150 = $10,150
2
I = ($10,150)(0.03)(0.5) = $152.25
$10,150 + $152.25 = $10,302.25
3
I = ($10,302.25)(0.03)(0.5) ≈ $154.53
$10,302.25 + $154.53 = $10,456.78
4
I = ($10,456.78)(0.03)(0.5) ≈ $156.85
$10,456.78 + $156.85 = $10,613.63
 0.08 
95. A = $850 1 +

4 

4⋅2
 0.05 
96. A = $2050 1 +

2 

1⋅6
 0.075 
97. A = $11,000  1 +

1 

≈ $995.91
2⋅5
12⋅4
 0.045 
98. A = $8200  1 +

12 

≈ $2624.17
242
≈ $16,976.32
≈ $9813.88
Chapter 6
Chapter 6
1.
Test
Test
22
= 22%
100
2.
(g) 100% = 100 ×
1
100
=
=1
100 100
(h) 150% = 150 ×
1
150 3
=
=
100 100 2
5. 2.8% = 2.8 × 0.01 = 0.028
1
28
1
7 1
7
2.8% = 2.8 ×
=
×
= ×
=
100 10 100 5 50 250
6.9.9% = 9.9 × 0.01 = 0.099
1
99 1
99
9.9% = 9.9 ×
= ×
=
100 10 100 1000
3. (a) 5.4% = 5.4 × 0.01 = 0.054
1
54
1
5.4% = 5.4 ×
=
×
100 10 100
27
1
27
=
×
=
5 100 500
7. Multiply the fraction by 100%.
(b) 0.15% = 0.15 × 0.01 = 0.0015
1
15
1
0.15% = 0.15 ×
=
×
100 100 100
3
1
3
=
×
=
20 100 2000
(c) 170% = 170 × 0.01 = 1.70
1
170 17
170% = 170 ×
=
=
100 100 10
9.
1
1
100
=
× 100% =
% = 0.4%
250 250
250
10.
7 7
700
= × 100% =
% = 175%
4 4
4
11.
5 5
500
= × 100% =
% ≈ 71.4%
7 7
7
13. 0.32 = 0.32 × 100% = 32%
14. 0.052 = 0.052 × 100% = 5.2%
1
25 1
=
=
100 100 4
15. 1.3 = 1.3 × 100% = 130%
1
100 1
1
(c) 33 % =
×
=
3
3 100 3
16. 0.006 = 0.006 × 100% = 0.6%
17. (0.24)(150) = 36
1
50 1
(d) 50% = 50 ×
=
=
100 100 2
18. (1.2)(16) = 19.2
2
200 1
200 2
(e) 66 % =
×
=
=
3
3 100 300 3
(f) 75% = 75 ×
3 3
300
= × 100% =
% = 60%
5 5
5
12. Multiply the decimal by 100%.
1
1
4. (a) 1% = 1×
=
100 100
(b) 25% = 25 ×
8.
19.
1
75 3
=
=
100 100 4
20.
243
21 (0.06) x
=
0.06
0.06
350 = x
(0.40) x 80
=
0.40
0.40
x = 200
Chapter 6
21.
22.
Percents
27. Commission = (0.06)($3500) = $210
$400 + $210 = $610
Charles will earn $610.
x ⋅ 220 198
=
220
220
x = 0.9
x = 90%
28. Discount = $45 − $18 = $27
Let x represent the discount rate.
x ⋅ $45 $27
=
$45
$45
x = 0.6
The discount rate of this product is 60%.
75 x ⋅ 150
=
150
150
0.50 = x
x = 50%
23. (a) 740 − 10 = 730 mg
29. Decrease = $240.00 − $169.00= $71.00
$71.00
Percent increase =
× 100%
$240.00
≈ 0.296 × 100%
= 29.6%
(b) Let x represent the percent.
730 x ⋅ 740
=
740
740
0.986 ≈ x
98.6% is from the dressing.
24. (0.78)(500) = 390 m
30. (a) I = Prt = ($5000)(0.08)(3) = $400(3)
= $1200
3
(b) $5000 + $1200 = $6200
25. (0.21)(2000) = 420 m 3
 0.045 
31. A = $25,000  1 +

4 

26. (a) $32.10 − $30.00 = $2.10
The amount of sales tax is $2.10.
(b) Let x represent the tax rate.
x ⋅ $30 $2.10
=
$30
$30
x = 0.07
The sales tax rate is 7%.
244
4⋅5
≈ $31, 268.76
Chapters 1–6
Chapters 1–6
Cumulative Review Exercises
1. Millions place
3
21 1 3
1
21
10.
÷7=
⋅ = or 1
2
2
2 7 2
2. (a) 3,539,245
(b) Eight hundred thirty thousand
1
(c) Four thousand, seven hundred
11. 16 ×
(d) 401,044
3.
8 10
18
+ +3= +3= 6+3
3 3
3
= 9 km
13. Perimeter =
14.
234
44
6
+ 2901
3185
7
7
= 13
10
10
2
4
+ $1
= +1
5
10
11
14
10
14
7. (a) Improper
(c) Proper
(d) Proper
8.
17. (a) 18, 36, 54, 72
4
3 32 4
1
×
= or 1
8 9
3
3
1
11
1
1
= 14 + 1 = 15
10
10
10
 1
16. A = 13  (17)
 2
27 17
=
⋅
2 1
459
1
or 229 in.2
=
2
2
(b) Improper
1
3 17
3
300 170
3
+
+
=
+
+
10 100 1000 1000 1000 1000
473
=
1000
15. $13
16 − 6 ÷ 3 + 32 = 4 − 6 ÷ 3 + 9
= 4−2+9
= 2+9
= 11
6.
1 16 2
=
=
24 24 3
 3  5  15 2
yd
12. A =    =
 8  4  32
34,882
×
100
3, 488, 200
87
4. 9 783
−72
63
−63
0
5.
Cumulative Review Exercises
(b) 1, 2, 3, 6, 9, 18
(c) 18 = 2 ⋅ 3 ⋅ 3 = 2 ⋅ 32
3
6
4
1
1
42 7
42 100
9.
÷
=
= 24
⋅
25 100 25 7
245
18. (a)
5
2
(b)
5
6
Chapter 6
Percents
19.
3
= 0.375
8
20.
4
1
= 1 = 1.3
3
3
31.
4 13
12
p 18
 1
12 p =  4  (18)
 3
=
6
7
21.
= 0.7
9
13 18
12 p = ⋅
3 1
1
3
22.
= 0.75
4
12 p 78
=
12 12
1
p=6
2
23. 90% − 24.7% = 65.3%
24. 66.8% − 24.7% = 42.1%
32.
25. 85 × 0.001 = 0.085
26. 85 × 100 = 8500
27. 85 ÷ 10 = 8.5
28. 85 ÷ 0.0001 = 850,000
29.
30.
p
21.87
=
100
81
81 p = (100)(21.87)
81 p 2187
=
81
81
p = 27
33. Let x represent the time to read 5 chapters.
1
2
3 15
=
4 p
3 p = (4)(15)
3 p 60
=
3
3
p = 20
x
1 5
1
1 ⋅ x =   (5)
2
5
x=
2
1
x=2
2
2.5 p
=
6
9
6 p = (2.5)(9)
6 p 22.5
=
6
6
p = 3.75
=
It will take 2
34.
1
hr
2
$2.30
= $0.25/oz
9.2 oz
The unit price is $0.25 per ounce.
35. Let x represent the time to download
4.6 MB.
1.6 MB 4.6 MB
=
x min
2.5 min
1.6 x = (2.5)(4.6)
1.6 x 11.5
=
1.6
1.6
x = 7.1875
It will take about 7.2 min.
246
Chapters 1–6
36.
39. I = Prt = ($13,000)(0.032)(5) = $416(5)
= $2080
$13,000 + $2080 = $15,080
Kevin will have $15,080.
1799 mi
= 514 mi/hr
3.5 hr
The DC-10 flew 514 mph.
37. Increase = 449,265 − 398,134 = 51,131
51,131
Percent increase =
× 100%
398,134
≈ 0.13 × 100%
= 13%
The increase was around 13%.
40.
 0.08 
A = $75,000  1+
12 

≈ $166, 473.02
12⋅10
$166,473.02 − $75,000 = $91,473.02
There is $91,473.02 paid in interest.
.
38. (a) 8.4 million − 3.4 million = 5million
(b)
Cumulative Review Exercises
5 million
= 0.045 million people per
110 years
year or 45,000 people per year
247
Chapter 7
Measurement
Chapter Opener Puzzle
Section 7.1
Converting U.S. Customary Units of Length
Section 7.1 Practice Exercises
1. (a) measure
9. b. 14 in.; 6 in. is too short, and 2 ft and 1 yd
are too long
(b) conversion
1
yd ; 2 ft and 20 in. are too short, and
2
10 yd is too long
2. The statement is ambiguous because no
units of measurement are given. For
example, it is not clear whether the table is
4.2 ft long, 4.2 in. long, 4.2 yd long, and
so on.
10. d. 1
3. 5280 ft = 1 mi
1
4.
ft = 1 in.
12
12. c. 8 ft; 5 ft and 65 inches are too short, and
7 yd is too tall
11. a. 72 in.; 18 in. is too short, and 18 ft and 3
yd are too tall
13. c. 15 ft; 30 ft and 20 yd are too tall, and 60
in. is too short
5. 1 yd = 3 ft
6. 1 ft = 12 in.
7. 1 ft =
14. a 5 in.; 10 in. and
1
yd
3
2
yd are too long, and 2
3
in. is too short
15. 2 yd = 2 × 1 yd = 2 × 3 ft = 6 ft
1
8. 1 ft =
mi
5280
16. 2 mi = 2 × 1 mi = 2 × 1760 yd = 3520 yd
17. 6 ft = 6 × 1 ft = 6 × 12 in. = 72 in.
248
Section 7.1
18. 1.25 mi = 1.25 × 1 mi = 1.25 × 5280 ft
= 6600 ft
Converting U.S. Customary Units of Length
3.5 ft 12 in.
⋅
= (3.5)(12) in.
1
1 ft
= 42 in.
33. 3.5 ft =
19. 2 mi = 2 × 1 mi = 2 × 5280 ft = 10,560 ft
4 1 in. 1 ft
1
9 1
34. 4 in. = 2
⋅
= ⋅
ft
2
12 in. 2 12
1
9
3
=
ft = ft
24
8
20. 5 ft = 5 × 1 ft = 5 × 12 in. = 60 in.
21. 24 ft = 24 × 1 ft = 24 ×
= 8 yd
24
1
yd =
yd
3
3
11,880 ft 1 mi
⋅
1
5280 ft
11,880
1
=
mi = 2 mi
5280
4
35. 11,880 ft =
36
1
ft =
ft
22. 36 in. = 36 × 1 in. = 36 ×
12
12
= 3 ft
23. 9 in. = 9 × 1 in. = 9 ×
1
9
3
ft =
ft = ft
12
12
4
24. 10 ft = 10 × 1 ft = 10 ×
1
= 3 yd
3
1
10
yd =
yd
3
3
25. 1760 ft = 1760 × 1 ft = 1760 ×
=
0.75 mi 5280 ft
⋅
1 mi
1
= (0.75)(5280) ft = 3960 ft
36. 0.75 mi =
1
1760
mi = mi
3
5280
26. 880 yd = 880 × 1 yd = 880 ×
1
880
mi = mi
=
2
1760
37. 6 yd =
5280
5280 yd 1 mi
⋅
=
mi
1760 yd 1760
1
= 3 mi
1
mi
5280
38. 5280 yd =
39. 14 ft =
1
mi
1760
14 ft 1 yd 14
2
yd = 4 yd
⋅
=
1 3 ft
3
3
40. 75 in. =
75 in. 1 ft
75
1
⋅
=
ft = 6 ft
1 12 in. 12
4
320 mi 1760 yd
⋅
1 mi
1
= (320)(1760) yd = 563,200 yd
41. 320 mi =
27. b
28. c
29. a
1
42. 3
30. d
31. 9 ft =
6 yd 3 ft
⋅
= (6)(3) ft = 18 ft
1 1 yd
3 ft 12 in.  1 
1
ft = 4 ⋅
=  3  (12) in.
4
1
1 ft  4 
3
9 ft 1 yd 9
⋅
= yd = 3 yd
1 3 ft 3
13 12
in. = 39 in.
= ⋅
4 1
1
1
2 yd 3 ft 7
1
⋅
= ⋅ 3 ft = 7 ft
32. 2 yd = 3
3
1
1 yd 3
171 in. 1 ft 1 yd
⋅
⋅
1
12 in. 3 ft
171
171
3
=
yd =
yd = 4 yd
(12)(3)
36
4
43. 171 in. =
249
Chapter 7
Measurement
1
2
ft = ft
12
3
2
2
10 ft 8 in. = 10 ft + ft = 10 ft
3
3
0.3 mi 5280 ft 12 in.
⋅
⋅
1 mi
1 ft
1
= (0.3)(5280)(12) in. = 19,008 in.
(b) 8 in. = 8 × 1 in. = 8 ×
44. 0.3 mi =
2 yd 3 ft 12 in.
⋅
⋅
= (2)(3)(12) in.
1 1 yd 1 ft
= 72 in.
45. 2 yd =
54. (a) 2 yd = 2 × 1 yd = 2 × 3 ft = 6 ft
2 yd 2 ft = 6 ft + 2 ft = 8 ft
1
2
yd = yd
3
3
2
2
2 yd 2 ft = 2 yd + yd = 2 yd
3
3
(b) 2 ft = 2 × 1 ft = 2 ×
12,672 in. 1 ft
1 mi
46. 12,672 in. =
⋅
⋅
1
12 in. 5280 ft
12,672
=
mi
(12)(5280)
1
= 0.2 mi or mi
5
1
ft = 0.5 ft
12
3'6" = 3 ft + 0.5 ft = 3.5 ft
55. (a) 6 in. = 6 × 1 in. = 6 ×
0.8 mi 5280 ft 12 in.
⋅
⋅
1 mi
1 ft
1
= (0.8)(5280)(12) in. = 50,688 in.
47. 0.8 mi =
(b) 3 ft = 3 × 1 ft = 3 × 12 in. = 36 in.
3'6" = 36 in. + 6 in. = 42 in.
900 in. 1 ft 1 yd
⋅
⋅
1
12 in. 3 ft
900
=
yd = 25 yd
(12)(3)
48. 900 in. =
12,672 in. 1 ft
1 mi
⋅
⋅
1
12 in. 5280 ft
12,672
1
=
= mi = 0.2 mi
(12)(5280) 5
56. 1'3"+ 6'4" =
1 ft 3 in.
+ 6 ft 4 in.
7 ft 7 in. or 7 '7"
57. 6'2"+ 4'6" =
6 ft 2 in.
+ 4 ft 6 in.
10 ft 8 in. or 10'8"
49. 12,672 in. =
6 yd 3 ft 12 in.
⋅
⋅
= (6)(3)(12) in.
1 1 yd 1 ft
= 216 in.
50. 6 yd =
1.6 mi 5280 ft 12 in.
⋅
⋅
1 mi
1 ft
1
= (1.6)(5280)(12) in. = 101,376 in.
58.
2 ft 8 in.
+ 3 ft 4 in.
5 ft 12 in. = 5 ft + 1 ft = 6 ft
59.
5 ft 2 in.
+ 6 ft 10 in.
11 ft 12 in. = 11 ft + 1 ft = 12 ft
60. 4'10"+ 6'4" =
4 ft 10 in.
+ 6 ft 4 in.
10 ft 14 in. = 10 ft + 1 ft + 2 in.
= 11 ft 2 in. or 11'2"
51. 1.6 mi =
52. (a) 6 ft = 6 × 1 ft = 6 × 12 in. = 72 in.
6'4" = 72 in. + 4 in. = 76 in.
61. 4'9"+ 3'9" =
4 ft 9 in.
+ 3 ft 9 in.
7 ft 18 in. = 7 ft + 1 ft + 6 in.
= 8 ft 6 in. or 8'6"
1
1
(b) 4 in. = 4 × 1 in. = 4 ×
ft = ft
12
3
1
1
6'4" = 6 ft + ft = 6 ft
3
3
53. (a) 10 ft = 10 × 1 ft = 10 × 12 in. = 120in.
10 ft 8 in. = 120 in. + 8 in. = 128 in.
250
Section 7.1
62.
8 ft 8 in.
− (5 ft 4 in.)
3 ft 4 in.
63.
3 ft 2 in. = 2 ft 14 in
− (1 ft 5 in.) = − (1 ft 5 in.)
1 ft 9 in.
Converting U.S. Customary Units of Length
75. 10 ft 6 in. = 10.5 ft
10.5 ft + 3 ft + 10.5 ft + 3 ft = 27 ft
27 ft
= 18
1.5 ft
18 pieces of border are needed.
76. 114 ft + 63 ft + 114 ft + 63 ft = 354 ft
= 354 × 1 ft
1
= 354 × yd
3
= 118 yd
64. 9'2"− 4'10" =
9 ft 2 in. = 8 ft 14 in.
− (4 ft 10 in.) − (4 ft 10 in.)
4 ft 4 in. or 4'4"
118 1
⋅ = 59
1 2
59 panels of fencing are needed.
118 yd ÷ 2 yd =
65. 2(4 ft 5 in.) = 8 ft 10 in.
66. 4(5 ft 1 in.) = 20 ft 4 in.
67. 6(4 ft 8 in.) = 24 ft 48 in.
= 24 ft + 4 ft = 28 ft
77. 8(6 ft 4 in.) = 48 ft 32 in.
= 48 ft + 2 ft + 8 in.
= 50 ft 8 in.
50 ft 8 in. is needed.
68. 8(2 ft 5 in.) = 16 ft 40 in.
= 16 ft + 3 ft + 4 in.
= 19 ft 4 in.
78. 10(3 ft 2 in.) = 30 ft 20 in.
= 30 ft + 1 ft + 8 in.
= 31 ft 8 in.
31 ft 8 in. is needed.
69. (6'4") ÷ 2 = 3'2"
70. (16'8") ÷ 8 = 2'1"
71.
18 ft 3 in.
= 6 ft 1 in.
3
72.
10 ft 10 in.
= 2 ft 2 in.
5
79. 4'6"+ 2'8" =
4 ft 6 in.
+ 2 ft 8 in.
6 ft 14 in. = 6 ft + 1 ft + 2 in. = 7 ft 2 in.
The plumber used 7 '2" of pipe.
1
3
ft = ft
2
4
3
3
6
1
2 ft + ft + 2 ft + ft = 4 + ft = 5 ft
4
4
4
2
73. 9 in. = 9 × 1 in. = 9 ×
80. 2(6'8") + 10' = 2(6 ft 8 in.) + 10 ft
= 12 ft + 16 in. + 10 ft
= 22 ft + 1 ft + 4 in.
= 23 ft 4 in.
The carpenter needs 23'4" of molding.
1
4
yd = yd
3
3
1
2
2 ft = 2 × 1 ft = 2 × yd = yd
3
3
74. 4 ft = 4 × 1 ft = 4 ×
81. 4 yd = 4 × 1 yd = 4 × 3 ft = 12 ft
4 yd − 5 ft = 12 ft − 5 ft = 7 ft
7 ft is left over.
4
4
2
yd + yd + yd
3
3
3
2
+ 4 yd + yd
3
12
= 12 yd +
yd
3
= 12 yd + 4 yd = 16 yd
2 yd + 6 yd +
82.
251
85 ft
= 84 ft 12 in.
− 82 ft 10 in. = − 82 ft 10 in.
2 ft 2 in.
The arena is 2 ft 2 in. shorter than
regulation.
Chapter 7
83.
Measurement
54 ft 2 1 yd 1 yd 54 2
⋅
⋅
=
yd
1
3 ft 3 ft 9
= 6 yd 2
6 ft 9 in.
= 2 ft 3 in.
3
Each piece is 2 ft 3 in. long.
84. (8'6") ÷ 2 =
89. 54 ft 2 =
8 ft 6 in.
+
= 4 ft + 3 in.
2
2
108 ft 2 1 yd 1 yd 108 2
⋅
⋅
=
yd
1
3 ft 3 ft
9
= 12 yd 2
90. 108 ft 2 =
Each piece is 4'3" long.
85. 5(6') + 4(3'3") + 2(18")
432 in.2 1 ft 1 ft
⋅
⋅
12 in. 12 in.
1
432 2
=
ft = 3 ft 2
144
91. 432 in.2 =
= 5(6 ft) + 4(3.25 ft) + 2(1.5 ft)
= 30 ft + 13 ft + 3 ft
= 46 ft
The total length is 46 '.
720 in.2 1 ft 1 ft
⋅
⋅
92. 720 in. =
1
12 in. 12 in.
720 2
=
ft = 5 ft 2
144
86. 60 yd − 12(2.5 ft) = 60 yd − 30 ft
= 60 yd − 10 yd
= 50 yd
2
There is 50 yd left over
( )
87. 32 in. − 2 4 in. = 32 in. − 8 in.
= 24 in. = 2 ft
14 2 2 ft = 56 ft
5 yd = 5 × 3 ft = 15 ft
11
56 ft
= 3 rolls
15
15 ft
( )( )
93. 5 ft 2 =
5 ft 2 12 in. 12 in.
⋅
⋅
= 720 in.2
1
1 ft 1 ft
94. 7 ft 2 =
7 ft 2 12 in. 12 in.
⋅
⋅
= 1008 in.2
1
1 ft 1 ft
95. 3 yd 2 =
She should buy 4 rolls.
(
)
3 yd 2 3 ft 3 ft
⋅
⋅
= 27 ft 2
1 1 yd 1 yd
10 yd 2 3 ft 3 ft
⋅
⋅
= 90 ft 2
96. 10 yd =
1
1 yd 1 yd
88. 100 1.5 in. = 150 in.
150
150 in. 1 ft
⋅
=
ft = 12.5 ft
12 in. 12
1
2
The moon’s orbit will move 12.5 ft.
252
Section 7.2
Converting U.S. Customary Units of Time, Weight and Capacity
Section 7.2 Converting U.S. Customary Units of Time, Weight, and
Capacity
Section 7.2 Practice Exercises
1. c.
3 ft
1 yd
8. 1
3 mi 1760 yd
⋅
= 5280 yd
1
1 mi
5280 yd 3 ft
=
⋅
= 15,840 ft
1
1 yd
15,840 ft 12 in.
=
⋅
= 190,080 in.
1
1 ft
2. 3 mi =
1
1.5 mi 1760 yd
mi =
⋅
= 2640 yd
2
1
1 mi
2640 yd 3 ft
=
⋅
= 7920 ft
1
1 yd
7920 ft 12 in.
=
⋅
= 95,040 in.
1
1 ft
9. 1 qt = 2 pt
10. 1 c = 8 fl oz
12 ft 1 yd
⋅
= 4 yd
1 3 ft
12 ft 12 in.
12 ft =
⋅
= 144 in.
1
1 ft
11. 1 lb = 16 oz
14 ft 1 yd 14
2
yd or 4 yd
⋅
=
1 3 ft
3
3
14 ft 12 in.
14 ft =
⋅
= 168 in.
1
1 ft
14. 1 ton = 2000 lb
3. 12 ft =
12. 1 pt = 2 c
13. 1 yr = 365 days
4. 14 ft =
15. 1 gal = 4 qt
16. 1 day = 24 hr
6 yd 3 ft
⋅
= 18 ft
1 1 yd
18 ft 12 in.
=
⋅
= 216 in.
1
1 ft
17. d. 112 oz
5. 6 yd =
18. c. 2000 lb
19. b. 8 fl oz
20. d. 10 fl oz
76 in. 1 ft
1
⋅
= 6 ft
1 12 in.
3
19 ft
1 yd 19
1
= 3 ⋅
=
yd = 2 yd
9
9
1 3 ft
6. 76 in. =
21. 2 yr =
22. 1
50 yd 3 ft
7. 50 yd =
⋅
= 150 ft
1 1 yd
150 ft 12 in.
=
⋅
= 1800 in.
1
1 ft
2 yr 365 days
⋅
= 730 days
1
1 yr
1
1.5 days 24 hr
days =
⋅
= 36 hr
2
1
1 day
90 min 1 hr
90
⋅
=
hr
1
60 min 60
1
3
= hr or 1 hr
2
2
23. 90 min =
24. 3 wk =
253
3 wk 7 days
⋅
= 21 days
1
1 wk
Chapter 7
Measurement
26. 3
35. 2:55:15 = 2 hr 55 min 15 sec
2 hr 60 min
2 hr =
⋅
= 120 min
1
1 hr
15sec 1min 1
15sec =
⋅
= min = 0.25min
1
60sec 4
2 hr 55 min 15 sec
= 120 min + 55 min + 0.25 min
= 175.25 min
180 sec 1 min
⋅
= 3 min
1
60 sec
25. 180 sec =
1
3.5 hr 60 min
hr =
⋅
= 210 min
2
1
1 hr
27. 72 hr =
72 hr 1 day 72
days = 3 days
⋅
=
1
24 hr 24
36. 1:40:30 = 1 hr 40 min 30 sec
1 hr = 60 min
30 sec = 0.5 min
1 hr 40 min 30 sec
= 60 min + 40 min + 0.5 min
= 100.5 min
28
28 days 1 wk
⋅
=
wk
7 days 7
1
= 4 wk
28. 28 days =
3600 sec 1 min 1 hr
⋅
⋅
1
60 sec 60 min
3600
=
hr = 1 hr
3600
29. 3600 sec =
37.
1 hr 10 min
45 min
1 hr 20 min
30 min
50 min
+ 1 hr
3 hr 155 min = 3 hr + 2 hr + 35 min
= 5 hr 35 min
Gil ran for 5 hr 35 min.
38.
1 hr 10 min = 70 min
50 min = −50 min
20 min
The difference is 20 min.
39.
15 min 30 sec
50 min 20 sec
+ 28 min 10 sec
93 min 60 sec = 93 min + 1 min
= 94 min or 1 hr 34 min
The total time is 1 hr 34 min.
40.
32 min 8 sec
+ 1 hr 2 min 40 sec
1 hr 34 min 48 sec
Keiji total time is 1 hr 34 min 48 sec.
168 hr 1 day 1 wk
⋅
⋅
1
24 hr 7 days
168
=
wk = 1 wk
168
30. 168 hr =
31. 9 wk =
9 wk 7 day 24 hr
⋅
⋅
= 1512 hr
1
1 wk 1 day
1680 hr 1 day 1 wk
⋅
⋅
1
24 hr 7 days
1680
=
wk = 10 wk
168
32. 1680 hr =
33. 1:20:30 = 1 hr 20 min 30 sec
1 hr = 60 min
30 sec = 0.5 min
1 hr 20 min 30 sec
= 60 min + 20 min + 0.5 min
= 80.5 min
34. 3:10:45 = 3 hr 10 min 45 sec
3 hr 60 min
3 hr =
⋅
= 180 min
1
1 hr
45 sec 1 min 3
⋅
= min
45 sec =
60 sec 4
1
= 0.75 min
3 hr 10 min 45 sec
= 180 min + 10 min + 0.75min
= 190.75min
−
41. 32 oz =
254
32 oz 1 lb
⋅
= 2 lb
1 16 oz
Section 7.2
Converting U.S. Customary Units of Time, Weight and Capacity
53.
2500 lb 1 ton
⋅
1
2000 lb
2500
=
tons
2000
1
= 1 tons or 1.25 tons
4
42. 2500 lb =
43. 2 tons =
8 oz 1 lb
1
⋅
= lb or 0.5 lb
1 16 oz 2
45. 4 lb =
4 lb 16 oz
⋅
= 64 oz
1 1 lb
46. 3
54. 20 lb 3 oz
15 oz
+
20 lb 18 oz = 20 lb + 1 lb + 2 oz
= 21 lb 2 oz
2 tons 2000 lb
⋅
= 4000 lb
1
1 ton
44. 8 oz =
55. 50(6 lb 4 oz) = 300 lb 200 oz
= 300 lb + 12 lb + 8 oz
= 312 lb 8 oz
The total weight is 312 lb 8 oz.
56. 6(2 lb 4 oz) = 12 lb 24 oz
= 12 lb + 1 lb + 8 oz
= 13 lb 8 oz
The total weight of 6 cans is 13 lb 8 oz.
1
3.25 tons 2000 lb
tons =
⋅
= 6500 lb
4
1
1 ton
1
2.5 tons 2000 lb
tons =
⋅
= 5000 lb
2
1
1 ton
5000 lb
=2
2500 lb
The truck will have to make 2 trips.
3000 lb 1 ton
⋅
1
2000 lb
3000
=
tons
2000
1
= 1 tons or 1.5 tons
2
57. 2
47. 3000 lb =
48. 6 lb =
49.
50.
18
12
lb +
oz
6
6
= 3 lb + 2 oz = 3 lbŹ2 oz
Each textbook weighs 3 lb 2 oz.
58. (18 lb 12 oz) ÷ 6 =
6 lb 16 oz
⋅
= 96 oz
1 1 lb
6 lb 10 oz
+ 3 lb 14 oz
9 lb 24 oz = 9 lb + 1 lb + 8 oz
= 10 lb 8 oz
59. 16 fl oz =
60. 5 pt =
12 lb 11 oz
+ 13 lb 7 oz
25 lb 18 oz = 25 lb + 1 lb + 2 oz
= 26 lb 2 oz
51.
30 lb 10 oz
− 22 lb 8 oz
8 lb 2 oz
52.
4 lb 16 oz
=
5 lb
− (2 lb 5 oz) − (2 lb 5 oz)
2 lb 11 oz
= 9 lb 16 oz
10 lb
− (3 lb 8 oz) − (3 lb 8 oz)
6 lb 8 oz
16 fl oz 1 c
⋅
=2c
1
8 fl oz
5 pt 2 c
⋅
= 10 c
1 1 pt
61. 6 gal =
6 gal 4 qt
⋅
= 24 qt
1 1 gal
62. 8 pt =
8 pt 1 qt
⋅
= 4 qt
1 2 pt
63. 1 gal =
1 gal 4 qt 2 pt 2 c
⋅
⋅
⋅
= 16 c
1 1 gal 1 qt 1 pt
64. 1 T = 3 tsp
255
Chapter 7
Measurement
65. 2 qt =
2 qt 1 gal 1
⋅
= gal
1 4 qt 2
66. 2 qt =
2 qt 2 pt 2 c
⋅
⋅
=8 c
1 1 qt 1 pt
12 pt 1 qt 1 gal
⋅
⋅
1 2 pt 4 qt
12
=
gal
8
= 1.5 gal
74. 12(1 pt) = 12 pt =
6(24 fl oz)
= 144 fl oz
144 fl oz 1 c 1 pt 1 qt 1 gal
=
⋅
⋅
⋅
⋅
1
8 fl oz 2c 2 pt 4 qt
144
=
gal
128
= 1.125 gal
The 12-pack of 1-pt bottles is 1.5 gal, and
the 6-pack is 1.125 gal. The 12-pack
contains the most water.
1 pt 2 c 8 fl oz
67. 1 pt =
⋅
⋅
= 16 fl oz
1 1 pt 1 c
32 fl oz 1 c 1 pt 1 qt
⋅
⋅
⋅
1
8 fl oz 2 c 2 pt
32
=
qt = 1 qt
32
68. 32 fl oz =
69. 2 T =
2 T 3 tsp
⋅
= 6 tsp
1 1T
70. 2 gal =
32 fl oz 1 c
⋅
=4c
1
8 fl oz
4 c 1 pt
⋅
= 2 pt
=
1 2c
= 1 qt
1 qt 1 gal
=
⋅
= 0.25 gal
1 4 qt
75. 32 fl oz =
2 gal 4 qt 2 pt
⋅
⋅
= 16 pt
1 1 gal 1 qt
3 c 8 fl oz
⋅
= 24 fl oz
1
1c
Yes, 3 c is 24 fl oz, so the 48-fl-oz jar will
suffice.
71. 3 c =
12 fl oz 1 c
12
⋅
=
c = 1.5 c
1
8 fl oz 8
1.5 c 1 pt
=
⋅
= 0.75 pt
1 2c
0.75 pt 1 qt
=
⋅
= 0.375 qt
1
2 pt
0.375 qt 1 gal
=
⋅
= 0.09375 gal
1
4 qt
76. 12 fl oz =
6 c 1 pt 1 qt 6
72. 6 c =
⋅
⋅
= qt = 1.5 qt
1 2 c 2 pt 4
Yes, 6 c is 1.5 qt, so the 2-qt bottle will
suffice.
1 qt 2 pt 2 c 8 fl oz
⋅
⋅
⋅
= 32 fl oz
1 1 qt 1 pt 1 c
$2.69
≈ $0.112 per fl oz
24 fl oz
$3.29
≈ $0.103 per fl oz
32 fl oz
The unit price for the 24-fl-oz jar is about
$0.112 per ounce, and the unit price for
the 1-qt jar is about $0.103 per ounce;
therefore the 1-qt jar is the better buy.
73. 1 qt =
77. 1 gal = 4 qt
4 qt 2 pt
=
⋅
= 8 pt
1 1 qt
8 pt 2 c
=
⋅
= 16 c
1 1 pt
16 c 8 fl oz
=
⋅
= 128 fl oz
1
1c
256
Section 7.2
Converting U.S. Customary Units of Time, Weight and Capacity
81. 1 c = 8 fl oz
1 c 1 pt
⋅
= 0.5 pt
1c=
1 2c
0.5 pt 1 qt
=
⋅
= 0.25 qt
1
2 pt
0.25 qt 1 gal
=
⋅
= 0.0625 gal
1
4 qt
5 gal 4 qt
⋅
= 20 qt
1 1 gal
20 qt 2 pt
=
⋅
= 40 pt
1 1 qt
40 pt 2 c
=
⋅
= 80 c
1 1 pt
80 c 8 fl oz
=
⋅
= 640 fl oz
1
1c
78. 5 gal =
79. 0.5 qt =
82. 1 pt = 2 c
2 c 8 fl oz
=
⋅
= 16 fl oz
1
1c
1 pt 1 qt
1 pt =
⋅
= 0.5 qt
1 2 pt
0.5 qt 1 gal
⋅
= 0.125 gal
=
4 qt
1
0.5 qt 1 gal
⋅
= 0.125 gal
1
4 qt
0.5 qt 2 pt
⋅
= 1 pt
1
1 qt
=2c
2 c 8 fl oz
⋅
= 16 fl oz
=
1c
1
0.5 qt =
80. 0.75 qt =
64 fl oz 1 c
⋅
=8 c
1
8 fl oz
8 c 1 pt
⋅
= 4 pt
=
1 2c
4 pt 1 qt
⋅
= 2 qt
=
1 2 pt
2 qt 1 gal
⋅
= 0.5 gal
=
1 4 qt
83. 64 fl oz =
0.75 qt 1 gal
⋅
= 0.1875 gal
1
4 qt
0.75 qt 2 pt
⋅
= 1.5 pt
1
1 qt
1.5 pt 2 c
=
⋅
=3c
1 1 pt
3 c 8 fl oz
=
⋅
= 24 fl oz
1
1c
0.75 qt =
Section 7.3
Metric Units of Length
Section 7.3 Practice Exercises
1. (a) metric
5. 48 oz =
(b) prefix
(c) meter; m
4.2 t 2000 lb
⋅
= 8, 400 lb
1
1t
2200 yd 1 mi
⋅
= 1.25 mi
3. 2200 yd =
1
1760 yd
2. 4.2 t =
4. 8 c =
8 c 1 pt
⋅
= 4 pt
1 2c
257
48 oz 1 lb
⋅
= 3 lb
1 16 oz
6. 2 yd =
2 yd 3 ft 12 in.
⋅
⋅
= 72 in.
1 1 yd 1 ft
7. 1 day =
1 day 24 hr 60 min
⋅
⋅
= 1440 min
1 1 day 1 hr
Chapter 7
Measurement
8. 160 fl oz
160 fl oz 1 c 1 pt 1 qt 1 gal
=
⋅
⋅
⋅
⋅
1
8 fl oz 2 c 2 pt 4 qt
160
=
gal
128
= 1.25 gal
9. 3.5 lb =
3.5 lb 16 oz
⋅
= 56 oz
1
1 lb
10. a, d, i, j
11. b, f, g
27.
1m
100 cm
28.
1m
1000 mm
29.
1m
10 dm
30.
1 dam
10 m
31. 2430 m =
12. c, e, h
32. 52 hm =
13. 3.2 cm or 32 mm
52 hm 100 m
⋅
= 5200 m
1
1 hm
14. 3.5 cm or 35 mm
33. 103 dm =
15. 2.1 cm or 21 mm
16. 2.7 cm or 27 mm
103 dm 1 m
⋅
= 10.3 m
1
10 dm
1251 mm
1m
⋅
1
1000 mm
= 1.251 m
34. 1251 mm =
17. (a) 5 cm
(b) 2 cm
(c) 2(2) = 10 + 4 = 14cm
(d) (5)(2) = 10 cm 2
18. (a)
(b)
(c)
(d)
2430 m 1 km
⋅
= 2.43 km
1
1000 m
3 cm
3 cm
4(3) = 12 cm
(3)(3) = 9 cm 2
35. 50 m =
50 m 1000 mm
⋅
= 50,000 mm
1
1m
36. 1.3 m =
1.3 m 1000 mm
⋅
= 1300 mm
1
1m
37. 4 km =
4 km 1000 m
⋅
= 4000 m
1
1 km
19. a
20. a
38. 5 m =
21. d
5 m 100 cm
⋅
= 500 cm
1
1m
39. 4.31 cm = 43.1 mm
22. b
40. 18 cm = 180 mm
23. d
41. 3328 dm = 0.3328 km
24. a
42. 128 hm = 12.8 km
1 km
25.
1000 m
43. 345 mm = 0.345 m
44. 450 mm = 4.5 dm
1 hm
26.
100 m
45. 0.25 km = 250 m
258
Section 7.3
46. 3 hm = 300 m
Metric Units of Length
61. 0.108 km = 108 m
108 m
= 24
4.5 m
There can be 24 parking spaces.
47. 4003 cm = 400.3 dm
48. 6.8 m = 680 cm
62. 2.1 m − 60 cm = 210 cm − 60 cm
= 150 cm or 1.5 m
49. 0.07 mm = 0.007 cm
50. 8 m = 800 cm
63. 30,000 mm 2
51. 20.91 m = 2091 cm
30,000 mm 2 1 cm 1 cm
⋅
⋅
1
10 mm 10 mm
= 300 cm 2
=
52. 16.24 mm = 1.624 cm
53. 2538 m = 2.538 km
54. 40 m = 0.04 km
64. 65,000,000 m 2
55. 270 m = 0.27 km
=
65,000,000 m 2 1 km
1 km
⋅
⋅
1
1000 m 1000 m
2
= 65 km
56. 283 m = 0.283 km
57. 2(12 cm) + 2(40 cm) = 24 cm + 80 cm
= 104 cm = 1.04 m
No, she needs 1.04 m of framing.
4.1 m 2 100 cm 100 cm
⋅
⋅
1
1m
1m
2
= 41,000 cm
65. 4.1 m 2 =
58. 90 cm + 10 cm + 12 cm = 112 cm = 1.12 m
Yes, she only needs 1.12 m of material.
5600 cm 2 1 m
1m
⋅
⋅
1
100 cm 100 cm
2
= 0.56 m
66. 5600 cm 2 =
59. 110 mm = 0.11 m
1.43 m
= 13
0.11 m
It will take 13 tiles.
60. 5 km = 5000 m
5000 m − 500 m = 4500 m
The difference is 4500 m.
Section 7.4
Metric Units of Mass, Capacity, and Medical
Applications
Section 7.4 Practice Exercises
3. 670 km = 670,000 m
= 67,000,000 cm
= 670,000,000 mm
1. (a) gram; g
(b) liter; L
(c) cubic
4. 3766 km = 3,766,000 m
= 376,600,000 cm
= 3,766,000,000 mm
(d) microgram
2. b. 40 km
259
Chapter 7
Measurement
5. 2.5 cm = 25 mm
= 0.025 m
= 0.000025 km
23. 42,500 cg = 425,000 mg
= 425 g
= 0.425 kg
6. 3.2 cm = 32 mm
= 0.032 m
= 0.000032 km
24. 17,000 cg = 170,000 mg
= 170 g
= 0.17 kg
7. 1.35 mm = 0.135 cm
= 0.00135 m
= 0.00000135 km
25. 325 mg = 32.5 cg
= 0.325 g
= 0.000325 kg
8. 24.3 mm = 2.43 cm
= 0.0243 m
= 0.0000243 km
26. 12 mg = 1.2 cg
= 0.012 g
= 0.000012 kg
9. 539 g = 0.539 kg
27. 1 cL < 1 L
10. 328 mg = 0.328 g
28. 1 L > 1 mL
11. 2.5 kg = 2500 g
29. 1 mL = 1 cc
12. 2011 g = 2.011 kg
30. 1 L > 1 cc
13. 0.0334 g = 33.4 mg
31. 1 cL < 1 kL
14. 0.38 dag = 38 dg
32. 1 mL < 1 cL
15. 90 hg = 9 kg
33. cubic centimeter
16. 0.003 kg = 3 g
34. b, c
17. 45 dg = 4.5 g
35. 3200 mL = 3.2 L
18. 409 cg = 4.09 g
36. 280 L = 0.28 kL
19. 1.58 kg = 1580 g
= 158,000 cg
= 1,580,000 mg
37. 7 L = 700 cL
38. 0.52 L = 520 mL
39. 42 mL = 0.42 dL
20. 2.26 kg = 2260 g
= 226,000 cg
= 2, 260,000 mg
40. 0.88 L = 0.0088 hL
41. 64 cc = 64 mL
21. 170 g = 0.17 kg
= 17,000 cg
= 170,000 mg
42. 125 mL = 125 cc
43. 0.04 L = 40 mL = 40 cc
22. 907 g = 0.907 kg
= 90,700 cg
= 907,000 mg
44. 38 cc = 38 mL = 0.038 L
260
Section 7.4
Metric Units of Mass, Capacity, and Medical Applications
45. 15 mL = 1.5 cL
63. 19 L = 0.019 kL
64. 0.25 L = 250 mL
= 0.015 L
= 0.000015 kL
65. 7(600 mg + 500 mg + 250 mg)
= 7(1350 mg)
= 9450 mg
= 9.45 g
Stacy gets 9.45 g per week.
= 0.015 L
= 0.000015 kL
46. 59 mL = 5.9 cL
= 0.059 L
= 0.000059 kL
66. 3(1800 m) + 2(6 km) = 5400 m + 12 km
= 5.4 km + 12 km
= 17.4 km
Cliff drives 17.4 km per week.
47. 35.5 cL = 355 mL
= 0.355 L
= 0.000355 kL
67.
48. 29.6 cL = 296 mL
= 0.296 L
= 0.000296 kL
$74.25
= $1.65 per L
45 L
The price is $1.65 per liter.
68. 8(120 L) = 960 L = 0.96 kL
8 cans hold 0.96 kL.
49. 2 L = 0.002 kL
= 200 cL
= 2000 mL
69. 6(710 mL) = 4260 mL = 4.26 L
A 6-pack contains 4.26 L.
70. 33(15 mL) = 495 mL = 49.5 cL
The bottle contains 49.5 cL.
50. 1 L = 0.001 kL
= 100 cL
= 1000 mL
1 qt 2 pt 2 c
⋅
⋅
=4c
1 1 qt 1 pt
4(130 mg) = 520 mg
520 mg of sodium per 1-qt bottle.
71. 1 qt =
51. 0.0377 kL = 37.7 L
= 3770 cL
= 37,700 mL
52. 0.0757 kL = 75.7 L
= 7570 cL
= 75,700 mL
72.
53. c
54. a
55. b
56. b, f
100%
= 20 servings
5%
1 
20  c  = 10 c
2 
10 cups of cereal give 100% of the
recommended daily allowance of
potassium.
73. 1 wk =
57. c, d
1 wk 7 days 24 hr
⋅
⋅
= 168 hr
1
1 wk 1 day
168 hr
= 21 doses
8 hr
21(250 mg) = 5250 mg = 5.25 g
5.25 g of the drug would be given in 1 wk.
58. a, f
59. 112.014 m = 11.2014 dam
60. 4669 km = 4,669,000 m
61. 600 mg = 0.6 g
62. 305 g = 305,000 mg
261
Chapter 7
Measurement
74. 3 days =
85. 0.2 mg = 200 mcg
3 days 24 hr
⋅
= 72 hr
1
1 day
86. 0.05 mg/day = 50 mcg/day
72 hr
= 12 doses
6 hr
12(325 mg) = 3900 mg = 3.9 g
3.9 g of the drug would be taken in a 3day period.
87. 1000 μ g = 1 mg
88. 800 μ g = 0.8 mg
89.
75. 2 cc = 2 mL
76. 11.5 mL + 1.5 mL = 13 mL = 13 cc
77. 2 cc = 2 mL
1L
1000 mL
=
= 500
2 mL
2 mL
500 people can be vaccinated.
78.
90.
45 mg
=3
15 mg
3 ml should be given.
500 mg
=2
250 mg
2 (5 ml ) = 10 ml
10 ml should be given.
100 cL
1L
=
=5
20 cL 20 cL
5 bottles can be made.
91.
18 g
18 g
=
= 0.18 g/mL
1 dL 100 mL
(20 mL )(0.18 g/mL ) = 3.6 g
79. 0.2 mg (48 ) = 9.6 mg
9.6 mg should be given to the patient.
92.
80. 0.05 mg (90 ) = 4.5 mg
4.5 mg should be given to the patient.
15 g
15 g
=
= 0.15 g/mL
1 dL 100 mL
(40 mL )(0.15 g/mL ) = 6 g
81. (a) 20 mg (20 ) = 400 mg
One dose is 400 mg.
3300 kg 1 metric ton
⋅
1
1000 kg
= 3.3 metric tons
93. 3300 kg =
(b) 5 (4 )(400 mg ) = 8000 mg or 8 g
8 g would be given over a 5-day
period.
5780 kg 1 metric ton
⋅
1
1000 kg
= 5.78 metric tons
94. 5780 kg =
82. (a) 40 mg (15 ) = 600 mg
One dose is 600 mg.
95. 10.9 metric tons
10.9 metric tons 1000 kg
=
⋅
1
1 metric ton
= 10,900 kg
(b) 10 (3)(600 mg ) = 18, 000 mg or 18 g
18 g would be given over a 10-day
period.
96. 8.5 metric tons
8.5 metric tons 1000 kg
=
⋅
1
1 metric ton
= 8500 kg
83. 0.01 mg = 10 mcg
84. 0.0004 cg = 4 mcg
262
Problem Recognition Exercises: U.S. Customary and Metric Conversions
Problem Recognition Exercises: U.S. Customary and Metric
Conversions
1. 36 c =
15. 4322 g = 4.322 kg
36 c 1 pt 1 qt
⋅
⋅
= 9 qt
1 2 c 2 pt
16. 5 m = 5000 mm
2. 220 cm = 2.2 m
3.
3
0.75 lb 16 oz
⋅
= 12 oz
lb =
4
1
1 lb
4. 0.3 L = 300 mL
5. 12 ft =
12 ft 1 yd
⋅
= 4 yd
1 3 ft
8.
20 fl oz 1 c
⋅
= 2.5 c
1
8 fl oz
18. 510 sec =
510 sec 1 min
⋅
= 8.5 min
1
60 sec
19. 4 pt =
6. 6.03 kg = 6030 g
7. 9 in. =
17. 20 fl oz =
4 pt 1 qt 1 gal 1
⋅
⋅
= gal
1 2 pt 4 qt 2
20. 26 fl oz =
9 in. 1 ft
3
⋅
= ft
1 12 in. 4
21. 5.46 kg = 5460 g
1
0.5 mi 5280 ft
⋅
= 2640 ft
mi =
2
1
1 mi
22. 9.02 L = 902 cL
6000 lb 1 ton
⋅
= 3 tons
9. 6000 lb =
1
2000 lb
8 pt 1 qt
10. 8 pt =
⋅
= 4 qt
1 2 pt
23. 9.1 mi =
9.1 mi 1760 yd
⋅
= 16,016 yd
1
1 mi
24. 48 oz =
48 oz 1 lb
⋅
= 3 lb
1 16 oz
25. 1.62 tons =
1.5 tsp 1 T 1
11. 1.5 tsp =
⋅
= T
1
3 tsp 2
27. 60 hr =
13. 36 mL = 36 cc
64 oz 1 lb
⋅
= 4 lb
1 16 oz
Section 7.5
1.62 tons 2000 lb
⋅
= 3240 lb
1
1 ton
26. 4.6 km = 4600 m
12. 21 m = 0.021 km
14. 64 oz =
26 fl oz 1 c
⋅
= 3.25 c
1
8 fl oz
60 hr 1 day
⋅
= 2.5 days
1 24 hr
28. 8 cc = 8 mL
Converting Between U.S. Customary and Metric Units
Section 7.5 Practice Exercises
1. (a) Fahrenheit; 32; 212
2. (a) Length
(b) Capacity
(b) Celsuis; 0; 100
263
Chapter 7
Measurement
(c) Mass
19. 400 ft ≈
(d) Capacity
400 ft 0.305 m
⋅
= 122 m
1
1 ft
3. 500 g = 0.5 kg
500 g = 50,000 cg
d, f
20. 0.75 m ≈
4. 500 mg = 0.5 g
500 mg = 50 cg
g, h
21. 45 in ≈
45 in 1 ft 0.305 m
⋅
⋅
≈ 1.1 m
1 12 in
1 ft
22. 150 cm ≈
5. 500 cg = 5 g
500 cg = 5000 mg
b, e
150 cm 1 in.
1 ft
⋅
⋅
≈ 4.9 ft
1
2.54 cm 12 in
0.5 ft 12 in 2.54 cm
⋅
⋅
≈ 15.2 cm
1
1 ft
1 in
23. 0.5 ft ≈
6. 500 kg = 500,000 g
500 kg = 500,000,000 mg
a, c
0.75 m 1 yd
⋅
≈ 0.8 yd
1
0.914 m
24. 6 oz ≈
6 oz 28 g
⋅
= 168 g
1 1 oz
25. 6 lb ≈
6 lb 0.45 kg
⋅
= 2.7 kg
1
1 lb
8. 200 kL = 20,000,000 cL
200 kL = 200,000 L
d, e
26. 4 kg ≈
4 kg 1 lb
⋅
≈ 8.9 lb
1 0.45 kg
9. 200 mL = 200 cc
200 mL = 0.2 L
b, g
27. 10 g ≈
10 g 1 oz
⋅
≈ 0.4 oz
1 28 g
28. 14 g ≈
14 g 1 oz
⋅
= 0.5 oz
1 28 g
7. 200 L = 0.2 kL
200 L = 200,000 mL
c, f
10. 200 cL = 2000 mL
200 cL = 2 L
a, h
29. 0.54 kg ≈
11. b
12. d
30. 0.3 lb ≈
13. a
31. 2.2 tons ≈
2 in. 2.54 cm
⋅
≈ 5.1 cm
1
1 in.
120 km 1 mi
⋅
≈ 74.5 mi
16. 120 km ≈
1
1.61 km
15. 2 in. ≈
8 m 1 yd
⋅
≈ 8.8 yd
1 0.914 m
18. 4 ft ≈
4 ft 0.305 m
⋅
≈ 1.2 m
1
1 ft
0.3 lb 0.45 kg
⋅
≈ 0.1 kg
1
1 lb
2.2 tons 2000 lb
⋅
= 4400 lb
1
1 ton
4400 lb 0.45 kg
=
⋅
= 1980 kg
1
1 lb
14. c
17. 8 m ≈
0.54 kg 1 lb
⋅
= 1.2 lb
1
0.45 kg
4500 kg 1 lb
⋅
= 10000 lb
1
0.45 kg
10000 lb 1 ton
⋅
= 5 tons
=
2000 lb
1
32. 4500 kg ≈
264
Section 7.5
Converting Between U.S. Customary and Metric Units
6 qt 0.95 L
⋅
= 5.7 L
1
1 qt
33. 6 qt ≈
34. 5 fl oz ≈
5 fl oz 30 mL
⋅
= 150 mL
1
1 fl oz
35. 120 mL ≈
36. 19 L ≈
18 mi 1.61 km
⋅
= 28.98 km
1
1 mi
18 mi is about 28.98 km. Therefore the
30-km race is longer than 18 mi.
41. 18 mi ≈
85 g 1 oz
⋅
≈ 3 oz
1 28 g
The can contains approximately 3 oz of
cat food.
42. 85 g ≈
120 mL 1 fl oz
⋅
= 4 fl oz
1
30 mL
19 L 1 qt
⋅
= 20 qt
1 0.95 L
97 lb 0.45 kg
⋅
= 43.65 kg
1
1 lb
97 lb is approximately 43.65 kg.
43. 97 lb ≈
37. 960 cc = 960 mL
960 mL 1 fl oz
=
⋅
1
30 mL
= 32 fl oz
100 yd 0.914 m
⋅
≈ 91.4 m
1
1 yd
(a) Warren runs the longer race.
(b) 100 m − 91.4 m ≈ 8.6 m longer.
44. 100 yd ≈
0.5 fl oz 30 mL
⋅
= 15 mL
1 fl oz
1
= 15 cc
38. 0.5 fl oz ≈
4 qt 0.95 L
⋅
= 3.8 L
1
1 qt
3.8($1.90) = $7.22
The price is approximately $7.22 per
gallon.
45. 1 gal = 4 qt ≈
2 lb 16 oz
⋅
= 32 oz
1 1 lb
$3.19 $3.19
=
≈ $0.100 per ounce
2 lb 32 oz
354 g 1 oz
354 g ≈
⋅
≈ 12.6 oz
1
28 g
$1.49 $1.49
≈
≈ $0.118 per ounce
354 g 12.6 oz
The box of sugar costs $0.100 per ounce,
and the packets cost $0.118 per ounce.
The 2-lb box is the better buy.
39. 2 lb =
(
8 oz 1 lb
⋅
= 0.5 lb
1 16 oz
2.5 lb 0.45 kg
2.5 lb ≈
⋅
≈ 1.13 kg
1
1 lb
This jar contains about 1.13 kg of
spaghetti sauce.
46. 8 oz =
47. 2.54 cm = 1 in.
A hockey puck is 1 in. thick.
)
40. 6 24 fl oz = 144 fl oz
144 fl oz 1 qt
⋅
= 4.5 qt
32 fl oz
1
$3.60
≈ $0.80 per qt
4.5 qt
2L≈
1.9 L 1 qt
⋅
1 0.95 L
= 2 qt
2 qt 2 pt 2 c 8 fl oz
=
⋅
⋅
⋅
1 1 qt 1 pt 1 c
= 64 fl oz
10(6 oz) = 60 oz
Yes, there is enough juice to fill 10 6-oz
glasses. There will be approximately 4 oz
left over.
48. 1.9 L ≈
2 L 1 qt
⋅
≈ 2.1 qt
1 0.95 L
$1.59
≈ $0.76 per qt
2.1 qt
The 2-L bottle costs $0.76/qt. The 6-pack
cost $0.80/qt. The 2-L bottle is the better
buy.
265
Chapter 7
Measurement
49. 99,790 g = 99.790 kg
99.790 kg 1 lb
≈
⋅
1
0.45 kg
≈ 222 lb
Tony weighs about 222 lb.
9
9
61. F = C + 32 = (4000) + 32
5
5
= 7200 + 32 = 7232
7232°F
9
9
62. F = C + 32 = (1085) + 32
5
5
= 1953 + 32 = 1985
1985°F
6m
1 ft
⋅
≈ 20 ft
50. 6 m ≈
1 0.305 m
The distance is approximately 20 ft.
51. 45 cc = 45 mL ≈
= 1.5 fl oz
45 cc is 1.5 fl oz.
9
9
63. F = C + 32 = (35) + 32 = 63 + 32 = 95
5
5
It is a hot day. The temperature is 95°F
45 mL 1 fl oz
⋅
30 mL
1
64. In Miami, the Fahrenheit scale is used so
25°F would be a cold day.
4 fl oz 30 mL
⋅
= 120 mL
52. 4 fl oz ≈
1
1 fl oz
4 fl oz is 120 mL.
(
9
9
65. F = C + 32 = (25) + 32 = 45 + 32 = 77
5
5
)
53. 2 2.9 + 2.2 + 2.9 + 2.2 + 5.8 + 4.4 ft
( )
In Italy, the Celsius scale is used.
Converting 25°C to Fahrenheit gives 77°F
which would be a warm day.
= 2 20.4 ft = 40.8 ft
54. 3 (2 (4.3) + 2 (3.2 )) ft = 3 (8.6 + 6.4 ) ft
= 3 (15 ) ft = 45 ft
9
9
F = C + 32
66. F = C + 32
5
5
9
9
= (18) + 32
= (13) + 32
5
5
= 32.4 + 32
= 23.4 + 32
= 64.4
= 55.4
The high temperature was 64.4°F, and the
low temperature was 55.4°F.
9
9
55. F = C + 32 = (25) + 32 = 45 + 32 = 77
5
5
77°F
5
5
5
56. C = ( F − 32) = (113 − 32) = (81) = 45
9
9
9
45°C
9
9
67. F = C + 32 = (100) + 32 = 9(20) + 32
5
5
= 180 + 32 = 212
5
5
5
57. C = ( F − 32) = (68 − 32) = (36) = 20
9
9
9
20°C
9
9
68. F = C + 32 = (0) + 32 = 0 + 32 = 32
5
5
9
9
58. F = C + 32 = (15) + 32 = 27 + 32 = 59
5
5
59°F
5700 lb 0.45 kg
⋅
= 2565 kg
1
1 lb
= 2.565 metric tons
The Navigator weighs approximately
2.565 metric tons.
69. 5700 lb ≈
9
9
59. F = C + 32 = (30) + 32 = 54 + 32 = 86
5
5
86°F
5
5
5
60. C = ( F − 32) = (104 − 32) = (72) = 40
9
9
9
40°C
266
Section 7.5
Converting Between U.S. Customary and Metric Units
72. 1.25 metric tons = 1250 kg
1250 kg 1 lb
≈
⋅
1
0.45 kg
≈ 2778 lb
The weight of the Mini-Cooper is
approximately 2778 lb
1200 lb 0.45 kg
⋅
= 540 kg
1 lb
1
= 0.54 metric ton
The maximum capacity is about 0.54
metric ton.
70. 1200 lb ≈
71. 108 metric tons = 108,000 kg
108,000 kg 1 lb
≈
⋅
1
0.45 kg
= 240,000 lb
The average weight of the blue whale is
approximately 240,000 lb.
Chapter 7
Review Exercises
Section 7.1
1. 48 in. =
2 mi 1760 yd
⋅
= 3520 yd
1
1 mi
4. 2200 yd =
1
mi =
2
7. 2 yd =
2200 yd 1 mi
1
⋅
= 1 mi
1
1760 yd
4
5 ft 3 in. = 4 ft 15 in.
− 2 ft 5 in. − 2 ft 5 in.
2 ft 10 in.
or 2'10"
1
2
12 ft 7 in. = 11 ft 19 in.
− 8 ft 10 in. − 8 ft 10 in.
3 ft 9 in.
13. 4 × (5'3") = 4(5 ft 3 in.)
= 20 ft 12 in.
= 21 ft or 21'
14. 2 × (4 ft 8 in.) = 8 ft 16 in.
= 8 ft + 1 ft + 4 in.
= 9 ft 4 in.
mi 5280 ft
⋅
= 2640 ft
1
1 mi
2 yd 3 ft 12 in.
⋅
⋅
= 72 in.
1 1 yd 1 ft
6336 in. 1 ft
1 mi
⋅
⋅
1
12 in. 5280 ft
= 0.1 mi
8. 6336 in. =
9.
11. 5'3"− 2'5" =
12.
7040 ft 1 mi
1
5. 7040 ft =
⋅
= 1 mi
1
5280 ft
3
6.
4 ft 11 in.
+ 1 ft 5 in.
5 ft 16 in.
= 5 ft + 1 ft + 4 in.
= 6 ft 4 in. or 6'4"
48 in. 1 ft
⋅
= 4 ft
1 12 in.
3 1 ft 12 in.  13 
1
=   (12) in.
2. 3 ft = 4 ⋅
1
1 ft
4
 4
= 39 in.
3. 2 mi =
10. 4'11"+ 1'5" =
3 ft 9 in.
+ 5 ft 6 in.
8 ft 15 in. = 8 ft + 1 ft + 3 in.
= 9 ft 3 in.
267
15. (6 ft 3 in.) ÷ 3 =
6 ft 3 in.
+
= 2 ft 1 in.
3
3
16. (6 yd 2 ft) ÷ 2 =
6 yd 2 ft
+
= 3 yd 1 ft
2
2
Chapter 7
17.
Measurement
1 ft 4 in.
1 ft 4 in.
2 ft
2 ft
+ 10 in.
6 ft 18 in. = 6 ft + 1 ft + 6 in.
1
= 7 ft 6 in. or 7 ft
2
Section 7.2
6 min 60 sec
⋅
= 360 sec
1
1 min
1 wk 7 days 24 hr
⋅
⋅
= 168 hr
1
1 wk 1 day
0.25 ton 2000 lb
⋅
= 500 lb
1
1 ton
16 qt 1 gal
⋅
= 4 gal
1
4 qt
3 lb 10 oz
4 lb 2 oz
+ 4 lb 1 oz
11 lb 13 oz
The total weight was 11 lb 13 oz.
Section 7.3
3500 lb 1 ton
⋅
1
2000 lb
3
= 1.75 tons or 1 tons
4
35. b
25. 3500 lb =
26. 150 min =
30. 16 qt =
34. 24(12 fl oz)
= 288 fl oz
288 fl oz 1 c 1 pt 1 qt 1 gal
=
⋅
⋅
⋅
⋅
1
8 fl oz 2 c 2 pt 4 qt
1
= 2.25 gal or 2 gal
4
12 fl oz 1 c
1
⋅
=1 c
1
8 fl oz
2
24. 0.25 ton =
12 oz 1 lb
3
⋅
= lb
1 16 oz 4
 1

33.  1 tons  ÷ 8 = (3000 lb) ÷ 8 = 375 lb
 2

375 lb will go to each location.
5 lb 16 oz
21. 5 lb =
⋅
= 80 oz
1 1 lb
23. 12 fl oz =
29. 12 oz =
32.
72 hr 1 day
19. 72 hr =
⋅
= 3 days
1
24 hr
22. 1 wk =
2 gal 4 qt 2 pt
⋅
⋅
= 16 pt
1 1 gal 1 qt
31. 2:24:30 = 2 hr + 24 min + 30 sec
2 hr 60 min
⋅
= 120 min
2 hr =
1
1 hr
30 sec 1 min
30 sec =
⋅
= 0.5 min
1
60 sec
2 hr + 24 min + 30 sec
= 120 min + 24 min + 0.5 min
= 144.5 min
50 yd 3 ft
⋅
= 150 ft
1 1 yd
150 ft − 48 ft = 102 ft
102 ft 1 yd
102 ft =
⋅
= 34 yd
1
3 ft
There are 102 ft or 34 yd of wire left.
18. 50 yd =
20. 6 min =
28. 2 gal =
36. a
37. c
150 min 1 hr
⋅
= 2.5 hr
1
60 min
38. d
39. 52 cm = 520 mm
1800 sec 1 min 1 hr
⋅
⋅
27. 1800 sec =
1
60 sec 60 min
= 0.5 hr
40. 91 mm = 9.1 cm
41. 2.338 km = 2338 m
268
Chapter 7
42. 93 m = 0.093 km
64.
43. 34 dm = 3.4 m
44. 2.1 m = 0.21 dam
Review Exercises
1.2 L = 120 cL
120 cL
=5
24 cL
5 glasses can be filled.
65. 3200 g = 3.2 kg
68 kg − 3.2 kg = 64.8 kg
The difference is 64.8 kg.
45. 4 cm = 0.04 m
46. 3 m = 300 cm
47. 1.2 m = 1200 mm
48. 4023 hm = 402.3 km
66. 2 m = 200 cm
3(75 cm) = 225 cm
No, the board is 25 cm too short.
49. 3.851 km − 163 m = 3851 m − 163 m
= 3688 m
The difference is 3688 m.
67. (a) 80 (0.04 mg ) = 3.2 mg
3.2 mg should be prescribed.
(b) 3.2 mg (2 )(7 ) = 44.8 mg
44.8 mg would be taken in a week.
50. 56 mm + 0.8 dm + 9 cm
= 5.6 cm + 8 cm + 9 cm
= 22.6 cm
68. 0.45 mg = 450 mcg
Section 7.4
69. 3 cc = 3 mL
3 mL − 1.8 mL = 1.2 mL = 1.2 cc
There are 1.2 cc or 1.2 mL of fluid left.
51. 6.1 g = 610 cg
52. 420 g = 0.42 kg
70. (250 mg)(3)(10) = 7500 mg = 7.5 g
Clayton took 7.5 g.
53. 3212 mg = 3.212 g
54. 0.7 hg = 70 g
Section 7.5
55. 5 cg = 50 mg
56. 0.1 dag = 100 cg
57. 300 mL = 0.3 L
71. 6.2 in. ≈
6.2 in. 2.54 cm
⋅
≈ 15.75 cm
1
1 in.
72. 75 mL ≈
75 mL 1 fl oz
⋅
= 2.5 fl oz
1
30 mL
58. 2.4 hL = 240 L
59. 830 cL = 8.3 L
73. 140 g ≈
140 g 1 oz
⋅
= 5 oz
1
28 g
60. 124 mL = 124 cc
74. 5 L ≈
61. 225 cc = 225 mL = 22.5 cL
5 L 1 qt
⋅
≈ 5.26 qt
1 0.95 L
62. 0.49 kL = 490 L
75. 3.4 ft ≈
63. 125 cm = 1.25 m
Perimeter = 2(2 m) + 2(1.25 m)
= 4 m + 2.5 m
= 6.5 m
Area = (1.25 m)(2 m) = 2.5 m 2
3.4 ft 0.305 m
⋅
≈ 1.04 m
1
1 ft
76. 100 lb ≈
100 lb 0.45 kg
⋅
= 45 kg
1
1 lb
77. 120 km ≈
269
120 km 1 mi
⋅
≈ 74.53 mi
1
1.61 km
Chapter 7
78. 6 qt ≈
Measurement
6 qt 0.95 L
⋅
= 5.7 L
1
1 qt
42.195 km 1 mi
⋅
1
1.61 km
≈ 26.2 mi
The marathon is approximately 26.2 mi.
84. 42.195 km ≈
1.5 fl oz 30 mL
⋅
= 45 mL
1
1 fl oz
= 45 cc
79. 1.5 fl oz ≈
5
85. C = ( F − 32)
9
12.5 tons 2000 lb
⋅
1
1 ton
= 25,000 lb
25,000 lb 0.45 kg
=
⋅
1
1 lb
= 11, 250 kg
80. 12.5 tons =
5
86. C = ( F − 32)
9
5
= (180 − 32)
9
5
= (148)
9
≈ 82.2
82.2°C to 85°C
30 in. 2.54 cm
⋅
= 76.2 cm
1
1 in.
76.2 cm − 38 cm = 38.2 cm
The difference in height is 38.2 cm.
81. 30 in. ≈
82. 7.2 oz ≈
5
C = ( F − 32)
9
5
= (185 − 32)
9
5
= (153)
9
= 85
9
87. F = C + 32
5
7.2 oz 28 g
⋅
= 201.6 g
1
1 oz
9
9
88. F = C + 32 = (8) + 32 = 14.4 + 32
5
5
= 46.4
46.4°F
201.6 g
= 6.72
30 g
There are approximately 6.72 servings.
83. (30 mL)(2)(7) = 420 mL = 0.42 L
The total amount of cough syrup is
approximately 0.42 L.
Chapter 7
Test
1. c, d, g, j
7.
2. f, h, i
3 c
3
8 fl oz
c= 4 ⋅
= 6 fl oz
4
1
1c
6 oz + 4 oz = 10 oz of liquid
3. a, b, e
4. 25 ft =
8. 1200 sec =
25 ft 1 yd
1
⋅
= 8 yd
1
3 ft
3
9. 2'+ 30"+ 2'+ 30" = 4'+ 60"
60 in. 1 ft
⋅
= 5 ft
60 in. =
1 12 in.
4'60" = 4 ft + 5 ft = 9 ft or 9'
11,000 lb 1 ton
5. 11,000 lb =
⋅
= 5.5 tons
1
2000 lb
6. 52,800 ft =
1200 sec 1 min
⋅
= 20 min
1
60 sec
52,800 ft 1 mi
⋅
= 10 mi
1
5280 ft
270
Chapter 7
10. 1'10"+ 2'4" =
11.
1 ft 10 in.
+ 2 ft 4 in.
3 ft 14 in.
= 3 ft + 1 ft + 2 in.
= 4 ft 2 in. or 4 '2"
22. 100 m ≈
8 lb 1 oz = 7 lb 17 oz
− 7 lb 10 oz − 7 lb 10 oz
7 oz
He lost 7 oz.
100 m 1 yd
⋅
≈ 109 yd
1
0.914 m
23. 4.5 km ≈
4.5 km 1 mi
⋅
≈ 2.8 mi
1
1.61 km
24. 9603 ft ≈
9603 ft 0.305 m
⋅
≈ 2929 m
1
1 ft
20 in. 2.54 cm
⋅
= 50.8 cm
1
1 in.
38 in. 2.54 cm
⋅
= 96.52 cm
38 in. ≈
1
1 in.
50.8 cm tall and 96.52-cm wingspan
25. 20 in. ≈
12. (3 ft 11 in.)(5) = 15 ft + 55 in.
= 15 ft + 4 ft + 7 in.
= 19 ft 7 in.
13. 1:15:15 = 1 hr 15 min 15 sec
1 hr = 60 min
15 sec = 0.25 min
1 hr 15 min 15 sec
= 60 min + 15 min + 0.25 min
= 75.25 min
26. 5000 g = 5 kg =
5 kg 1 lb
⋅
≈ 11 lb
1 0.45 kg
5
5
27. C = (F − 32) = (375 − 32)
9
9
5
= (343) ≈ 190.6
9
14. 2.4 cm or 24 mm
15. c
190.6°C
16. 1158 m = 1.158 km
9
9
28. F = C + 32 = (2) + 32 = 3.6 + 32 = 35.6
5
5
17. 0.015 L = 15 mL
18. (a) Cubic centimeters
35.6°F
(b) 235 mL = 235 cc
(c) 1 L = 1000 mL = 1000 cc
19. 411 g = 41,100 cg
29.
(0.1 mg )(70 )(4 ) = 28 mg
30.
(0.125 mg )(2 )(7 ) = 1.75 mg
20. 210 g = 210,000 mg
210,000 mg
= 7 servings
30,000 mg
21. 2 L ≈
= 1750 mcg per week
2 L 1 qt
⋅
≈ 2.1 qt
1 0.95 L
Chapter 1–7
Test
Cumulative Review Exercises
1. (a) 2499 ≈ 2000
3. Area = (10 cm)(18 cm) = 180 cm 2
(b) 42,099 ≈ 42,100
2. P = 18 cm + 10 cm + 18 cm + 10 cm
= 56 cm
271
Chapter 7
Measurement
oatmeal. this is less than 1 c, so Keesha
does have enough.
4. 144 ÷ 9 ÷ (17 − 3 ⋅ 5)2 = 144 ÷ 9 ÷ (17 − 15) 2
= 144 ÷ 9 ÷ (2)2
= 144 ÷ 9 ÷ 4
= 16 ÷ 4
=4
11. The LCD of
12.
5. (a) Ford Motor Company spends the
most. That amount is $7400 million or
$7,400,000,000.
6.
(b)
4620
− 4318
302
The difference between IBM and
Motorola is $302 million or
$302,000,000.
(c)
7400
6200
4620
4379
+ 4318
26,917
The total amount spent is $26,917
million or $26,917,000,000.
10
2 5 20 17 170
8
⋅ =
or 18
14. 6 ⋅ 2 =
3 6
3 6
9
9
3
2
2
5 20 17 20 6 40
6
÷ =
⋅ =
or 2
15. 6 ÷ 2 =
3
6 3
6
3 17 17
17
1
2
5 20 17 40 17 23
5
16. 6 − 2 =
− =
− =
or 3
3
6 3
6
6
6
6
6
7. The number 32,542 is not divisible by 3
because the sum of the digits (16) is not
divisible by 3.
17.
3
9
1
= 0.3
3
18. 0.45 =
3 27
45
9
=
100 20
1 5
19. 1.25 = 1 =
4 4
2 54
2 108
108 = 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 3
1
9. Area = (54 in.)(20 in.) = 540 in.2
2
10.
1 6 3
5 12 3 14 7
+ − = + − = =
2 5 10 10 10 10 10 5
2
5 20 17
+
13. 6 + 2 =
3
6 3
6
40 17
=
+
6
6
57
=
6
19
1
or 9
=
2
2
2
2⋅3
6
=
=
13 13 ⋅ 3 39
8. 3
1 6
3
is 10.
, , and
2 5
10
1
3
(3 c) = c
4
4
1
3
of the recipe would call for c of
4
4
20.
7
= 3.5
2
21.
3
= 0.375
8
22. 0.04 =
272
4
1
=
100 25
Chapter 1–7
23. (a)
31. x = 0.45(60)
x = 27 people
6
5
(b) 6 + 5 = 11;
32. 1.26 = x(21)
1.26 21x
=
21
21
0.06 = x
6%
6
11
24. 1 L = 1000 mL
1000 mL
= 40 bottles
25 mL
25.
26.
27.
Cumulative Review Exercises
33.
18 x
=
5 25
5 x = (18)(25)
5 x = 450
5 x 450
=
5
5
x = 90 cars
2100 = 0.14( x )
2100 0.14 x
=
0.14
0.14
15,000 = x
$15,000 in sales
34. I = Prt
= ($5000)(0.034)(6)
= $1020 in interest
40 beds
≈ 6.7 beds per nurse
6 nurses
35. 5800 g = 5.8 kg
6 ? 2
=
8 3
36. 5.8 kg ≈
5.8 kg 1 lb
⋅
≈ 12.9 lb
1
0.45 kg
37. 72 in. ≈
72 in. 2.54 cm
⋅
≈ 182.9 cm
1
1 in.
38. 72 in. =
72 in. 1 ft
⋅
= 6 ft
1 12 in.
?
(6)(3) = (8)(2)
18 ≠ 16
6 2
No, because ≠ .
8 3
28.
29.
4 4 ⋅ 20 80
=
=
= 80%
5 5 ⋅ 20 100
1
3 qt 2 pt
1
= 7 pt
39. 3 qt = 2 ⋅
2
1
1 qt
3 4
=
4 x
3 x = (4)(4)
3 x = 16
3 x 16
=
3
3
16
yd
x=
3
1
40. 3
30. 1420 = 62%( x)
1420 0.62 x
=
0.62 0.62
2290 ≈ x
2290 trees
273
3 qt 0.95 L
1
qt ≈ 2 ⋅
≈ 3.3 L
2
1
1 qt
Chapter 8
Geometry
Chapter Opener Puzzle
1. e
4. a
2. d
5. f
3. c
6. b
Section 8.1
Lines and Angles
Section 8.1 Practice Exercises
1. (a) point
9. The double arrowheads indicate that the
figure is a line.
(b) line
.
(c) segment
10. The dot represents a point.
(d) P; Q
11.
(e) angle; vertex
(f) right; 180
12.
(g) protractor
(h) acute; obtuse
13.
(i) complementary; supplementary
14.
(j) parallel
(k) perpendicular
2. The vertex of ∠ABC is point B.
15.
3. A line extends forever in both directions.
A line segment is a portion of a line
between and including two endpoints.
16.
4. A line extends forever in both directions.
A ray begins at an endpoint and extends
forever in one direction.
17.
5. The single arrowhead indicates that the
figure is a ray.
18.
19. 20°
6. The endpoints indicate that the figure is a
line segment.
20. 52°
7. The dot represents a point.
21. 90°
8. The single arrowhead indicates that the
figure is a ray.
22. 115°
274
Section 8.1
Lines and Angles
50. No, because the sum of two angles that are
both less than 90º will be less than 180º.
23. 148°
24. 170°
51. Yes. For two angles to add to 90º, the
angles themselves must both be less than
90º.
25. Right
26. Obtuse
52. No, because the sum of complementary
angles is 90º, neither angle can be more
than 90º.
27. Obtuse
28. Acute
29. Acute
53. A 90° angle; 90° + 90° = 180°
30. Obtuse
54. A 45° angle; 45° + 45° = 90°
31. Straight
55.
32. Acute
33. 90° − 80° = 10°
56.
34. 90° − 5° = 85°
35. 90° − 27° = 63°
36. 90° − 64° = 26°
57.
37. 90° − 29.5° = 60.5°
38. 90° − 13.2° = 76.8°
39. 90° − 89° = 1°
58.
40. 90° − 1° = 89°
59. m(∠a) = 41° since ∠a and the given angle
are vertical angles.
m(∠b) = 180° − 41° = 139° since ∠b is
the supplement to the given angle.
m(∠c) = 180° − 41° = 139° since ∠c is the
supplement to the given angle.
41. 180° − 80° = 100°
42. 180° − 5° = 175°
43. 180° − 127° = 53°
60. m(∠a) = 180° − 58° = 122° since ∠a is
the supplement to the given angle.
m(∠b) = 58° since ∠b and the given angle
are vertical angles.
m(∠c) = 180° − 58° = 122° since ∠c is the
supplement to the given angle.
44. 180° − 124° = 56°
45. 180° − 37.4° = 142.6°
46. 180° − 173.9° = 6.1°
47. 180° − 179° = 1°
48. 180° − 1° = 179°
49. No, because the sum of two angles that are
both greater than 90º will be more than
180º.
275
Chapter 8
Geometry
61. m(∠a ) = 180° − (42° + 112°)
= 180° − 154° = 26°
since ∠a together with the two given
angles form a straight angle.
m(∠b) = 112° since ∠b and the 112° angle
are vertical angles.
m(∠c) = 26° since ∠c and ∠a are vertical
angles.
m(∠d) = 42° since ∠d and the 42° angle
are vertical angles.
72. m(∠a) = 180° − 50° = 130° since ∠a and
the given angle are supplementary.
m(∠b) = 50° since ∠b and the given angle
are vertical angles.
m(∠c) = 130° since ∠c and ∠a are
vertical angles.
m(∠d) = 130° since ∠d and ∠a are
corresponding angles.
m(∠e) = 50° since ∠e and the given angle
are alternate exterior angles.
m(∠f) = 130° since ∠f and ∠d are vertical
angles.
m(∠g) = 50° since ∠g and the given angle
are corresponding angles.
62. m(∠a) = 180° − (21° + 20°)
= 180° − 41° = 139°
since ∠a together with the two given
angles form a straight angle.
m(∠b) = 20° since ∠b and the 20° angle
are vertical angles.
m(∠c) = 21° since ∠c and the 21° angle
are vertical angles.
m(∠d) = 139° since ∠d and ∠a are
vertical angles.
73. m(∠a) = 180° − 60° = 120° since ∠a and
the given angle are supplementary.
m(∠b) = 60° since ∠b and the given angle
are vertical angles.
m(∠c) = 120° since ∠c and ∠a are
vertical angles.
m(∠d) = 120° since ∠d and ∠a are
corresponding angles.
m(∠e) = 60° since ∠e and the given angle
are corresponding angles.
m(∠f) = 120° since ∠f and ∠a are
alternate exterior angles.
m(∠g) = 60° since ∠g and the given angle
are alternate interior angles.
63. The two lines are perpendicular.
64. Supplementary angles
65. Vertical angles
66. Supplementary angles
67. a, c or b, h or e, g or f, d
74. m(∠e) = 180° − 110° = 70° since ∠e and
the given angle are supplementary.
m(∠a) = 70° since ∠a and ∠e are
corresponding angles.
m(∠b) = 110° since ∠b and the given
angle are corresponding angles.
m(∠c) = 70° since ∠c and ∠a are vertical
angles.
m(∠d) = 110° since ∠d and the given
angle are alternate interior angles.
m(∠f) = 110° since ∠f and the given angle
are vertical angles.
m(∠g) = 180° − 110° = 70° since ∠g and
the given angle are supplementary angles.
68. d, h or g, c
69. a, e or f, b
70. f, h or g, a or b, d or c, e
71. m(∠a) = 180° − 125° = 55° since ∠a and
the given angle are supplementary.
m(∠b) = 125° since ∠b and the given
angle are vertical angles.
m(∠c) = 55° since ∠c and ∠a are vertical
angles.
m(∠d) = 55° since ∠d and ∠a are
corresponding angles.
m(∠e) = 125° since ∠e and ∠b are
corresponding angles.
m(∠f) = 55° since ∠f and ∠a are alternate
exterior angles.
m(∠g) = 125° since ∠g and ∠e are
vertical angles.
75. True
76. False
77. True
276
Section 8.1
Lines and Angles
78. True
(b) 48°
79. False
(c) 42° + 90° = 132°
90. (a) 90°
80. False
(b) 90° − 67° = 23°
81. True
(c) 90°
82. True
91. 180°, since 30 sec =
83. True
84. True
92. 90°, since 15 sec =
85. 22° + 48° = 70°
86. 71° + 62° = 133°
1
min
4
93. 120°, since 20 sec =
1
min
3
94. 270°, since 45 sec =
3
min
4
87. 69° + 21° = 90°
88. 39° + 6° = 45°
1
min
2
89. (a) 180° − (42° + 90°) = 180° − 132° =
48°
Section 8.2
Triangles and the Pythagorean Theorem
Section 8.2 Practice Exercises
1. (a) 180
8. No
(b) acute; right; obtuse
9. 90° + 36° + m(∠a ) = 180°
126° + m(∠a ) = 180°
m(∠a ) = 180° − 126°
m(∠a ) = 54°
(c) equilateral
(d) isosceles
(e) scalene
10. 30° + m(∠b) + 45° = 180°
m(∠b) + 75° = 180°
m(∠b) = 180° − 75°
m(∠b) = 105°
(f) hypotenuse; legs
(g) Pythagorean; c2.
2. (a) The supplement of a 75° angle is 105°.
11. 62° + m(∠b) + 40° = 180°
m(∠b) + 102° = 180°
m(∠b) = 180° − 102°
m(∠b) = 78°
(b) The complement of a 75° angle is 15°.
3. Yes
4. Yes
12. m(∠a) + 90° + 38° = 180°
m(∠a) + 128° = 180°
m(∠a ) = 180° − 128°
m(∠a ) = 52°
5. No
6. Yes
7. No
277
Chapter 8
Geometry
13. m(∠b) = 180° − 100° = 80°
m(∠a) + 40° + 80° = 180°
m(∠a) + 120° = 180°
m(∠a) = 180° − 120°
m(∠a) = 60°
30.
25 = 5
31.
36 = 6
32.
100 = 10
14. m(∠a) = 180° − 90° = 90°
90° + m(∠b) + 49° = 180°
m(∠b) + 139° = 180°
m(∠b) = 180° − 139°
m(∠b) = 41°
33. 62 = 36
15. m(∠a) = 40°
40° + m(∠b) + 68° = 180°
m(∠b) + 108° = 180°
m(∠b) = 180° − 108°
m(∠b) = 72°
36. 32 = 9
16. m(∠b) = 59°
m(∠a) + 59° + 85° = 180°
m(∠a) + 144° = 180°
m(∠a) = 180° − 144°
m(∠a) = 36°
39. a 2 + b 2 = c 2
34. 102 = 100
35. 9 2 = 81
37.
81 = 9
38.
9 =3
32 + 42 = c 2
9 + 16 = c 2
25 = c 2
25 = c
5=c
c=5m
17. c, f
18. a, e
40.
19. b, d
7 2 + 242 = c 2
20. a, d, f
49 + 576 = c 2
625 = c 2
625 = c
25 = c
c = 25 cm
21. b, c, e
22. c
23.
49 = 7
24.
64 = 8
41. a 2 + b 2 = c 2
92 + b 2 = 152
25. 7 2 = 49
81 + b 2 = 2252
b 2 = 225 − 81
26. 82 = 64
b 2 = 144
b = 144
b = 12 yd
27. 42 = 16
28. 52 = 25
29.
a 2 + b2 = c2
16 = 4
278
Section 8.2
42.
a 2 + b2 = c2
47.
162 + 122 = c 2
a 2 + 225 = 289
256 + 144 = c 2
400 = c 2
400 = c
20 = c
The brace is 20 in. long.
a 2 = 64
a = 64
a = 8 in.
48.
a 2 + b2 = c2
144 + 25 = c 2
576 + b 2 = 676
169 = c 2
169 = c
13 = c
The ramp is 13 ft long.
b 2 = 676 − 576
b 2 = 100
b = 100
b = 10
Leg = 10 ft
49.
144 + b 2 = 225
92 + b 2 = 412
b 2 = 225 − 144
81 + b 2 = 1681
b 2 = 81
b = 81
b=9
The height is 9 km.
b 2 = 1681 − 81
b 2 = 1600
b = 1600
b = 40
Leg = 40 km
50.
a2 + b2 = c2
202 + b 2 = 252
a2 + b2 = c2
400 + b 2 = 625
322 + 242 = c 2
b 2 = 625 − 400
1024 + 576 = c 2
b 2 = 225
b = 225
b = 15
The height is 15 in.
1600 = c 2
1600 = c
40 = c
Hypotenuse = 40 in.
51. The distances traveled form the legs of a
right triangle.
a 2 + b2 = c 2
162 + 302 = c 2
256 + 900 = c
a 2 + b2 = c2
122 + b 2 = 152
44. a 2 + b 2 = c 2
46.
a 2 + b2 = c2
122 + 52 = c 2
242 + b 2 = 262
45.
a 2 + b2 = c 2
a 2 + 152 = 17 2
a 2 = 289 − 225
43.
Triangles and the Pythagorean Theorem
242 + 7 2 = c 2
2
576 + 49 = c 2
1156 = c 2
1156 = c
34 = c
Hypotenuse = 34 m
625 = c 2
625 = c
25 = c
The car is 25 mi from the starting point.
279
Chapter 8
Geometry
57. The right triangle has legs a = 4 in. and
b = 11 in. − 8 in. = 3 in.
52. a 2 + 242 = 262
a 2 + 576 = 676
a 2 + b2 = c2
a 2 = 676 − 576
42 + 32 = c 2
a 2 = 100
a = 100
a = 10
The stake should be 10 ft from the pole.
16 + 9 = c 2
25 = c 2
25 = c
5=c
c = 5 in.
P = 5 in. + 11 in. + 4 in. + 8 in. = 28 in.
53. a 2 + b2 = c 2
a 2 + 62 = 102
a 2 + 36 = 100
a 2 = 100 − 36
a 2 = 64
a = 64
a=8
P = 8 m + 6 m + 10 m = 24 m
54.
58. The unknown side c is the hypotenuse of a
right triangle with legs a = 12 m and
b = 25 m − 20 m = 5 m.
a 2 + b2 = c2
122 + 52 = c 2
144 + 25 = c 2
a2 + b2 = c2
169 = c 2
169 = c
13 = c
c = 13 m
P = 13 m + 20 m + 12 m + 25 m = 70 m
a 2 + 242 = 262
a 2 + 576 = 676
a 2 = 676 − 576
a 2 = 100
a = 100
a = 10
P = 10 ft + 24 ft + 26 ft = 60 ft
55.
59. The unknown side c is the hypotenuse of a
right triangle with legs
a = 18 − (7 + 5) = 18 − 12 = 6 ft and
b = 20 − (6 + 6) = 20 − 12 = 8 ft.
a 2 + b2 = c2
a 2 + b2 = c 2
62 + 82 = c 2
a 2 + 122 = 132
36 + 64 = c 2
a 2 + 144 = 169
100 = c 2
100 = c
10 = c
P = 10 + 7 + 6 + 18 + 20 + 5 + 6 = 72 ft
a 2 = 169 − 144
a 2 = 25
a = 25
a=5
P = 5 km + 12 km + 13 km = 30 km
56.
a 2 + b2 = c2
9 2 + 122 = c 2
81 + 144 = c 2
225 = c 2
225 = c
15 = c
P = 15 cm + 9 cm + 12 cm = 36 cm
280
Section 8.2
60. The distance is the hypotenuse c of a right
triangle with legs
a = 20 mi − 4 mi = 16 mi and
b = 8 mi + 4 mi = 12 mi.
76. a 2 + 122 = 202
a 2 + 144 = 400
a 2 = 400 − 144
a 2 = 256
a = 256
a = 16 cm
a 2 + b2 = c 2
162 + 122 = c 2
256 + 144 = c 2
400 = c 2
400 = c
20 = c
Tyler is 20 mi from the starting point.
61.
10 is between 3 and 4; 3.162
62.
90 is between 9 and 10; 9.487
63.
116 is between 10 and 11; 10.770
64.
65 is between 8 and 9; 8.062
65.
5 is between 2 and 3; 2.236
66.
48 is between 6 and 7; 6.928
67.
427.75 ≈ 20.682
68.
3184.75 ≈ 56.434
69.
1, 246,000 ≈ 1116.244
70.
50, 416,000 ≈ 7100.423
71.
0.49 = 0.7
72.
0.25 = 0.5
73.
0.56 ≈ 0.748
74.
0.82 ≈ 0.906
Triangles and the Pythagorean Theorem
77. 52 + 102 = c 2
25 + 100 = c 2
125 = c 2
125 = c
11.180 ≈ c
Hypotenuse = 11.180 mi
78. 22 + 82 = c 2
4 + 64 = c 2
68 = c 2
68 = c
8.246 ≈ c
Hypotenuse = 8.246 m
79. 122 + b 2 = 222
144 + b 2 = 484
b 2 = 484 − 144
b 2 = 340
b = 340
b ≈ 18.439
Leg = 18.439 in.
80. 152 + b 2 = 182
225 + b 2 = 324
b 2 = 324 − 225
b 2 = 99
b = 99
b ≈ 9.950
Leg = 9.950 ft
81. 12 + 12 = c 2
75. 202 + b 2 = 292
1 + 1 = c2
400 + b 2 = 841
2 = c2
2 =c
1.41 ≈ c
The diagonal length is 1.41 ft.
b 2 = 841 − 400
b 2 = 441
b = 441
b = 21 ft
281
Chapter 8
82.
Geometry
1202 + 602 = c 2
83. 252 + 252 = c 2
625 + 625 = c 2
14, 400 + 3600 = c 2
18,000 = c 2
18,000 = c
134.16 ≈ c
The length of the diagonal is 134.16 ft.
Section 8.3
1250 = c 2
1250 = c
35.36 ≈ c
The diagonal length is 35.36 ft.
Quadrilaterals, Perimeter, and Area
Section 8.3 Practice Exercises
15. P = 2l + 2w = 2(25 cm) + 2(15 cm)
= 50 cm + 30 cm = 80 cm
1. (a) perimeter
(b) area
16. P = 4s = 4(32 in.) = 128 in.
2. a, d, e.
17. P = 4s = 4(65 mm) = 260 mm
3. (a) acute triangle
18. P = 2l + 2w = 2(3.4 yd) + 2(5.8 yd)
= 6.8 yd + 11.6 yd = 18.4 yd
(b) scalene triangle
4. (a) obtuse triangle
19. P = 1.8 m + 3 m + 2 m + 3.9 m = 10.7 m
(b) scalene triangle
20. P = 46 cm + 46 cm + 60 cm + 85 cm
= 237 cm
5. (a) right triangle
21. P = 3 ft 8 in.
2 ft 10 in.
+ 4 ft
9 ft 18 in. = 9 ft + 1 ft + 6 in.
= 10 ft 6 in.
(b) isosceles triangle
6. (a) acute triangle
(b) isosceles triangle
7. (a) obtuse triangle
22. P = 4 ft 2 in.
3 ft
+ 2 ft 9 in.
9 ft 11 in.
(b) isosceles triangle
8. (a) acute triangle
23. P = 2l + 2 w = 2(2 ft) + 2(6 in.)
= 4 ft + 12 in. = 4 ft + 1 ft = 5 ft or 60 in.
(b) equilateral triangle
9. a, b, c, d, e, h
24. P = 2l + 2w = 2(4 m) + 2(85 cm)
= 8 m + 170 cm = 8 m + 1.7 m
= 9.7 m or 970 cm
10. a, b, d, e
11. a, b, c
12. a, b, g
13. a, b, e, h
14. b, f
282
Section 8.3
25. x = 300 mm + 250 mm = 550 mm
y = 7.5 dm − 4.5 dm = 3 dm
P = 550 mm + 3 dm + 300 mm + 4.5 dm
+250 mm + 7.5 dm
= 1100 mm + 15 dm = 11 dm + 15 dm
= 26 dm or 2600 mm
Quadrilaterals, Perimeter, and Area
1
1
33. A = ( a + b) h = (47 in. + 35 in.)16 in.
2
2
1
= (82 in.)16 in. = 656 in.2
2
1
1
34. A = bh = (3 km)(1 km) = 1.5 km 2
2
2
3
3
1
1
ft − 3 in. = 1 ft − ft = 1 ft
4
4
4
2
1
1
3
3
b = 1 ft − 9 in. = 1 ft − ft = ft
2
2
4
4
1
3
1
P = 9 in. + 1 ft + ft + 3 in. + 1 ft
2
4
2
3
+1 ft
4
1
1
1
= 12 in. + 5 ft = 1 ft + 5 ft = 6 ft
2
2
2
26. a = 1
35. A = bh = (4.6 ft)(4 ft) = 18.4 ft 2
1
1
( a + b) h = (40 cm + 69 cm)12 cm
2
2
1
= (109 cm)12 cm = 654 cm 2
2
36. A =
37. A = bh = (5 ft 6 in.)(2 ft 3 in.)
= (5.5 ft)(2.25 ft) = 12.375 ft 2
27. The total length of the unknown vertical
sides is 80 ft. The total length of the
unknown horizontal sides is 40 ft + 20 ft =
60 ft.
P = 80 ft + 60 ft + 40 ft + 20 ft + 80 ft
= 280 ft
280 ft of rain gutters are needed.
38. A = s 2 = (3'6") 2 = (3.5 ft)2 = 12.25 ft 2
39. Area of large rectangle on left:
A = lw = (7.5 yd)(18.2 yd) = 136.5 yd 2
Area of smaller rectangle on right:
A = lw = (4.0 yd)(3.0 yd) = 12.0 yd 2
Total area of shaded figure
= 136.5 yd 2 + 12.0 yd 2 = 148.5 yd 2
28. The total length of the unknown horizontal
side is 12 yd + 1.5 yd = 13.5 yd. The total
length of the unknown vertical sides
(minus the doorway) is 14 yd − 1.5 yd =
12.5 yd.
P = 13.5 yd + 12.5 yd + 14 yd + 12 yd
+1.5 yd
= 53.5 yd
53.5 yd of molding is needed.
40. Area of outer rectangle:
A = lw = (18.4 ft)(12.8 ft) = 235.52 ft 2
Area of smaller removed rectangle:
A = lw = (3.1 ft)(5.8 ft) = 17.98 ft 2
Area of shaded region
= 235.52 ft 2 − 17.98 ft 2 = 217.54 ft 2
29. A = s 2 = (24 yd) 2 = 576 yd 2
7
 1 
30. A = lw = (6 ft)  2 ft  = 6 ⋅ ft 2 = 14 ft 2
3
3


1
1
31. A = bh = (12 m)(9 m) = 54 m 2
2
2
32. A = bh = (78 cm)(70 cm) = 5460 cm 2
283
Chapter 8
Geometry
41. Area of outer rectangle:
A = lw = (22 mm)(16 mm) = 352 mm 2
Area of removed triangle:
1
1
A = bh = (16 mm)(9 mm) = 72 mm 2
2
2
Area of shaded region
= 352 mm 2 − 72 mm 2 = 280 mm 2
The area to be carpeted is 382.5 ft 2 .
The area to be tiled is 13.5 ft 2 .
46. Area of triangle:
1
1
A = bh = (12 m)(4 m) = 24 m 2
2
2
Area of small upper rectangle:
A = lw = (12 m)(15 m − 12 m)
= (12 m)(3 m) = 36 m 2
Area of large lower rectangle:
A = lw = (18 m)(12 m) = 216 m 2
42. Area of outer rectangle:
A = lw = (24 in.)(16 in.) = 384 in.2
Area of one small removed triangle:
1
1
A = bh = (12 in.)(3 in.) = 18 in.2
2
2
Area of shaded region
Total area = 24 m 2 + 36 m 2 + 216 m 2
= 276 m 2
The area is 276 m 2 .
= 384 in.2 − 2(18 in.2 )
= 384 in.2 − 36 in.2 = 348 in.2
47. Area of one triangle:
1
1
A = bh = (3 yd)(1.5 yd) = 2.25 yd 2
2
2
Area of rectangle:
A = lw = (4 yd)(3 yd) = 12 yd 2
43. Area of outer rectangle:
A = lw = (8 in.)(10 in.) = 80 in.2
Area of one small removed rectangle:
A = lw = (5 in.)(2 in.) = 10 in.2
Area of shaded region
Total area = 12 yd 2 + 2(2.25 yd 2 )
= 12 yd 2 + 4.5 yd 2
= 80 in.2 − 2(10 in.2 )
= 16.5 yd 2
= 80 in.2 − 20 in.2 = 60 in.2
The area of the sign is 16.5 yd 2 .
44. Area of outer square:
A = s 2 = (12 in.) 2 = 144 in.2
Area of one removed triangle:
1
1
A = bh = (2.2 in.)(2.2 in.) = 2.42 in.2
2
2
Area of shaded region
48. Area of upper triangle:
1
1
A = bh = (1 ft)(1.25 ft) = 0.625 ft 2
2
2
Area of lower triangle:
1
1
A = bh = (1 ft)(2 ft) = 1 ft 2
2
2
Total area = 0.625 ft 2 + 1 ft 2 = 1.625 ft 2
= 144 in.2 − 2(2.42 in.2 )
= 144 in.2 − 4.84 in.2 = 139.16 in.2
The area is 1.625 ft 2 .
45. The area to be tiled is a trapezoid:
1
1
A = ( a + b) h = (8 ft + 10 ft)1.5 ft
2
2
1
= (18 ft)1.5 ft = 13.5 ft 2
2
The area to be carpeted is a rectangle with
the tiled area removed.
(
)(
)
49. (a) A = lw = 23 ft 21 ft = 483 ft 2
The area is 483 ft 2 .
(b)
A = lw − 13.5 ft 2 = (22 ft)(18 ft) − 13.5 ft 2
483 ft 2
≈ 1.932
250 ft 2
They will need 2 paint kits.
50. A = (2 s ) 2 = 4 s 2
The area is increased by 4 times.
= 396 ft 2 − 13.5 ft 2 = 382.5 ft 2
284
Section 8.3
Quadrilaterals, Perimeter, and Area
51. A = (3s ) 2 = 9 s 2
The area is increased by 9 times.
53. False
52. True
55. True
Section 8.4
54. False
Circles, Circumference, and Area
Section 8.4 Practice Exercises
1. (a) radius
9. d = 2r = 2(6 in.) = 12 in.
(b) diameter
10. d = 2r = 2(44 mm) = 88 mm
(c) circumference
3 
11. d = 2r = 2  m  = 3 m
2 
(d) diameter
(e) 3.14;
22
7
1
 1
12. d = 2r = 2  yd  = yd
4
 2
(f) Either formula can be used.
2. The formula for circumference C = 2πr
shows the radius raised to the first power
which is consistent with linear units of
measurement such as yd, ft, and m. The
formula for area A = πr 2 shows the radius
squared which is consistent with square
units of measurement such as yd2, ft2, and
m2.
3. A = lw = (42 cm)(30 cm) = 1260 cm 2
13. r =
d 8 in.
=
= 4 in.
2
2
14. r =
d 20 cm
=
= 10 cm
2
2
15. r =
d 16.6 m
=
= 8.3 m
2
2
16. r =
d 52.2 mm
=
= 26.1 mm
2
2
17. c
4. A = bh = (42 cm)(30 cm) = 1260 cm 2
18. a, d, g, h
1
1
5. A = bh = (42 cm)(30 cm) = 630 cm 2
2
2
19. π is the circumference divided by the
C
diameter. That is, π = .
d
6. The areas of the rectangle and the
parallelogram are the same, and the area
of the triangle is one-half that area.
20. a, d
21. (a) C = 2πr = 2π (2 m) = 4π m
7. Yes. Since a rectangle is a special type of
parallelogram (one that contains four right
angles), the area formula for a
parallelogram applies to a rectangle.
(b) C = 2πr ≈ 2(3.14)(2 m) = 12.56 m
22. (a) C = 2πr = 2π (5 ft) = 10π ft
8. The length of a radius is one-half the
length of a diameter.
(b) C = 2πr ≈ 2(3.14)(5 ft) = 31.4 ft
285
Chapter 8
Geometry
23. (a) C = πd = π(20 cm) = 20π cm
34. C = πd = π(5 mm) ≈ (3.14)(5 mm)
= 15.7 mm
(b) C = πd ≈ (3.14)(20 cm) = 62.8 cm
35. (a) A = πr 2 ≈ π(7 m)2 = 49π m 2
24. (a) C = πd = π(12 yd) = 12π yd
 22 
(b) A = πr 2 ≈   (7 m)2
 7
 22 
=   (49 m 2 ) = 154 m 2
 7
(b) C = πd ≈ (3.14)(12 yd) = 37.68 yd
25. (a) C = 2πr = 2π(2.1 cm) = 4.2π cm
(b) C = 2πr ≈ 2(3.14)(2.1 cm)
= 13.188 cm
2
49
7

π km2
36. (a) A = πr 2 ≈ π  km =
4

16
26. (a) C = 2πr = 2π(6.3 in.) = 12.6π in.
(b) C = 2πr ≈ 2(3.14)(6.3 in.)
= 39.564 in.
 22   7

(b) A = πr ≈    km
 7 4

 22   49

= 
km 2 
 7   16

77
5
=
km 2 or 9 km 2
8
8
2
 1

27. (a) C = 2πr = 2π  2 km  = 2π 2.5 km
 2

= 5π km
(
)
 1

(b) C = 2πr ≈ 2(3.14)  2 km 
 2

= 15.7 km
 1 
28. (a) C = 2πr = 2π  1 m  = 2π 1.25 m
 4 
= 2.5π m
(
37. r =
2
d 42 in.
=
= 21 in.
2
2
(a) A = πr 2 ≈ π(21 in.)2 = 441π in.2
)
 22 
(b) A = πr 2 ≈   (21 in.)2
 7
 22 
=   (441 in.2 ) = 1386 in.2
 7
 1 
(b) C = 2πr ≈ 2(3.14)  1 m  = 7.85 m
 4 
38. r =
29. C = πd = π(6 cm) ≈ (3.14)(6 cm)
= 18.84 cm
d 21 cm 21
=
=
cm
2
2
2
2
 21

441
(a) A = πr ≈ π 
cm  =
π cm 2
4
 2

2
30. C = πd = π(8.5 cm) ≈ (3.14)(8.5 cm)
= 26.69 cm
(b)
31. C = πd = π(4.5 in.) ≈ (3.14)(4.5 in.)
= 14.13 in.
32. C = πd = π(3.5 in.) ≈ (3.14)(3.5 in.)
= 10.99 in.
39. r =
33. C = πd = π(2.2 cm) ≈ (3.14)(2.2 cm)
= 6.908 cm
2
 22  21

cm 
A = πr ≈  
 7  2

22
441
 

cm 2  = 346.5 cm 2
=  
7
4
 

2
d 25 mm
=
= 12.5 mm
2
2
(a) A = πr 2 ≈ π(12.5 mm)2
= 156.25π mm 2
286
Section 8.4
46. A = πr 2 − lw
(b) A = πr 2 ≈ (3.14)(12.5 mm) 2
≈ (3.14)(5 cm) 2 − (8 cm)(6 cm)
= (3.14)(156.25 mm 2 ) ≈ 491 mm 2
40. r =
= (3.14)(25 cm 2 ) − (8 cm)(6 cm)
d 10 ft
=
= 5 ft
2
2
= 78.5 cm 2 − 48 cm 2
= 30.5 cm 2
(a) A = πr 2 ≈ π(5 ft)2 = 25π ft 2
2
(b) A = πr ≈ (3.14)(5 ft)
Circles, Circumference, and Area
1
1 d 
47. A = bh + π  
2
2 2
2
= (3.14)(25 ft 2 ) ≈ 79 ft 2
2
1
1
 4 in. 
≈ (4 in.)(6 in.) + (3.14) 

2
2
 2 
1
1
= (4 in.)(6 in.) + (3.14)(2 in.)2
2
2
1
1
= (4 in.)(6 in.) + (3.14)(4 in.2 )
2
2
2
2
= 12 in. + 6.28 in.
41. (a) A = πr 2 ≈ π(6.2 ft)2 = 38.44π ft 2
(b) A = πr 2 ≈ (3.14)(6.2 ft)2
= (3.14)(38.44 ft 2 ) ≈ 121 ft 2
42. (a) A = πr 2 ≈ π(2.9 m)2 = 8.41π m 2
= 18.28 in.2
 1  d 2 
48. A = s 2 + 4  π   
2 2
   
(b) A = πr 2 ≈ (3.14)(2.9 m)2
= (3.14)(8.41 m 2 ) ≈ 26 m 2
2
1
6 m 
≈ (6 m)2 + 4  (3.14) 
 
 2  
 2
1

= 36 m 2 + 4  (3.14)(9 m 2 ) 
2

1
43. A = (a + b)h − 2πr 2
2
1
≈ (5 ft + 4 ft)(2 ft) − 2(3.14)(1 ft) 2
2
1
= (9 ft)(2 ft) − 2(3.14)(1 ft 2 )
2
= 9 ft 2 − 6.28 ft 2
= 36 m 2 + 56.52 m 2
= 92.52 m 2
= 2.72 ft 2
49. Area of outer circle = πr 2
≈ (3.14)(10 mm)2
44. A = bh − πr 2
= (3.14)(100 mm 2 )
≈ (5.2 cm)(3.2 cm) − (3.14)(1 cm)2
= 314 mm 2
= 16.64 cm 2 − 3.14 cm 2
Area of inner circle = πr 2
= 13.5 cm 2
d
45. A = s 2 − π  
2
2
≈ (3.14)(8 mm)2
2
 16 in. 
≈ (16 in.)2 − (3.14) 

 2 
= (16 in.)2 − (3.14)(8 in.)2
= (3.14)(64 mm 2 )
= 200.96 mm 2
2
Area = 314 mm 2 − 200.96 mm 2
= 113.04 mm 2
= 256 in.2 − (3.14)(64 in.2 )
= 256 in.2 − 200.96 in.2
= 55.04 in.2
287
Chapter 8
Geometry
1 d
50. A = lw − π  
2 2
53. C = π d ≈ (3.14 )(5.3 mi) = 16.642 mi
2
54. A = πr 2
2
1
 8 ft 
≈ (8 ft)(4 ft) − (3.14) 

2
 2 
1
= (8 ft)(4 ft) − (3.14)(4 ft) 2
2
1
= (8 ft)(4 ft) − (3.14)(16 ft 2 )
2
2
= 32 ft − 25.12 ft 2
= 6.88 ft
≈ (3.14)(2 ft)2
= (3.14)(4 ft 2 )
= 12.56 ft 2
55. A = πr 2
≈ (3.14)(30 ft)2
2
= (3.14)(900 ft 2 )
= 2826 ft 2
2

1  d
51. A = lw + 3  π   
 2  2  
56. A = πr 2
2

 6 in.  
1

≈ (18 in.)(10 in.) + 3  (3.14) 
 2  
 2
1

= (18 in.)(10 in.) + 3  (3.14)(3 in.)2 
2

1

= (18 in.)(10 in.) + 3  (3.14)(9 in.2 )
2

= π(3 m)2
= π(6 m)2
= 9π m 2
= 36π m2
36π m 2
=4
9π m 2
The area of the larger circle is 4 times
larger.
2
 d
 32 mi 
57. (a) A = π   ≈ 3.14 
 2
 2 
= 180 in.2 + 42.39 in.2
= 222.39 in.2
2
24 yd 3 ft
⋅
= 72 ft
1 1 yd
30 yd 3 ft
30 yd =
⋅
= 90 ft
1 1 yd
≈ 804 mi
 1   24 yd 
(b) A = (24 yd)(30 yd) + 2   π 
 2   2 
2
(
2
 d
 10 mi 
(b) A = π   ≈ 3.14 
 2
 2 
2
≈ 79 mi
2
(
)
2
= 3.14 (5 mi ) = 3.14 25 mi2
1
P = 90 ft + 90 ft + 2   π(72 ft)
2
≈ 90 ft + 90 ft + 226.08 ft
= 406.08 ft
Cost = (406.08 ft)($2.59) ≈ $1051.75
)
58. (a) C = πd = (3.14)(20 in.) = 62.8 in.
(b) 40 yd =
2
40 yd 3 ft 12 in.
⋅
⋅
= 1440 in.
1 1 yd 1 ft
1440 in.
≈ 23 times
62.8 in.
 1
≈ (24 yd)(30 yd) + 2   π(12 yd)2
 2
59. (a) C = πd ≈ (3.14)(26 in.) = 81.64 in.
= (24 yd)(30 yd) + (3.14)(144 yd 2 )
= 720 yd + 452.16 yd
2
= 3.14 (16 mi) = 3.14 256 mi2
52. (a) 24 yd =
2
A = πr 2
(b)
2
= 1172.16 yd 2
Cost = (1172.16 yd 2 )($8.00) = $9377.28
288
12,000 in.
≈ 147 times
81.64 in.
Section 8.4
60. C = πd ≈ (3.14)(22 in.) = 69.08 in.
Distance = (69.08 in.)(1000)
= 69,080 in.
69,080 in. 1 ft
=
⋅
1
12 in.
≈ 5757 ft
2
66. 8-in. pizza: r = 4 in.
A = πr 2 = π(4 in.)2 = 50.27 in.2
)
Cost per in.2 =
(b) 25 21 in. = 525 in.
525 in. 1 ft
⋅
≈ 43.8 ft
12 in.
1
$6.50
50.27 in.2
≈ $0.129
12-in. pizza: r = 6 in.
A = πr 2 = π(6 in.)2 = 113.10 in.2
2
62. A = π r 2 = π (12.83 cm ) ≈ 517.1341 cm 2
C = 2π r = 2π (12.83 cm ) ≈ 80.6133 cm
Cost per in.2 =
$12.40
≈ $0.110
113.10 in.2
The 12-in. is the better buy.
2
63. A = π r 2 = π (5.1 ft ) ≈ 81.7128 ft 2
C = 2π r = 2π (5.1 ft ) ≈ 32.0442 ft
2
2
 d
 103.24 mm 
65. A = π   = π 

 2

2
≈ 8371.1644 mm 2
C = π d = π (103.24 mm ) ≈ 324.3380 mm
61. (a) C = πd ≈ (3.14)(6.75 in.) = 21 in.
(
Circles, Circumference, and Area
2
 d
 9.5 in. 
64. A = π   = π 
≈ 70.8822 in.2
 2
 2 
C = π d = π (9.5 in.) ≈ 29.8451 in.
Problem Recognition Exercises: Area, Perimeter, and Circumference
2
1 
4. A = lw = (2 ft ) ft  = 1 ft 2 or
2 
A = lw = (24 in.)(6 in.) = 144 in.2
1 
P = 2l + 2w = 2 (2 ft ) + 2  ft 
2 
= 4 ft + 1 ft = 5 ft or
P = 2l + 2w = 2 (24 in.) + 2 (6 in.)
= 48 in. + 12 in. = 60 in.
1. A = s 2 = (5 ft ) = 25 ft 2
P = 4 s = 4 (5 ft ) = 20 ft
2
2. A = s 2 = (12 m ) = 144 m 2
P = 4 s = 4 (12 m ) = 48 m
3. A = lw = (4 m )(3 m ) = 12 m 2 or
A = lw = (400 cm )(300 cm )
= 120, 000 cm 2
P = 2l + 2 w = 2 (4 m ) + 2 (3 m )
= 8 m + 6 m = 14 m or
P = 2l + 2 w = 2 (400 cm ) + 2 (300 cm )
= 800 cm + 600 cm = 1400 cm
1  1
5. A = bh = (1 yd ) yd  = yd 2 or
3  3
A = bh = (3 ft )(1 ft ) = 3 ft 2
1

P = 2a + 2b = 2 (1 yd ) + 2  yd 
2

= 2 yd + 1 yd = 3 yd or
P = 2l + 2w = 2 (6 ft ) + 2 (1.5 ft )
= 12 ft + 3 ft = 9 ft
289
Chapter 8
Geometry
2
6. A = bh = (1.1 km )(0.43 km ) = 0.473 km
or
A = bh = (1100 m )(430 m ) = 473, 000 m 2
P = 2a + 2b = 2 (1.1 km ) + 2 (0.52 km )
= 2.2 km + 1.04 km = 3.24 km or
P = 2l + 2w = 2 (1100 m ) + 2 (520 m )
= 2200 m + 1040 m = 3240 m
7.
8.
2
(
2
= 1256 cm
C = π d = 3.14 ( 40 cm ) = 125.6 cm
13. A = π r 2 =
1
1
bh = (5 cm )(12 cm ) = 30 cm 2
2
2
c 2 = a 2 + b 2 = 5 2 + 12 2 = 25 + 144 = 169
c = 169 = 13 cm
P = a + b + c = 5 cm + 12 cm + 13 cm
= 30 cm
14. A = π r 2 =
A=
= 616 ft
(
)
(
)
22
22
(14 ft )2 = 196 ft 2
7
7
2
 22 
C = 2π r = 2   (14 ft ) = 88 ft
 7
1  d
1
1
9. A = (a + b )h = (14 m + 8 m )(4 m )
2
2
1
= (22 m )(4 m ) = 44 m 2
2
P = 14 m + 5 m + 8 m + 5 m = 32 m
2
1
 4 ft 
2
15. A = π   + lw = 3.14 
+ (4ft)(8ft)
 2 
2  2
2
1
2
= 3.14 ( 2ft ) + 32ft 2
2
1
= 3.14 4 ft 2 + 32ft 2 = 6.28 yd 2 + 32ft 2
2
= 38.28ft 2
1
1
C = π d + 8 + 4 + 8 = 3.14 ( 4 ft ) + 20 ft
2
2
= 6.28 ft + 20 ft = 26.28 ft
(
1
1
(a + b )h = (14 in. + 8 in.)(8 in.)
2
2
1
= (22 in.)(8 in.) = 88 in.2
2
P = 8 in. + 8 in. + 14 in. + 10 in. = 40 in.
10. A =
)
2
11. A = π  d  = 3.14  6 yd  = 3.14 (3yd )2
 2
(
= 3.14 9 yd
2
 2 
= 28.26 yd 2
)
C = π d = 3.14 (6 yd ) = 18.84 yd
Section 8.5
)
22
22
(7 cm )2 = 49 cm 2
7
7
= 154 cm 2
 22 
C = 2π r = 2   (7 cm ) = 44 cm
 7
1
1
bh = (3 yd )(4 yd ) = 6 yd 2
2
2
c 2 = a 2 + b 2 = 32 + 4 2 = 9 + 16 = 25
c = 25 = 5 yd
P = a + b + c = 3 yd + 4 yd + 5 yd = 12 yd
A=
2
2
d
 40 cm 
12. A = π   = 3.14 

2
 2 
2
= 3.14 ( 20 cm ) = 3.14 400 cm 2
Volume
Section 8.5 Practice Exercises
1. (a) s3
2. 90° and 78°
(b) lwh
3. C = 2πr ≈ 2(3.14)(4 in.) = 25.12 in.
(c) πr 2 h
A = πr 2 ≈ (3.14)(4 in.)2
= (3.14)(16 in.2 ) = 50.24 in.2
(d) cone; sphere.
290
Section 8.5
4. C = 2πr ≈ 2(3.14)(10 cm) = 62.8 cm
17. V = πr 2 h ≈ (3.14)(2 mm)2 (1 mm)
A = πr 2 ≈ (3.14)(10 cm)2
= (3.14)(4 mm 2 )(1 mm) = 12.56 mm 3
= (3.14)(100 cm 2 ) = 314 cm 2
d
5. A = lw − 2π  
2
18. V = πr 2 h ≈ (3.14)(3 m) 2 (6 m)
2
= (3.14)(9 m 2 )(6 m) = 169.56 m 3
 8 cm 
≈ (24 cm)(12 cm) − 2(3.14) 

 2 
= (24 cm)(12 cm) − 2(3.14)(4 cm)2
4 3 4
πr ≈ (3.14)(9 yd)3
3
3
4
= (3.14)(729 yd 3 ) = 3052.08 yd 3
3
2
19. V =
= (24 cm)(12 cm) − 2(3.14)(16 cm 2 )
3
= 288 cm 2 − 100.48 cm 2
 12 in. 
4
4
20. V = πr 3 ≈ (3.14) 
3
3
 2 
4
4
= (3.14)(6 in.)3 = (3.14)(216 in.3 )
3
3
= 904.32 in.3
= 187.52 cm 2
1
1
a + b h = 24 in. + 14 in. 12 in.
2
2
1
= 38 in. 12 in. = 228 in 2
2
6. A =
(
)
(
)(
Volume
(
)(
)
)
1
21. V = πr 2 h
3
1
≈ (3.14)(5 cm)2 (9 cm)
3
1
= (3.14)(25 cm 2 )(9 cm)
3
= 235.5 cm3
7. b, d
8. c, e
9. A = s 2 = (1 ft) 2 = 1 ft 2
V = s 3 = (1 ft)3 = 1 ft 3
2
 12 ft 
1
1
22. V = πr 2 h ≈ (3.14) 
(10 ft)
3
3
 2 
1
= (3.14)(6 ft)2 (10 ft)
3
1
= (3.14)(36 ft 2 )(10 ft) = 376.8 ft 3
3
10. A = s 2 = (1 m) 2 = 1 m 2
V = s 3 = (1 m)3 = 1 m3
11. A = s 2 = (1 km) 2 = 1 km 2
V = s 3 = (1 km)3 = 1 km3
12. A = s 2 = (1 mi) 2 = 1 mi 2
3
 12 ft 
1  4 3 1 4
⋅  πr  ≈ ⋅ (3.14) 
2 3
 2 3
 2 
1 4
= ⋅ (3.14)(6 ft)3
2 3
1 4
= ⋅ (3.14)(216 ft 3 ) = 452.16 ft 3
2 3
23. V =
V = s 3 = (1 mi)3 = 1 mi3
13. V = s 3 = (1.4 cm)3 = 2.744 cm3
14. V = s 3 = (4.5 m)3 = 91.125 m3
15. V = lwh = (8 ft)(12 ft)(6 in.)
= (8 ft)(12 ft)(0.5 ft) = 48 ft
1  4 3 1 4
⋅ πr  ≈ ⋅ (3.14)(15 cm)3
2  3
 2 3
1 4
= ⋅ (3.14)(3375 cm 3 ) = 7065 cm 3
2 3
24. V =
3
16. V = lwh = (2.5 yd)(0.8 ft)(0.8 ft)
= (7.5 ft)(0.8 ft)(0.8 ft) = 4.8 ft 3
291
Chapter 8
Geometry
d 8.2 in.
=
= 4.1 in.
2
2
4
4
V = πr 3 ≈ (3.14)(4.1 in.)3
3
3
4
= (3.14)(68.921 in.3 ) ≈ 289 in.3
3
25. r =
(b)
(b)
2 ft 3
≈ 38 bags
≈ (3.14)(3 mm)2 (20 mm)
− (3.14)(1 mm)2 (20 mm)
≈ 565.2 mm3 − 62.8 mm3 ≈ 502 mm3
d 6 in.
=
= 3 in.
2
2
d 4 in.
r= =
= 2 in.
2
2
V = πR 2 h − πr 2 h
34. R =
2
 12 cm 
1
1
28. V = πr 2 h ≈ (3.14) 
(20 cm)
3
3
 2 
1
= (3.14)(6 cm)2 (20 cm)
3
1
= (3.14)(36 cm 2 )(20 cm) ≈ 754 cm 3
3
≈ (3.14)(3 in.)2 (30 in.)
− (3.14)(2 in)2 (30 in)
≈ 847.8 in 3 − 376.8 in 3 ≈ 471 in 3
d 6 in.
=
= 3 in. = 0.25 ft
2
2
d 2 ft
=
= 1 ft
2
2
4
V = πr 2 h + πr 3
3
35. r =
V = πr 2 h ≈ (3.14)(0.25 ft)2 (50 ft)
= (3.14)(0.0625 ft 2 )(50 ft) ≈ 10 ft 3
4
≈ (3.14)(1 ft)2 (9 ft) + (3.14)(1 ft)3
3
4
= (3.14)(1 ft 2 )(9 ft) + (3.14)(1 ft 3 )
3
3
3
≈ 28.26 ft + 4.19 ft ≈ 32 ft 3
2
2
30. V = π r h ≈ (3.14)(1 ft) (3 ft)
≈ 9.4 ft 2
d 27 ft
=
= 13.5 ft
2
2
2
V = π r 2 h ≈ 3.14 (13.5 ft ) (54 in.)
31. (a) r =
2
= 3.14 (13.5 ft ) (4.5 ft )
)
= 3.14 182.25 ft 2 (4.5 ft )
≈ 2575 ft
75 ft 3
d 6 mm
=
= 3 mm
2
2
d 2 mm
r= =
= 1 mm
2
2
V = πR 2 h − πr 2 h
2
3
≈ 19, 260 gal
33. R =
 10 ft 
1
1
27. V = πr 2 h ≈ (3.14) 
(12 ft)
3
3
 2 
1
= (3.14)(5 ft)2 (12 ft)
3
1
= (3.14)(25 ft 2 )(12 ft) = 314 ft 3
3
(
0.1337 ft 3
32. (a) V = lwh = 15 ft (20 ft )(3 in.)
= 15 ft (20 ft )(0.25 ft ) = 75 ft 3
d 9 in.
26. r = =
= 4.5 in.
2
2
4
4
V = πr 3 ≈ (3.14)(4.5 in.)3
3
3
4
= (3.14)(91.125 in.3 ) ≈ 382 in.3
3
29. r =
2575 ft 3
292
Section 8.5
d 6 ft
=
= 3 ft
2
2
1 4

V = πr 2 h + ⋅  πr 3 
2 3

40. V = (3 ft)(1 ft)(1 ft) − (9 in.)(2.75 ft)(1 ft)
 3   11 
= (3 ft)(1 ft)(1 ft) −  ft  
ft (1 ft)
 4   4 
33 3 15 3
= 3 ft 3 −
ft =
ft or 0.9375 ft 3
16
16
or
V = (3 ft)(1 ft)(1 ft) − (9 in.)(2.75 ft)(1 ft)
= (36 in.)(12 in.)(12 in.)
− (9 in.)(33 in.)(12 in.)
36. r =
≈ (3.14)(3 ft) 2 (20 ft) +
1 4
⋅ (3.14)(3 ft)3
2 3
= (3.14)(9 ft 2 )(20 ft)
1 4
+ ⋅ (3.14)(27 ft 3 )
2 3
= 565.2 ft 3 + 56.52 ft 3 ≈ 622 ft 3
= 5184 in.3 − 3564 in.3 = 1620 in.3
41. V = s3 − πr 2 h
1
4
37. V = πr 2 h + πr 3
3
3
1
≈ (3.14)(2 in.)2 (5.5 in.)
3
4
+ (3.14)(2 in.)3
3
1
2
= (3.14)(4 in. )(5.5 in.)
3
4
+ (3.14)(8 in.3 )
3
3
3
≈ 23 in. + 33 in. ≈ 56 in.3
≈ (5 in.)3 − (3.14)(1 in.)2 (5 in.)
= 125 in.3 − 15.7 in.3 = 109.3 in.3
6 in.
= 3 in.
2
2 in.
Small radius r =
= 1 in.
2
42. Large radius R =
V ≈ (3.14)(3 in.) 2 (1.5 in.)
+ (3.14)(1 in.)2 (4.5 in.)
= (3.14)(9 in.2 )(1.5 in.)
38. r = 24 cm − 2 cm = 22 cm
 1 4
1 4
V = ⋅  πr 3  ≈ ⋅ (3.14)(22 cm)3
2 3
 2 3
1 4
= ⋅ (3.14)(10,648 cm 3 )
2 3
≈ 22,290 cm 3
+ (3.14)(1 in.2 )(4.5 in.)
= 42.39 in.3 + 14.13 in.3 = 56.52 in.3
43. 6 in. = 0.5 ft
V = (40 ft)(15 ft)(0.5 ft)
+ (35 ft − 15 ft)(40 ft − 25 ft)(0.5 ft)
= (40 ft)(15 ft)(0.5 ft)
+ (20 ft)(15 ft)(0.5 ft)
39. V = s3 − lwh
= (1 ft)3 − (10 in.)(10 in.)(1 ft)
 10   10 
= (1 ft)3 − 
ft
ft (1 ft)
 12   12 
= 1 ft 3 −
=
Volume
= 300 ft 3 + 150 ft 3 = 450 ft 3
44. 6 in. = 0.5 ft
1
1
V = π(40 ft)2 (0.5 ft) − π(20 ft) 2 (0.5 ft)
2
2
1
2
≈ (3.14)(1600 ft )(0.5 ft)
2
1
− (3.14)(400 ft 2 )(0.5 ft)
2
3
= 1256 ft − 314 ft 3 = 942 ft 3
100 3 44 3
ft =
ft
144
144
11 3
ft or approximately 0.306 ft 3
36
or
V = s3 − lwh
= (1 ft)3 − (10 in.)(10 in.)(1 ft)
1
1
45. V = πr 2 h ≈ (3.14)(3 in.)2 (9 in.)
3
3
1
= (3.14)(9 in.2 )(9 in.) = 84.78 in.3
3
= (12 in.)3 − (10 in.)(10 in.)(12 in.)
= 1728 in.3 − 1200 in.3 = 528 in.3
293
Chapter 8
Geometry
46. V = πr 2 h ≈ (3.14)(5 mm)2 (12 mm)
2
= (3.14)(25 mm )(12 mm) = 942 mm
1
1
48. V = πr 2 h ≈ (3.14)(2 ft)2 (6 ft)
3
3
1
= (3.14)(4 ft 2 )(6 ft) = 25.12 ft 3
3
3
47. V = πr 2 h ≈ (3.14)(20 cm)2 (40 cm)
= (3.14)(400 cm 2 )(40 cm)
= 50,240 cm 3
Chapter 8
Review Exercises
Section 8.1
1. d
18. m(∠a) = 62° since ∠a and the given angle
are alternate exterior angles.
2. a
19. m(∠b) = 118° since ∠a and ∠b are
supplementary angles.
3. c
4. b
20. m(∠c) = 118° since ∠c and ∠b are
vertical angles.
5. The measure of an acute angle is between
0° and 90°.
21. m(∠d) = 62° since ∠d and ∠a are vertical
angles.
6. The measure of an obtuse angle is
between 90° and 180°.
22. m(∠e) = 62° since ∠e and the given angle
are vertical angles.
7. The measure of a straight angle is 180°.
23. m(∠f) = 118° since ∠f and the given angle
are supplementary angles.
8. The measure of a right angle is 90°.
9. (a) 90° − 33° = 57°
24. m(∠g) = 118° since ∠g and the given
angle are supplementary angles.
(b) 180° − 33° = 147°
Section 8.2
10. (a) 90° − 20° = 70°
25. 66° + 74° + x = 180°
140° + x = 180°
x = 180° − 140°
x = 40°
m(∠x) = 40°
(b) 180° − 20° = 160°
11. m(∠ABE) = 90° − 30° = 60°
12. m(∠DBC) = 90°
13. m(∠ABG) = 180° − 5° = 175°
15. b
26. m(∠x) = 80°
m(∠y ) = 180° − (68° + 80°)
= 180° − 148°
= 32°
16. b, c
27. An obtuse triangle has one obtuse angle.
14. m(∠ABC) = 180°
17. a, c
294
Chapter 8
28. An equilateral triangle has three sides of
equal length and three angles of equal
measure.
Review Exercises
40. c 2 = a 2 + b 2
c 2 = 52 + 122
c 2 = 25 + 144
29. A right triangle has a right (90°) angle.
c 2 = 169
c = 169
c = 13
13 m of string is extended.
30. An acute triangle has three acute angles.
31. An isosceles triangle has two sides of
equal length and two angles of equal
measure.
Section 8.3
32. A scalene triangle has no sides or angles
of equal measure.
41. They both have sides of equal length, but
a square also has four right angles.
33.
25 = 5
34.
49 = 7
35.
100 = 10
43. A square is a rectangle with four sides of
equal length.
36.
64 = 8
44. A rectangle is a parallelogram with four
right angles.
42. A parallelogram must have both pairs of
opposite sides parallel.
37. The sum of the squares of the legs of a
right triangle equals the square of the
hypotenuse.
38.
45. P = 29 cm + 10 cm + 21 cm + 30 cm
= 90 cm
46. P = 4.2 m + 6.1 m + 7.0 m = 17.3 m
a2 + b2 = c2
47. P = 2l + 2w = 2(16 mi) + 2(12 mi)
= 32 mi + 24 mi = 56 mi
242 + b 2 = 252
576 + b 2 = 625
48. P = 2l + 2 w = 2(120 yd) + 2(80 yd)
= 240 yd + 160 yd = 400 yd
b 2 = 625 − 576
b 2 = 49
b = 49
b = 7 cm
49.
P = 2l + 2 w
120 ft = 36 ft + 2 w
120 ft − 36 ft = 2 w
84 ft = 2 w
84 ft 2 w
=
2
2
42 ft = w
50.
P = 4s
62 ft = 4s
62 ft 4s
=
4
4
15.5 ft = s
39. c 2 = a 2 + b 2
c 2 = 122 + 162
c 2 = 144 + 256
c 2 = 400
c = 400
c = 20 ft
1
1
51. A = bh = (8 in.)(5 in.) = 20 in.2
2
2
295
Chapter 8
Geometry
52. A = lw = (8.5 ft)(6 ft) = 51 ft 2
 22 
60. C = 2πr ≈ 2   (2.1 yd) = 13.2 yd
 7 
 22 
A = πr 2 ≈   (2.1 yd)2
 7
53. A = (150 ft − 2 ⋅12 ft)(80 ft − 2 ⋅12 ft)
= (150 ft − 24 ft)(80 ft − 24 ft)
= (126 ft)(56 ft) = 7056 ft 2
 22 
=   (4.41 yd 2 ) = 13.86 yd 2
 7
54. 62 + h 2 = 102
36 + h 2 = 100
 22 
61. C = πd ≈   (140 in.) = 440 in.
 7 
d 140 in.
r= =
= 70 in.
2
2
 22 
A = πr 2 ≈   (70 in.)2
 7
 22 
=   (4900 in.2 ) = 15, 400 in.2
 7
h 2 = 100 − 36
h 2 = 64
h = 64
h=8 m
1
1
A = bh = (9 m)(8 m) = 36 m 2
2
2
c 2 = 82 + (9 + 6) 2
c 2 = 82 + 152
62. C = πd ≈ (3.14)(40 ft) = 125.6 ft
d 40 ft
r= =
= 20 ft
2
2
c 2 = 64 + 225
c 2 = 289
c = 289
c = 17 m
A = πr 2 ≈ (3.14)(20 ft)2
= (3.14)(400 ft 2 ) = 1256 ft 2
P = 17 m + 9 m + 10 m = 36 m
The area is 36 m 2 . The perimeter is 36 m.
1
63. A = lw − (πr 2 )
2
1
≈ (20 in.)(8 in.) − (3.14)(4 in.)2
2
1
= (20 in.)(8 in.) − (3.14)(16 in.2 )
2
2
= 160 in. − 25.12 in.2 = 134.88 in.2
Section 8.4
55. d = 2r = 2(45 mm) = 90 mm
56. d = 2r = 2(3.2 ft) = 6.4 ft
57. r =
d 45 mm
=
= 22.5 mm
2
2
58. r =
d 3.2 ft
=
= 1.6 ft
2
2
64. (a) r =
A = πr 2 ≈ (3.14)(3 cm)2
= (3.14)(9 cm 2 ) = 28.26 cm 2
59. C = 2πr = 2π(8 m) ≈ 2(3.14)(8 m)
= 50.24 m
2
2
A = πr = π(8 m) ≈ (3.14)(8 m)
2
= (3.14)(64 m ) = 200.96 m
d 6 cm
=
= 3 cm
2
2
(b) r =
d 3 cm
=
= 1.5 cm
2
2
A = πr 2 ≈ (3.14)(1.5 cm)2
2
= (3.14)(2.25 cm 2 ) = 7.065 cm 2
2
(c) No
296
Chapter 8
1
65. A = s2 + (πr 2 )
2
d 6 in.
=
= 3 in.
2
2
4
4
V = πr 3 ≈ (3.14)(3 in.)3
3
3
4
= (3.14)(27 in.3 ) ≈ 113 in.3
3
71. r =
2
 2 yd 
1
≈ (2 yd) + (3.14) 
2
 2 
1
= (2 yd)2 + (3.14)(1 yd)2
2
2 1
= 4 yd + (3.14)(1 yd 2 )
2
2
= 4 yd + 1.57 yd 2 = 5.57 yd 2
2
1
72. V = πr 2 h − πr 2 h
3
≈ (3.14)(4 cm) 2 (10 cm)
1
− (3.14)(4 cm)2 (10 cm)
3
= (3.14)(16 cm 2 )(10 cm)
1
− (3.14)(16 cm 2 )(10 cm)
3
3
≈ 502.4 cm − 167.5 cm3 ≈ 335 cm3
Section 8.5
66. V = lwh = (25 cm)(25 cm)(40 cm)
= 25,000 cm 3
67. r =
d 6 ft
=
= 3 ft
2
2
2
Review Exercises
73. V = (54 in.)(10 in.)(50 in.) = 27,000 in.3
5 ft 12 in.
5 ft =
⋅
= 60 in.
1 1 ft
60 in. − 50 in. = 10 in.
V = (10 in.)(15 in.)(10 in.) = 1500 in.3
2
V = πr h ≈ (3.14)(3 ft) (8 ft)
= (3.14)(9 ft 2 )(8 ft) = 226.08 ft 3
d 30 in.
=
= 15 in.
2
2
4
4
V = πr 3 ≈ (3.14)(15 in.)3
3
3
4
= (3.14)(3375 in.3 ) = 14,130 in.3
3
68. r =
Total volume = 27,000 in.3 + 1500 in.3
= 28,500 in.3
74. V = (1 ft)(1 ft 9 in.)(1 ft 4 in.)
 7  4 
= (1 ft)  ft  ft 
 4  3 
7 3
1
= ft or 2 ft 3
3
3
1
1
69. V = πr 2 h ≈ (3.14)(3 km)2 (4 km)
3
3
1
= (3.14)(9 km 2 )(4 km) = 37.68 km 3
3
70. V = πr 2 h ≈ (3.14)(6.5 in.)2 (7.5 in.)
= (3.14)(42.25 in.2 )(7.5 in.) ≈ 995 in.3
Chapter 8
Test
5. 37º + 40º + x = 180º
77º + x = 180º
x = 180º −77º
x = 103º
1. d
2. c
3. 90º −16º = 74º
1
5
 1 
6. d = 2 r = 2  1 ft  = 2 ft or ft
 4 
2
2
4. 180º −147º = 33º
297
Chapter 8
Geometry
20. a 2 + b 2 = c 2
6
 22   5  55
ft or 7 ft
7. C = π d =    ft  =
 7 2  7
7
2
8. A = π r = 3.14 (150 ft ) ≈ 70, 650 ft
2
a 2 + 52 = 132
a 2 + 25 = 169
2
a 2 = 169 − 25
a 2 = 144
a = 144
a = 12 ft
1
1
(a + b )h = (4 ft + 12 ft )(6 ft )
2
2
1
= (16 ft )(6 ft ) = 48 ft 2
2
9. A =
10. (a)
(80 m) 2 + (60 m) 2 = c 2
21.
4 =2
6400 m 2 + 3600 m 2 = c 2
10,000 m 2 = c 2
(b) 4 2 = 16
10,000 m 2 = c
100 m = c
11. Obtuse
12. Acute
22. d
13. Right
23. c
14. Straight
24. f
15. m(∠x) = 180° − 55° = 125°
m(∠y) = 55°
25. b
26. a
16. The triangle is an isosceles triangle, so
m(∠A) = m(∠C).
m(∠A) + m(∠C ) + 90° = 180°
m(∠A) + m(∠A) = 90°
2 m(∠A) 90°
=
2
2
m(∠A) = 45°
They are each 45°.
27. e
28. P = 8s = 8(12 in.) = 96 in.
29. P = 2l + 2 w
= 2(12 ft) + 2(15 ft)
= 24 ft + 30 ft
= 54 ft
54 ft 1 yd
54 ft =
⋅
= 18 yd
1 3 ft
18 yd
=3
6 yd
3 rolls are needed.
17. m(∠S) = 90° − 41° = 49°
18. 180°
19. m(∠A) + (180° − 120°)
+ (180° − 140°) = 180°
m(∠A) + 60° + 40° = 180°
m(∠A) + 100° = 180°
m(∠A) = 180° − 100°
m(∠A) = 80°
1
1
A = bh
30. A = bh
2
2
1
1
= (6 in.)(20 in.)
= (6 in.)(4 in.)
2
2
= 60 in.2
= 12 in.2
Total area = 60 in.2 + 12 in.2 = 72 in.2
The area is 72 in.2
298
Chapter 8
33. V = lwh
= (18 in.)(5 in.)(14 in.)
31. A = lw = (12 in.)(8 in.) = 96 in.2
d 12 in.
r= =
= 6 in.
2
2
= 1260 in.3
The volume is 1260 in.3
A = πr 2
≈ (3.14)(6 in.) 2
1 2
2
34. V = πr h + πr h
3
 22 
≈   (7 cm)2 (10 cm)
 7 
1  22 
+   (7 cm) 2 (9 cm)
3 7 
 22 
=   (49 cm 2 )(10 cm)
 7 
1  22 
+   (49 cm 2 )(9 cm)
3 7 
3
= 1540 cm + 462 cm3 = 2002 cm3
2
= (3.14)(36 in. )
= 113.04 in.2
113.04 in.2 − 96 in.2 = 17.04 in.2
The area of the rectangular pizza is
96 in.2 The area of the round pizza is
approximately 113.04 in.2 The round
pizza is larger by about 17 in.2
32. r =
Test
d 8 ft
=
= 4 ft
2
2
V = πr 2 h ≈ (3.14)(4 ft)2 (3 ft)
= (3.14)(16 ft 2 )(3 ft) = 150.72 ft 3
The volume is about 151 ft 3 .
Chapters 1–8
Cumulative Review Exercises
3835
1. 21 80,535
−63
17 5
−16 8
73
−63
105
−105
0
6. LCD: 30
2 20
=
3 30
5 25
=
6 30
3 18
=
5 30
3 2 5
, ,
5 3 6
2. 0 ÷ 21 = 0
7.
3. 21 ÷ 0 is undefined.
4. 1, 275,000
− 609,000
666,000
5.
1
3
is used, so
is left.
4
4
3
3 14
42
1
(14 oz) = ⋅
oz =
oz = 10 oz
4
4 1
4
2
1
There is 10 fl oz left.
2
1 6 3
8. 6 ÷ = ⋅ = 18
3 1 1
1, 275,000
+ 1, 236,000
2,511,000
9.
299
1
1 6 1 1 1
÷6 = ÷ = ⋅ =
3
3 1 3 6 18
Chapter 8
Geometry
1
3
2 3 9 2 7 9 6
÷ ⋅ = ⋅ ⋅ =
10.
7 7 5 7 3 5 5
1
21.
1
11. 6 = 2 ⋅ 3
4=2⋅2
10 = 2 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 3 ⋅ 5 = 60
12.
1 1 7 10 15 42 67
+ + =
+
+
=
6 4 10 60 60 60 60
13.
13 3 3 130 45 18 67
− − =
−
−
=
6 4 10 60 60 60 60
22.
23.
2 12
3 34
=
5
2
15
4
1
1
2 8.3
=
9 n
2n = (9)(8.3)
2n = 74.7
2n 74.7
=
2
2
n = 37.35
25 60
=
7
x
25 x = (7)(60)
25 x = 420
25 x 420
=
25
25
x = 16.8
17 pizzas
16
14. 8 132
−8
52
−48
4
132
4
1
= 16 = 16
8
8
2
24.
408 mi
= 34 mpg
12 gal
1 5 ⋅ 9 + 1 46
15. 5 =
=
9
9
9
25.
$8590
= $3436 per hour
2.5 hr
16. $11.99
×
4
$47.96
Four glasses cost $47.96.
17.
18.
26. (0.22)(240) = 52.8
27. 46.8 = 0.65( x )
46.8 0.65 x
=
0.65 0.65
72 = x
$26.98
= $13.49
2
Geraldo will save the cost of one shirt
which is $13.49.
28. 65 = x(50)
65 50 x
=
50 50
1.3 = x
130%
3
= 0.375
8
19. 0.2 =
2
9
20. 0.02 =
29. $180 − $150 = $30
$30 = x($150)
$30 $150 x
=
$150 $150
0.2 = x
20% markup
2
1
=
100 50
300
2
5 15 5 4 2
=
= ÷ = ⋅
2 4 2 15 3
3
Chapter 1–8
30. $85 − $71.40 = $13.60
$13.60 = x($85)
$13.60 $85 x
=
$85
$85
0.16 = x
16% discount
100 km 1 mi
⋅
≈ 62 mi
1
1.61 km
100 kph ≈ 62 mph
34. 100 km =
9
9
35. º F = º C + 32 = ( 5 ) + 32 = 9 + 32 = 41º F
5
5
36. P = 38 dm + 5 m + 450 cm
= 3.8 m + 5 m + 4.5 m = 13.3 m
8 in.
1 ft 9 in.
16 in.
1 ft 3 in.
2 ft
31.
37. P = 1.75 ft + 2 ft + 2 ft + 1.75 ft + 3.5 ft
= 11 ft
+ 1 yd
1 yd 4 ft 36 in. = 3 ft + 4 ft + 3 ft
= 10 ft
38. r =
d 40 cm
=
= 20 cm
2
2
A = πr 2 ≈ (3.14)(20 cm)2
4 12
ft 12 in.
1
⋅
= 54 in.
ft =
2
1
1 ft
1
Yes, 4 ft is 54 in.
2
32. 4
Cumulative Review Exercises
= (3.14)(400 cm 2 ) = 1256 cm 2
39. A = bh = (3 yd)(3 ft) = (3 yd)(1 yd) = 3 yd 2
or
A = bh = (3 yd)(3 ft) = (9 ft)(3 ft) = 27 ft 2
1 c
1
8 fl oz
33.
c= 2 ⋅
= 4 fl oz
2
1
1c
4 fl oz + 6 fl oz = 10 fl oz
6 fl oz 1 c
3
6 fl oz =
⋅
= c
1
8 fl oz 4
1
3
1
c + c =1 c
2
4
4
1
There is a total of 1 c or 10 fl oz of
4
liquid.
1 4 3 1 4
⋅ πr ≈ ⋅ (3.14)(6 in.)3
2 3
2 3
1 4
= ⋅ (3.14)(216 in.3 ) ≈ 452 in.3
2 3
40. V =
301
Chapter 9
Introduction to Statistics
Chapter Opener Puzzle
a. 6-month
b. 0.46%
c. 1.02%
d. 0.60%
e. 1-year
Section 9.1
Tables, Bar Graphs, Pictographs, and Line Graphs
Section 9.1 Practice Exercises
1. (a) Statistics
10. 26.8 − 25.1 = 1.7 yr
(b) table; cells
11. Men
(c) pictograph
12. Women
2. Since Mt. Everest is the highest mountain,
look at the 2nd column to see that Mt.
Everest is in Asia.
Dog
Cat
Neither
Boy
4
1
3
Girl
3
4
5
13.
3. Since Mt. Kosciusko is the shortest
mountain, look at the 2nd column to see
that Mt. Koscuisko is in Australia.
Yes
No
Drug (D)
7
3
Placebo (P)
4
6
14.
4. The highest mountain is 29.035 ft. The
lowest mountain is 7.310 ft. 29,035 –
7,310 = 21,725 ft.
15. (a) The 18—29 year-old age group has
the greatest percentage of Internet
users.
5. Heights are in the 3rd column. Mt.
Aconcagua is 22,834 ft − 20,320 ft
= 2514 ft higher than Denali.
6. The highest mountain in Europe is Elbrus.
The highest mountain in Australia is Mt.
Kosciusko. 18,510 ft − 7,310 ft = 11,200 ft
(b)
7. 25.1 − 21.5 = 3.6 yr
8. 26.8 − 24.3 = 2.5 yr
9. 24.3 − 21.5 = 2.8 yr
302
Section 9.1
Tables, Bar Graphs, Pictographs, and Line Graphs
16. (a) The health care industry has 219,400
new jobs, which is the greatest
number.
20. (a) Each computer icon represents 10% of
adults who access the Internet.
(b) The length of the “bar” for weather is
given by 6 computer icons. About
60% of adult users access the Internet
for weather.
(b)
1
computer
2
icons corresponds to International
news.
(c) The “bar” containing 4
21. (a) Each book icon represents 1 billion
dollars. The “bar” of greatest length
corresponds to Barnes & Noble/B.
Dalton. There are 4 12 book icons, so
17.
Barnes & Noble/B. Dalton has
approximately $4.5 billion in book
sales.
(b) There are 10 full book icons, and a
total of about 1 1 partial icons. There
2
is approximately $11.5 billion in book
sales.
22. (a) The “bar” for Texas contains about
18.
2 1 icons, and each icon corresponds
4
to 1 million senior citizens. Texas has
about 2.25 million senior citizens.
(b) The “bar” for California has almost
2 more full icons than the “bar” for
Pennsylvania. About 2 million more
senior citizens live in California than
in Pennsylvania.
19. (a) One icon represents 100 servings sold.
23. In 1920, 55.6% of men and 7.2% of
women were in the labor force.
55.6% − 7.2% = 48.4%
(b) The length of the “bar” for Saturday is
1
given by 4 ice cream cones. There
2
were about 450 servings sold on
Saturday.
3
(c) The “bar” containing 2 ice cream
4
cones corresponds to Sunday.
24. In 2000, 18.6% of men and 10.0% of
women were in the labor force.
18.6% − 10.0% = 8.6%
25. The trend for women over 65 in the labor
force shows a slight increase.
303
Chapter 9
Introduction to Statistics
26. The trend for men over 65 in the labor
force shows a significant decrease until
1980 and then levels off.
36. (a)
27. For example: 18%
28. For example: 10.5%
29. The most cars were sold in 2008. 22,400
cars were sold.
(b) Extend the graph to find that the height
of a 10-year-old boy is about 56 in.
30. The fewest cars were sold in 2009. 15,400
cars were sold.
37. There are 14 servings per container. Each
serving has 8 g of fat, so there are
8 × 14 = 112 g of fat in one container.
31. 22,400 − 17,600 = 4800 cars
32. 21,800 − 19,500 = 2300 cars
38. One serving has 45 mg of sodium, so there
are 14 × 45 = 630 mg of sodium in one
container.
33. Between 2007 and 2008
34. Between 2008 and 2009;
22,400 − 15,400 = 7,000
39.
8 g = (0.13) x
8 g 0.13x
=
0.13 0.13
61.5 g ≈ x
The daily value of fat is approximately
61.5 g.
40.
50 mg = (0.17) x
50 mg 0.17 x
=
0.17
0.17
294 mg ≈ x
The daily value of cholesterol is about
294 mg.
35. (a)
(b) Extend the graph to find that the height
of a 10-year-old girl is about 56 in.
Section 9.2
Frequency Distributions and Histograms
Section 9.2 Practice Exercises
1. (a) frequency
4. 29 + 12 + 6 + 22 + 56 + 60 = 185
There are 185 data.
(b) histogram
5. The 9−12 category had 24 data values,
which is the highest frequency.
2. (a) True
(b) False
6. The 251−300 category had 60 data values,
which is the highest frequency.
(c) True
3. 14 + 18 + 24 + 10 + 6 = 72
There are 72 data.
304
Section 9.2
7.
Class
Intervals
(Age Group)
Tally
Frequency
(Number of
Professors)
56−58
||
2
59−61
|
1
62−64
|
1
65−67
|||| ||
7
68−70
||||
5
71−73
||||
4
Frequency Distributions and Histograms
9.
Class
Intervals
(Amount
Purchased)
Tally
Frequency
(Number of
Customers)
8.0−9.9
||||
4
10.0−11.9
|
1
12.0−13.9
||||
5
14.0−15.9
||||
4
0
16.0−17.9
18.0−19.9
||
2
(a) The class of 65−67 has the most
values.
(a) The 12.0−13.9 class has the highest
frequency.
(b) 2 + 1 + 1 + 7 + 5 + 4 = 20
(b) 4 + 1 + 5 + 4 + 0 + 2 = 16
There are 16 data values represented
in the table.
There are 20 values represented in the
table
(c)
8.
5
= 0.25 or 25%
20
Of the professors, 25% retire when
they are 68 to 70 years old.
(c)
Class
Intervals
(Hourly
Salary, $)
Tally
Frequency
(Number of
Employees)
2
7.50−7.99
|||
3
|||| |||
8
8.00−8.49
||||
4
5−6
||
2
8.50−8.99
||
2
7−8
|||
3
9.00−9.49
|||
3
9−10
|
1
9.50−9.99
|
1
10.00−10.49
||
2
Class
Intervals
(Number of
Miles)
Tally
Frequency
(Number of
Runners)
1−2
||
3−4
10.
(a) The 3−4 class has the highest
frequency.
(a) The 8.00−8.49 class has the highest
frequency.
(b) 2 + 8 + 2 + 3 + 1 = 16
There are 16 data values represented
in the table.
(c)
2
= 0.125 or 12.5%
16
Of the customers, 12.5% purchase 18
to 19.9 gal of gas.
(b) 3 + 4 + 2 + 3 + 1 + 2 = 15
There are 15 data values represented
in the table.
8
= 0.5 or 50%
16
Of the runners, 50% run 3 to 4 mi/day.
305
Chapter 9
(c)
Introduction to Statistics
3 + 1+ 2 6
=
= 0.4 or 40%
15
15
Of the employees, 40% earn $9.00 or
more.
19.
11. The class widths are not the same.
12. The class widths are not the same.
13. There are too few classes.
14. There are too few classes.
20.
15. The class intervals overlap. For example,
it is unclear whether the data value 12
should be placed in the first class or the
second class.
16. The class intervals overlap. For example,
it is unclear whether the data value 5
should be placed in the first class or the
second class.
17.
18.
Class Interval
(Height, in.)
Frequency
(No. of Students)
62−63
2
64−65
3
66−67
4
68−69
4
70−71
4
72−73
3
Class Interval
(Amount, $)
Frequency
(No. of Customers)
0−49
3
50−99
4
100−149
7
150−199
1
200−249
5
21.
306
Class
Tally
Frequency
20−39
|||| |
6
40−59
|||| |||| |
11
60−79
||||
4
80−99
|
1
100−119
|
1
Section 9.2
22.
Class
Tally
Frequency
0−1
|||| |||
8
2−3
|||| |||| |||| |
16
4−5
|||| ||||
10
6−7
|||| ||
7
8−9
||
2
10−11
|
1
Section 9.3
Frequency Distributions and Histograms
Circle Graphs
Section 9.3 Practice Exercises
1. circle; sectors
2. (a)
8.
90 1 25
= =
360 4 100
25%
9.
180 1 50
(b)
= =
360 2 100
50%
3. Total number of traffic fatalities
= 16, 000 + 11, 520 + 10, 880 + 9600
+ 6400 + 9600
= 64, 000 fatalities
10.
4. The 15–24 years age group has the most
fatalities.
9600
= 0.15
64, 000
15% of the deaths were from the 65 and
older age group.
16,000
= 2.5 times
6400
There are 2.5 times as many deaths in the
15–24 age group than in the 55–64 age
group.
11,520
= 1.2 times
9600
There are 1.2 times as many deaths in the
25–34 age group than in the 65 and older
age group.
5. 11, 520 − 10, 880 = 640
640 more people died in the 25–34 age
group than in the 35–44 age group.
11. Total viewers (in millions)
= 5 + 2.2 + 3.3 + 2.5 + 2.6 = 15.6
There are 15.6 million viewers
represented.
6. 9600 − 6400 = 3200
3200 more people died in the 45–54 age
group than in the 55–64 age group.
12. The daytime drama corresponding to the
largest piece of the graph has 5 million
(5,000,000) viewers.
7.
16, 000
= 0.25
64, 000
25% of the deaths were from the 15–24
age group.
307
Chapter 9
Introduction to Statistics
13. The Young and the Restless has 5 million
viewers and The Days of Our Lives has
2.5 million viewers.
5
=2
2.5
20. Since 50% of the CDs are Pop/R&B, 50%
are not. Find 50% of 8000.
x = (0.50)(8000) = 4000
There are 4000 CDs that are not
Pop/R&B.
There are 2 times as many viewers who
watch The Young and the Restless as The
Days of Our Lives.
21. From the graph, 54% of the games sold
were Wii systems. Find 54% of 25
million.
x = (0.54)(25) = 13.5
There were 13.5 million Wii systems sold.
14. The Bold and the Beautiful has 3.3 million
viewers and As the World Turns has
2.2 million viewers.
3.3
= 1.5
2.2
There are 1.5 times as many viewers who
watch The Bold and the Beautiful as As
the World Turns.
22. From the graph, 24% of the games sold
were Xbox. Find 24% of 25 million.
x = (0.24)(25) = 6
There were 6 million Xbox systems sold.
23. From the graph, 22% of the games sold
were Play Station 3 systems. Find 22% of
25 million.
x = (0.22)(25) = 5.5
There were 5.5 million Play Station 3
systems sold.
15. General Hospital has 2.6 million viewers.
2.6
= 0.16 ≈ 0.17
15.6
Of the viewers, approximately 17% watch
General Hospital.
24. There were 25 million systems sold, and
13.5 million of those were Wii.
25 million – 13.5 million = 11.5 million
11.5 million of the systems were not Wii.
16. The Young and the Restless has 5 million
viewers.
5
≈ 0.3205 ≈ 0.32
15.6
Of the viewers, approximately 32% watch
The Young and the Restless.
25.
26.
17. The store carries 8000 CDs. From the
graph, 12% are musica Latina. Find 12%
of 8000.
x = (0.12)(8000) = 960
There are 960 Latina CDs.
27.
18. The store carries 8000 CDs. From the
graph, 20% are rap. Find 20% of 8000.
x = (0.20)(8000) = 1600
There are 1600 rap CDs.
28.
29.
19. From the graph, 5% of the CDs are jazz
and 3% are classical. This accounts for 8%
of the CDs. Find 8% of 8000.
x = (0.08)(8000) = 640
There are 640 CDs that are either classical
or jazz.
30.
31.
308
Section 9.3
35.
32.
33.
36.
34.
37.
(a) Total Expenses = 9000 + 600 + 2400 = $12,000
Expenses
Percent
Number of Degrees
Tuition
$9000
9000
= 0.75 or 75%
12,000
(0.75)(360°) = 270°
Books
600
600
= 0.05 or 5%
12,000
(0.05)(360°) = 18°
Housing
2400
2400
= 0.20 or 20%
12,000
(0.20)(360°) = 72°
(b)
309
Circle Graphs
Chapter 9
Introduction to Statistics
38. (a) Total Stores = 8100 + 7200 + 2700 = 18,000
Number of
Stores
Percent
Number of Degrees
Pizza Hut
8100
8100
= 0.45 or 45%
18,000
(0.45)(360°) = 162°
Domino’s
7200
7200
= 0.40 or 40%
18,000
(0.40)(360°) = 144°
Papa
John’s
2700
2700
= 0.15 or 15%
18,000
(0.15)(360°) = 54°
(b)
Section 9.4
Mean, Median, and Mode
Section 9.4 Practice Exercises
1. (a) mean
0 + 5 + 7 + 4 + 7 + 2 + 4 + 3 32
=
8
8
=4
5. Mean =
(b) median
(c) mean
7 + 6 + 5 + 10 + 8 + 4 + 8 + 6 + 0
9
54
=
=6
9
6. Mean =
(d) mode
(e) weighted
2. Answers will vary.
3. Mean =
4 + 6 + 5 + 10 + 4 + 5 + 8 42
=
=6
7
7
3+ 8 + 5 + 7 + 4 + 2 + 7 + 4 40
=
8
8
=5
4. Mean =
7. Mean =
10 + 13 + 18 + 20 + 15 76
=
= 15.2
5
5
8. Mean =
22 + 14 + 12 + 16 + 15 79
=
= 15.8
5
5
11.0 + 9.1+ 8.3+ 7.9 + 7.5 43.8
=
5
5
= 8.76 in.
9. Mean =
310
Section 9.4
(c)
96 + 90 + 62 + 78 + 95 421
=
5
5
= 84.2 wins
10. Mean =
5.5 + 6.0 + 5.8 + 5.8 + 6.0 + 5.6
6
34.7
=
≈ 5.8 hr
6
11. Mean =
98 + 80 + 78 + 90 346
=
4
4
= 86.5%
Zach’s mean test score was 86.5%.
98 + 98.4 + 98.9 + 100.1 + 99.2
5
494.6
=
= 98.92º F
5
(b)
(c)
360
370
380
400
400
+ 470
2380
Mean =
310
325
350
390
440
+ 500
2315
Mean =
79.0
− 77.9
1.1
The mean height for the 76ers is
slightly higher by1.1 in.
15. (a) Mean =
12. Mean =
13. (a)
Mean, Median, and Mode
98 + 80 + 78 + 90 + 59 405
=
5
5
= 81%
The mean of all five tests was 81%.
(b) Mean =
2380
≈ 397 Cal
6
(c) The low score of 59% decreased Zach’s
average by 86.5% − 81% = 5.5%.
$50 + $30 + $25 + $45 $150
=
4
4
= $37.50
16. (a) Mean =
$50 + $30 + $25 + $45 + $140
5
$290
=
= $58
5
2315
≈ 386 Cal
6
(b) Mean =
(c) Including the iron for $140 the mean
increased by $58 − $37.50 = $20.50.
17. Arrange the numbers in order from least to
greatest.
13 14 16 17 19 20 22
Median = 17
397
− 386
11
There is only an 11-Cal difference in
the means.
18. Arrange the numbers in order from least to
greatest.
22 30 31 32 35 36 38
Median = 32
83 + 83 + 72 + 79 + 77 + 84
+75 + 76 + 82 + 79
14. (a) Mean =
10
790
=
= 79 in.
10
19. Arrange the numbers in order from least to
greatest.
100 109 110 111 118 123


70 + 83 + 82 + 72 + 82 + 85
+75 + 75 + 78 + 77
(b) Mean =
10
779
=
= 77.9 in.
10
Median =
311
110 + 111 221
=
= 110.5
2
2
Chapter 9
Introduction to Statistics
20. Arrange the numbers in order from least to
greatest.
118 120 132 134 135 140


Median =
28. Arrange the numbers in order from least to
greatest.
2.6 2.7 3.0 3.0 3.0 3.2 3.4 3.9 4.8 7.4


3.0 + 3.2 6.2
Median =
=
2
2
= 3.1 million albums
132 + 134 266
=
= 133
2
3
21. Arrange the numbers in order from least to
greatest.
40 40 50 55 55 58

Median =
29. The data value 4 appears most often. The
mode is 4.
30. The data value 17 appears most often. The
mode is 17.
50 + 55 105
=
= 52.5
2
2
31. No data value occurs most often. There is
no mode.
22. Arrange the numbers in order from least to
greatest.
82 82 87 88 90 99

Median =
32. No data value occurs most often. There is
no mode.
87 + 88 175
=
= 87.5
2
2
33. There are 2 modes: 21 and 24.
34. There are 2 modes: 42 and 49.
23. Arrange the numbers in order from least to
greatest.
3.82 3.87 3.93 4.09 4.10
Median = 3.93 deaths per 1000
35. $300
36. 39
37. 5.2%
24. Arrange the numbers in order from least to
greatest.
80 103 133 311 1700
Median = 133%
38. 2 children
39. These data are bimodal: $2.49 and $2.51
40. These data are bimodal: 1.00 and 0.50
days
25. Arrange the numbers in order from least to
greatest.
43 46 52 55 56 60 61 62 64 69

56 + 60 116
Median =
=
= 58 years old
2
2
92 + 98 + 43 + 98 + 97 + 85 513
=
6
6
= 85.5%
41. Mean =
26. Arrange the numbers in order from least to
greatest.
18 49 108 121 124 167 177 227


121+124 245
=
= 122.5 stations
Median =
2
2
Arrange the numbers in order from least to
greatest.
43 85 92 97 98 98

92 + 97 189
Median =
=
= 94.5%
2
2
27. Arrange the numbers in order from least to
greatest.
42.4 45.4 46.5 48.3 51.7 56.4 71.2 86.8 91.6
Median = 51.7 million passengers
The median gave Jonathan a better overall
score.
312
Section 9.4
312 + 225 + 221 + 256 + 308
+ 280 + 147
43. Mean =
7
1749
=
≈ $250
7
52 + 85 + 89 + 90 + 83 + 89 488
=
6
6
= 81.3%
42. Mean =
Arrange the numbers in order from least to
greatest.
52 83 85 89 89 90

85 + 89 174
=
= 87%
Median =
2
2
Arrange the numbers in order from least to
greatest.
147 221 225 256 280 308 312
Median = $256
There is no mode.
The median gave Nora a better overall
score.
44. Mean
104,000 + 107,000 + 67,750 + 82,500
+ 73,500 + 88,300 + 104,000
=
7
627,050
=
≈ $89,579
7
Arrange the numbers in order from least to greatest.
6 7 ,7 5 0 7 3 ,5 0 0 8 2 ,5 0 0 8 8 ,3 0 0 1 0 4 ,0 0 0 1 0 4 ,0 0 0 1 0 7 ,0 0 0
Median = $88,300
Mode = $104,000
850 + 835 + 839 + 829 + 850 + 850
+ 850 + 847 + 1850 + 825
45. Mean =
10
9425
=
= $942,500
10
Arrange the numbers in order from least to greatest.
825 829 835 839
Median =
847 8
50
Mean, Median, and Mode
850 850 850 1850
847 + 850 1697
=
= $848,500
2
2
Mode = $850,000
313
Chapter 9
Introduction to Statistics
300 + 2495 + 2120 + 220 + 194
+ 391 + 315 + 330 + 435 + 250
46. Mean =
10
7050
=
= $705,000
10
Arrange the numbers in order from least to greatest.
194 220 250 300 315
330 391 435 2120 2495

Median =
315 + 330 645
=
= $322,500
2
2
There is no mode.
Age (yr)
Number
of
Students
Product
16
7
17
47.
(16)(7) = 112
Number of
Students in
Each Class
Number
of
Classes
Product
9
(17)(9) = 153
18
5
(18)(5) = 90
18
6
(18)(6) = 108
20
6
(20)(6) = 120
19
3
(19)(3) = 57
25
15
(25)(15) = 375
Total:
25
430
30
18
(30)(18) = 540
35
12
(35)(12) = 420
Total:
56
1545
49.
430
= 17.2
25
The mean age is approximately 17.2 years.
Mean =
48.
1545
≈ 28
56
The mean number of students per class is
approximately 28.
Mean =
Number of
Residents
in Each
House
Number
of
Houses
Product
1
3
(1)(3) = 3
2
9
(2)(9) = 18
3
10
(3)(10) = 30
4
9
(4)(9) = 36
5
6
(5)(6) = 30
Total:
37
117
Grade
CreditHours
Product
B = 3.0
4
(3.0)(4) = 12.0
C = 2.0
1
(2.0)(1) = 2.0
A = 4.0
3
(4.0)(3) = 12.0
D = 1.0
5
(1.0)(5) = 5.0
Total:
13
31
50.
117
Mean =
≈ 3.2
37
The mean number of residents is
approximately 3.2.
GPA =
314
31
≈ 2.38
13
Section 9.4
Grade
CreditHours
Product
B+ = 3.5
3
(3.5)(3) = 10.5
A = 4.0
4
(4.0)(4) = 16.0
A = 4.0
1
B = 3.0
Total:
51.
GPA =
GPA =
Grade
CreditHours
Product
(4.0)(1) = 4.0
C+ = 2.5
5
(2.5)(5) = 12.5
3
(3.0)(3) = 9.0
A = 4.0
4
(4.0)(4) = 16.0
11
39.5
D = 1.0
3
(1.0)(3) = 3.0
A = 4.0
1
(4.0)(1) = 4.0
Total:
13
35.5
Grade
CreditHours
Product
B+ = 3.5
3
(3.5)(3) = 10.5
C = 2.0
4
(2.0)(4) = 8.0
F = 0.0
1
(0.0)(1) = 0.0
A = 4.0
3
(4.0)(3) = 12.0
Total:
11
30.5
Section 9.5
30.5
≈ 2.77
11
53.
39.5
≈ 3.59
11
52.
Mean, Median, and Mode
GPA =
35.5
≈ 2.73
13
Introduction to Probability
Section 9.5 Practice Exercises
1. (a) experiment
greatest.
40 62 62 64 67
Median = 62
The mode is 62.
(b) sample
(c) probability
(d) complement
(e) 1
2. The mean will be most affected. The very
large number will greatly increase the sum
of the data values.
13 + 16 + 22 + 25 + 10 86
3. Mean =
=
= 17.2
5
5
Arrange the numbers in order from least to
greatest.
10 13 16 22 25
Median = 16
There is no mode.
62 + 64 + 62 + 67 + 40 295
=
= 59
5
5
Arrange the numbers in order from least to
4. Mean =
315
Chapter 9
Introduction to Statistics
9. {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}
8 + 9 + 10 + 7 + 8 + 8 + 11 + 10
8
71
=
= 8.875
8
Arrange the numbers in order from least to
greatest.
7 8 8 8 9 10 10 11
8 + 9 17
Median =
=
= 8.5
2
2
The mode is 8.
5. Mean =
10. {yellow, red, blue, green, white}
11. The sample space consists of all possible
sums of the numbers of dots.
{2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12}
12. {TT, TH, HT, HH}
13. in 3 ways {1, 3, 5}
14. in 5 ways {1, 2, 3, 4, 5}
96 + 88 + 89 + 90 + 88 + 50 501
=
6. Mean =
6
6
= 83.5
The mean is 83.5%.
Arrange the numbers in order from least to
greatest.
50 88 88 89 90 96
15. c, d, g, h
16. b, c, d, g, h
17. The event can occur in 2 ways:
The die lands as a 1 or 2.
The sample space has 6 elements:
1, 2, 3, 4, 5, and 6.
2 1
=
6 3
88 + 89 177
=
= 88.5
2
2
The median is 88.5%.
The mode is 88%.
Median =
18. The event can occur in 1 way: The die
lands as a 6.
The sample space has 6 elements:
1, 2, 3, 4, 5, and 6.
1
6
20 + 20 + 18 + 17 + 19 + 5 99
=
6
6
= 16.5
The mean is 16.5.
Arrange the numbers in order from least to
greatest.
5 17 18 19 20 20
7. Mean =
19. The event can occur in 3 ways:
The die lands as a 2, 4, or 6.
The sample space has 6 elements.
3 1
=
6 2
18 + 19 37
=
= 18.5
2
2
The median is 18.5.
The mode is 20.
Median =
20. The event can occur in 3 ways:
The die lands as a 1, 3, or 5.
The sample space has 6 elements.
3 1
=
6 2
100 + 90 + 95 + 98 + 90 + 10
6
483
=
= 80.5
6
Arrange the numbers in order from least to
greatest.
10 90 90 95 98 100
8. Mean =
21. There are 5 black socks. There are 8 socks
in the drawer.
5
8
90 + 95 185
=
= 92.5
2
2
The median is 92.5.
The mode is 90.
Median =
316
Section 9.5
Introduction to Probability
22. There are 2 white socks. There are 8 socks
in the drawer.
2 1
=
8 4
(b) 14 vacationers stay 2 days and
13 vacationers stay 3 days.
14 + 13 27
9
=
=
120
120 40
23. There is 1 blue sock. There are 8 socks in
the drawer.
1
8
(c) 30 vacationers stay 7 days and
6 vacationers stay 8 days.
30 + 6 36
=
= 0.30 or 30%
120
120
24. There are no purple socks in the drawer.
0
or 0.
The probability is
8
34. (a) There are 32 + 88 = 120 dorm
residents. The total number of
students is 32 + 88 + 59 + 26 = 205.
120 24
=
205 41
25. The probability is 1 because a number
from 1−6 will definitely come up.
(b) The total students surveyed is
32 + 88 + 59 + 26 = 205.
The total number of students who do
not own a car is 88 + 26 = 114.
114
205
26. There is no way to obtain a 7. The
probability is 0.
27. An impossible event is one in which the
probability is 0.
28. The probability is 1 because the only two
possibilities are for an event to occur or
not occur.
35. (a) 21 cars are manufactured in America.
21 7
=
60 20
29. There are 12 + 40 = 52 cards.
12 3
=
52 13
(b) There are 21 + 9 = 30 cars
manufactured in a country other than
Japan.
30
= 0.50 or 50%
60
30. There are 13 + 13 + 13 + 13 = 52 cards.
13 1
=
52 4
36. The total number of representatives is
4 + 2 + 14 + 10 + 16 + 18 + 10 + 6 = 80.
31. There are 7 + 5 + 4 = 16 marbles in the
jar. There are 5 red and 7 yellow marbles.
5 + 7 12 3
= =
16
16 4
(a) 10 representatives received
3 complaints.
10 1
=
80 8
32. There are 10 + 12 + 4 = 26 marbles in the
jar. There are 4 blue and 10 black marbles.
4 + 10 14 7
=
=
26
26 13
(b) A total of 2 + 14 + 10 + 16 + 18 = 60
representatives received between 1
and 5 complaints, inclusive.
60 3
=
80 4
33. The total number of vacationers is
14 + 13 + 18 + 28 + 11 + 30 + 6 = 120.
(c) A total of
16 + 18 + 10 + 6 = 50 representatives
received at least 4 complaints.
50 5
=
80 8
(a) 18 vacationers stayed 4 days.
18
3
=
120 20
317
Chapter 9
Introduction to Statistics
(d) The number of representatives who
received more than 5 complaints or
less than 2 complaints is
4 + 2 + 10 + 6 = 22.
22 11
=
80 40
(b) 1 person won the TV set.
1
60
(c) A total of 1 + 4 + 2 + 2 = 9 prizes
were won.
9
= 0.15 or 15%
60
37. The total number of students is
1 + 6 + 11 + 7 + 3 + 1 = 29.
(a) A total of 1 + 6 = 7 students are early.
7
29
39. Subtract the probability of winning from
1.
2 11 2
9
1− = − =
11 11 11 11
(b) A total of 7 + 3 + 1 = 11 students are
late.
11
29
40. 1 −
(c) A total of 1 + 6 + 11 = 18 students
arrive on time or early.
18
≈ 0.62 or 62%
29
41. 100% − 1.2% = 98.8%
1
1,000,000
1
=
−
1,000,000 1,000,000 1,000,000
999,999
=
1,000,000
42. 100% − 88% = 12%
38. (a) A total of
51 + 1 + 4 + 2 + 2 = 60 people bought
raffle tickets.
Chapter 9
Review Exercises
Section 9.1
1. From the 2nd column, Godiva has the
most calories.
5. The bar corresponding to 1970 has height
374. The average size of the farms in 1970
was 374 acres.
2. From the 4th column, Breyers has the least
amount of cholesterol.
6. The average size in 2000 was 434 acres.
The average size in 1940 was 174 acres.
434 − 174 = 260
The difference is 260 acres.
3. Blue Bell has 70 mg of sodium. Edy’s
Grand has 35 mg of sodium.
70
=2
35
Blue Bell has 2 times more sodium than
Edy’s Grand.
7. The average size in 1990 was 430 acres.
The average size in 1980 was 426 acres.
430 − 426 = 4
The difference is 4 acres.
4. Godiva has 28 g carbohydrate. Blue Bell
has 18 g carbohydrate.
28 g − 18 g = 10 g
There is a 10-g difference.
8. The greatest increase was between 1950
and 1960.
9. 1 icon represents 50 tornados.
318
Chapter 9
Review Exercises
Section 9.2
10. The May bar has 6 icons which represents
6 × 50 = 300 tornados.
Class
Intervals
(Age)
Tally
Frequency
18−21
||||
4
22−25
||||
5
26−29
||||
4
30−33
|||
3
14. 4900 liver transplants
34−37
|
1
15. Increasing
38−41
|
1
42−45
||
2
18.
11. The June bar has 4 icons which represents
200 tornados.
12. The difference in the lengths of the “bars”
is about
11
2
icons. The difference is
approximately 75 tornados.
13. The number of liver transplants was
greatest in 2010.
16. The number of liver transplants for 2007
would be about 7000.
19.
17.
Section 9.3
20. There are 4 parts of the circle representing
2 + 2 + 4 + 16 = 24 types of subs.
21.
22.
319
16 2
= of the subs contain beef.
24 3
2+2+4 8 1
=
= of the subs do not
24
24 3
contain beef.
Chapter 9
23.
(a)
Introduction to Statistics
Education Level
No. of
People
Grade School
Percent
No. of Degrees
10
10
= 0.05 or 5%
200
(0.05)(360°) = 18°
High School
50
50
= 0.25 or 25%
200
(0.25)(360°) = 90°
Some College
60
60
= 0.30 or 30%
200
(0.30)(360°) = 108°
Four-year degree
40
40
= 0.20 or 20%
200
(0.20)(360°) = 72°
Post graduate
40
40
= 0.20 or 20%
200
(0.20)(360°) = 72°
(b)
Section 9.4
26.
20 + 20 + 18 + 16 + 18 + 17 + 16
+ 10 + 20 + 20 + 15 + 20
24. Mean =
12
210
=
= 17.5
12
Arrange the numbers in order from
smallest to largest.
10 15 16 16 17 18 18 20 20 20 20 20
27.
The median is 18.
The mode is 20.
28.
800 + 1000 + 1200 + 1300 + 900
+ 1200 + 1000
25. Mean =
7
7400
=
≈ 1060
7
The mean daily calcium intake is
approximately 1060 mg.
320
Arrange the numbers in order from
smallest to largest.
18,624
20,000
20,562 ←
21,500
23,799
The median is 20,562 seats.
The mode is 4.
Age (yr)
No. of
Children
Product
10
8
80
11
6
66
12
5
60
13
1
13
Chapter 9
80 + 66 + 60 + 13 219
=
≈ 10.95
20
20
The mean age is 10.95 years.
31. a, c, d, e, g
Mean =
32. (a) There are 8 + 6 + 2 = 16 tricycles in
the warehouse. There are 8 red
tricycles.
8 1
=
16 2
Section 9.5
29. {blue, green, brown, black, gray, white}
30. There is one pair of gray socks out of 6
pairs of socks.
1
6
Chapter 9
Review Exercises
(b)
6+2 8 1
=
=
16
16 2
(c) There are no green tricycles. The
probability is 0.
Test
7. The “bar” with 5 icons corresponds to
February.
1.
8. Seattle
9. 2.5 − 0.77 = 1.73 in.
10. In May, Salt Lake City had 2.09 in. of
rain, while Seattle had 2.03 in.
11.
2. The year 1820 had the greatest percent of
workers employed in farm
occupations. This was 72%.
3.
4.
It appears that 10% of U.S. workers
were employed in farm occupations in
the year 1960.
5. $1000
6. The “bar” corresponding to April has 4 1
2
icons. Sales for April were $4500.
321
Number of
Minutes
Used
Monthly
Tally
Frequency
51−100
|||| |
6
101−150
||
2
151−200
|||
3
201−250
||
2
251−300
||||
4
301−350
|||
3
Chapter 9
Introduction to Statistics
11 + 14 + 11 + 16 + 15 + 16 + 12
+ 16 + 15 + 20
18. Mean =
10
146
=
= $14.60
10
Arrange the numbers in order from
smallest to largest.
11 11 12 14 15 15 16 16 16 20

The median is $15.
The mode is $16.
12. 44% have carpet on their living room
floors. Find 44% of 150.
x = (0.44)(150) = 66
66 people would have carpet.
19. (a) {1, 2, 3, 4, 5, 6, 7, 8}
(b)
13. 20% have tile on their living room floor.
Find 20% of 200.
x = (0.20)(200) = 40
40 people would have tile.
(c) There are 4 ways of obtaining an even
number: 2, 4, 6, or 8.
4 1
=
8 2
14. 10% have linoleum, so 90% do not have
linoleum. Find 90% of 300.
x = (0.90)(300) = 270
270 people would have something other
than linoleum.
15.
1
8
(d) There are 2 ways of obtaining number
less than 3 : 1 or 2.
2 1
=
8 4
19,340
18,510
22,834
20,320
16,864
7,310
+ 29,035
134, 213
134, 213
Mean =
≈ 19,173 ft
7
20. There are 6 + 4 + 2 + 2 = 14 cans in the
cooler.
(a) There are 4 cans of ginger ale.
4 2
=
14 7
(b) 1 −
2 7 2 5
= − =
7 7 7 7
Grade
Number
of CreditHours
Product
B = 3.0
4
(3.0)(4) = 12.0
A = 4.0
3
(4.0)(3) = 12.0
C = 2.0
3
(2.0)(3) = 6.0
A = 4.0
1
(4.0)(1) = 4.0
Total:
11
34.0
21.
16. Arrange the numbers in order from
smallest to largest.
7,310
16,864
18,510
19,340 ←
20,320
22,834
29,035
The median height is 19,340 ft.
17. There is no mode.
GPA =
22. c
322
34.0
≈ 3.09
11
Chapters 1–9
Chapters 1–9
Cumulative Review Exercises
1. (a) Millions
10.
3
7
30
7
37
+
=
+
=
10 100 100 100 100
11.
1 5 1 3 10 1 12
+ − = + − =
=2
2 3 6 6 6 6 6
(b) Ten-thousands
(c) Hundreds
2.
3.
1
7 9 1
12. 2 ÷ 17 + ⋅ −
8
12 14 3
17 17 7 9 1
= ÷ + ⋅ −
8 1 12 14 3
2 087
53
10,499
+
6
12,645
1
700
× 1 200
840,000
1
6.
14. 68.412 × 100 = 6841.2
15. 68.412 × 0.1 = 6.8412
16. 68.412 ÷ 0.001 = 68,412
2
17. (a)
3
3,700,000
− 2,950,000
750,000 km 2 or 0.75 million km 2
15
1
6
9.
2
13. 13.28 + 0.27 = 13.55
9.51 − 0.17 = 9.34
14.35 + 0.10 = 14.45
18.09 + 0.09 = 18.18
21.63 − 0.37 = 21.26
105 7 105 16 15 5
÷ =
⋅
= =
7.
96 16 96 7
6 2
8.
4
1 3 1 4 1 1 1
+ − = − = −
8 8 3 8 3 2 3
3 2 1
= − =
6 6 6
12 14 2
⋅
=
7 36 3
1
3
=
3
8
1
1
17 1
7 9 1
=
⋅
+
⋅
−
8 17 12 14 3
4. Divisor: 23
dividend: 651
28
23 651
−46
191
−184
7
quotient: 28
remainder: 7
5.
Cumulative Review Exercises
(b)
1
3
4
1
5
750,000
= 0.2027 or 20.3%
3,700,000
18. Quick Cut Lawn Company:
2.75
= 0.55 hr per customer
5
Speedy Lawn Company:
3
= 0.5 hr per customer
6
Speedy Lawn Company is faster.
5 6 24 5 15 24 60 3
÷ ⋅
= ⋅
⋅
=
=
8 15 25 8 6 25 40 2
97
63
34 1
−
=
=
102 102 102 3
323
Chapter 9
19.
20.
21.
Introduction to Statistics
4 10
=
50 x
4 x = (50)(10)
4 x = 500
4 x 500
=
4
4
x = 125 min or 2 hr 5 min
26. 4
27.
3 yd 2 ft
+ 5 yd 2 ft
8 yd 4 ft = 8 yd + 1 yd + 1 ft
= 9 yd 1 ft
x 15
=
8 24
24 x = (8)(15)
24 x = 120
24 x 120
=
24
24
x=5 m
28. 12 km − 2360 m = 12,000 m − 2360 m
= 9640 m
y 24
=
14 15
15 y = (14)(24)
15 y = 336
15 y 336
=
15
15
y = 22.4 m
30. Obtuse
29.
16 lb 12 oz 16 lb 12 oz
=
+
4
4
4
= 4 lb + 3 oz
= 4 lb 3 oz
31. Right
32. Acute
33. Area = bh = (4 ft)(2 ft) = 8 ft 2
1
34. V = πr 2 h
3
1  22 
≈   (3 m) 2 (7 m)
3 7 
1  22 
=   (9 m 2 )(7 m)
3 7 
= 66 m3
95 = (0.78) x
95 0.78 x
=
0.78 0.78
122 ≈ x
122 people
22. x = (2.30)(7.4 million) = 17.02 million
23.
4 1 gal 4 qt
1
gal = 2
⋅
= 18 qt
2
1
1 gal
78 = x (120)
78 120 x
=
120 120
0.65 = x
65%
35.
24. I = Prt
= ($1200)(0.034)(5)
= ($40.80)(5)
= $204
$204 + $1200 = $1404
2 ft 12 in.
⋅
= 24 in.
1
1 ft
2 ft 5 in. = 24 in. + 5 in. = 29 in.
25. 2 ft =
324
Chapters 1–9
Cumulative Review Exercises
36. Mean
33 + 10 + 62 + 132 + 123 + 316 + 138
+123 + 133 + 18 + 150 + 26
=
12
1264
=
≈ 105
12
Arrange the numbers in order from smallest to largest.
10 18 26 33 62 123
123 132 133 138 150 316

The median is 123.
2
2
37. 30 − 3 (5 − 2 ) = 30 − 3 (3) = 30 − 3⋅ 9
= 30 − 27 = 3
39.
38. {yellow, blue, red, green}
1
4
40. 1 −
325
1 4 1 3
= − =
4 4 4 4
Chapter 10
Real Numbers
Chapter Opener Puzzle
Section 10.1
Real Numbers and the Real Number Line
Section 10.1 Practice Exercises
11. 140,000
1. (a) positive; negative
(b) integers
12. −$20,000
(c) rational; irrational
13.
(d) absolute
(e) opposites
14.
2. −340 ft
15.
. 3. −86 m
16.
4. −$45
5. $3800
17.
6. 5
18.
7. −$500
8. $23
19.
9. −14 lb
20.
10. 5000
326
Section 10.1
Real Numbers and the Real Number Line
21.
1
4
5
1
46. − = − > − = −
5
20
20
4
22.
47.
7
1
>−
8
9
48.
1
3
>−
3
2
23.
24.
49. 0 <
25. Rational
26. Rational
50. −
27. Rational
1
10
8
<1
7
6
5
51. − < − = −1
5
5
28. Rational
29. Rational
11
10
<−
11
11
30. Rational
52. −1 = −
31. Irrational
53. |−2| = 2
32. Irrational
54. |−6| = 6
33. Irrational
55. |4.5| = 4.5
34. Irrational
56. |2.9| = 2.9
35. Rational
36. Rational
37. 0 > −3
38. −1 < 0
57. −
5 5
=
2 2
58. −
4 4
=
9 9
39. −8 > −9
59. |0| = 0
40. −5 < −2
60. |6| = 6
41. −9.1 < 2.2
61. |−3.2| = 3.2
42. −1.5 < 1.5
62. |−0.4| = 0.4
43. −3.35 < −3.3
63. |21| = 21
44. 0.9 > −0.5
64. |8| = 8
2
4
5
45. − = − > −
3
6
6
65. (a) −8
(b) |−12| = 12
|−8| = 8
|−12| is greater.
327
Chapter 10
Real Numbers
66. (a) −14
(b) |−14| = 14
|−20| = 20
|−20| is greater.
84. 2.25
67. (a) 7.8
(b) |5.2| = 5.2
|7.8| = 7.8
|7.8| is greater.
87. −(−2)
85. −6
86. −23
88. −(−9)
89. |7|
68. (a) 4.29
(b) |3.89| = 3.89
|4.29| = 4.29
|4.29| is greater.
69. −
4
5
70. −
3
8
90. |11|
91. |−3|
92. |−10|
93. −|14|
94. −|42|
95. − −30
71. Neither, they are equal.
96. − −5
72. Neither, they are equal.
97. −|2| = −2
73. −5
98. −|9| = −9
74. −31
99. −|−5.3| = −5.3
75. 12
100. −|−6.9| = −6.9
76. 25
77.
1
6
78.
4
7
101. −(−15) = 15
102. −(−4) = 4
103. |−4.7| = 4.7
104. |−9.5| = 9.5
79. −
2
11
105. − −
12
12
=−
17
17
80. −
14
15
106. − −
1
1
=−
7
7
81. −8.1
 3 3
107. −  −  =
 8 8
82. −9.5
 4 4
108. −  −  =
 9 9
83. 1.14
328
Section 10.2
Section 10.2
Addition of Real Numbers
Addition of Real Numbers
Section 10.2 Practice Exercises
1. (a) 0
(b) negative; positive
(c) Subtract the smaller absolute value
from the larger absolute value. The
sum takes on the sign of the addend
with the greater absolute value.
22. 23 + 12 = 35
2. −6 < −5
26. −6 + (−10) = −(6 + 10) = −16
2
8
11
3. − = − > −
3
12
12
27. −100 + (−24) = −(100 + 24) = −124
23. 12 + 3 = 15
24. −70 + (−15) = −(70 + 15) = −85
25. −40 + (−33) = −(40 + 33) = −73
28. 23 + 50 = 73
4. 2.4 = |−2.4| > −|2.4| = −2.4
29. 44 + 45 = 89
5. |6| = |−6|
30. To add two numbers with different signs,
subtract the smaller absolute value from
the larger absolute value. Then apply the
sign of the number having the larger
absolute value.
6. 0 > −0.6
7. −10 = −|−10| < 10
8. −(−2) = 2
31. 75 + (−23) = 75 − 23 = 52
9. 2 + (−4) = −2
32. 12 + (−7) = 12 − 7 = 5
10. 5 + (−1) = 4
33. −34 + 12 = −(34 − 12) = −22
11. −3 + 5 = 2
34. −88 + 35 = −(88 − 35) = −53
12. −6 + 3 = −3
35. −90 + 66 = −(90 − 66) = −24
13. −4 + (−4) = −8
36. −23 + 49 = 49 − 23 = 26
14. −2 + (−5) = −7
37. 78 + (−33) = 78 − 33 = 45
15. −3 + 9 = 6
38. 10 + (−23) = −(23 − 10) = −13
16. −1 + 5 = 4
39. 2 + (−2) = 2 − 2 = 0
17. 0 + (−7) = −7
40. −6 + 6 = 6 − 6 = 0
18. (−5) + 0 = −5
41. −1.3 + 1.3 = 1.3 − 1.3 = 0
19. −1 + (−3) = −4
42. 4.5 + (−4.5) = 4.5 − 4.5 = 0
20. −4 + 3 = −1
43. 12 + (−3) = 9
21. To add two numbers with the same sign,
add their absolute values and apply the
common sign.
44. −33 + (−1) = −34
329
Chapter 10
Real Numbers
45. −23 + (−3) = −26
67. 23.9 + 2.1 = 26
46. −5 + 15 = 10
68. 10.9 + 6.3 = 17.2
47. 4 + (−45) = −41
69. −34.2 + (−4.1) = −(34.2 + 4.1) = −38.3
48. −13 + (−12) = −25
70. −8.6 + (−12) = −(8.6 + 12) = −20.6
49. (−103) + (−47) = −150
3  5
8
3 5
71. − +  −  = −  +  = − = −2
4  4
4
4 4
50. 119 + (−59) = 60
2 1 
 4 1
2  1
72. − +  −  = −  +  = −  + 
5  10 
 5 10 
 10 10 
5
1
=− =−
10
2
51. 0 + (−17) = −17
52. −29 + 0 = −29
53. −19 + (−22) = −41
1  1
7  7   7 1
73. −1 +  −2  = − +  −  −  + 
6  3
6  3   6 3
21
 7 14 
= − +  = −
6 6
6
7
1
= − = −3
2
2
54. −300 + (−24) = −324
55. 6 + (−12) + 8 = −6 + 8 = 2
56. 20 + (−12) + (−5) = 8 + (−5) = 3
57. −33 + (−15) + 18 = −48 + 18 = −30
58. 3 + 5 + (−1) = 8 + (−1) = 7
74. −3 2 +  −4 1  = − 47 +  − 21 = −  47 + 21
 15 5 
15  5 
15  5 
110
 47 63 
= − +  = −
 15 15 
15
5
1
= −7 = −7
15
3
59. 7 + (−3) + 6 = 4 + 6 = 10
60. 12 + (−6) + (−9) = 6 + (−9) = −3
61. −10 + (−3) + 5 = −13 + 5 = −8
62. −23 + (−4) + (−12) + (−5)
= −27 + (−12) + (−5)
= −39 + (−5)
= −44
75. 34.8 + (−45) = −(45 − 34.8) = −10.2
76. 90 + (−12.3) = 90 − 12.3 = 77.7
77. −23.1 + 24.5 = 24.5 − 23.1 = 1.4
63. −18 + (−5) + 23 = −23 + 23 = 0
78. −12.2 + 10.9 = −(12.2 − 10.9) = −1.3
64. 14 + ( −15) + 20 + ( −42) = −1 + 20 + ( −42)
= 19 + ( −42)
= −23
65. 4 + (−12) + (−30) + 16 + 10
= −8 + (−30) + 16 + 10
= −38 + 16 + 10
= −22 + 10
= −12
79.
3  3 6  3 6 3
3
+−  = +−  = − =
8  16  16  16  16 16 16
80.
1  7 3  7
4
7 3
+  −  = +  −  = − −  = −
3  9 9  9
9
9 9
 5  1  10  3
 10 3 
81.  −  + =  −  + = −  − 
 6  4  12  12
 12 12 
7
=−
12
66. 24 + (−5) + (–19) = 19 + (–19) = 0
330
Section 10.2
3
3 24 21
94. − + 6 = − +
=
4
4 4
4
 4  7  16  7
 16 7 
82.  −  +
= −  +
= − − 
 5  20  20  20
 20 20 
9
=−
20
83. 1
Addition of Real Numbers
1
1 5 4
95. − + 1 = − + =
5
5 5 5
3  4
3
4
+  −2  = 1 − −2
10  5 
10
5
13 14 13 28
= − = −
10 5 10 10
1
 15 
= −   = −1
 10 
2
96. −4°F + 12°F = 12°F − 4°F = 8°F
97. 14°F − 20°F = –(20ºF – 14ºF) = –6°F
98. −$56.52 + $389.81 = $389.81 − $56.52
= $333.29
99. $23.89 − $40.00 = −($40.00 − $23.89)
= −$16.11
3
1
31 61
93 61
84. −7 + 5 = − + = − +
4
12
4 12
12 12
32
8
2
= − = −2 = −2
12
12
3
100. −$320.50 + $150.00
= −($320.50 − $150.00)
= −$170.50
85. Sum, added to, increased by, more than,
plus, total
101. $570.32 − $250.00 = $320.32
86. −23 + 49 = 49 − 23 = 26
102. For example: −12 + 2
87. 89 + (−11) = 89 − 11 = 78
103. For example: −6 + (−8)
88. 3 + (−10) + 5 = −7 + 5 = −2
104. For example: −1 + (−1)
89. −2 + (−4) + 14 + 20 = −6 + 14 + 20
= 8 + 20 = 28
105. For example: 5 + (−5)
106. 302 + (−422) = −120
90. −2.2 + (−4.2) = −6.4
107. −900 + 334 = −566
91. −12 + (−4.5) = −16.5
108. −23.991 + (−44.23) = −68.221
1
1 32 31
92. − + 8 = − +
=
4
4 4
4
109. −103.4 + (−229.1) = −332.5
1
1 6 5
93. − + 2 = − + =
3
3 3 3
110. 23 + (−125) + 912 + (−99) = 711
111. 891 + 12 + (−223) + (−341) = 339
331
Chapter 10
Real Numbers
Section 10.3
Subtraction of Real Numbers
Section 10.3 Practice Exercises
1. (a) ( −b)
16. −7 − 21 = −7 + (−21) = −28
(b) –5 + 4
17. −11 − (−13) = −11 + 13 = 2
2. 34 + (−13) = 34 − 13 = 21
18. −23 − (−9) = −23 + 9 = −14
3. −34 + (−13) = −(34 + 13) = −47
19. 35 − (−17) = 35 + 17 = 52
4. −34 + 13 = −(34 − 13) = −21
20. 23 − (−12) = 23 + 12 = 35
5 7
20 21 21 20 1
5. − + = − +
=
−
=
9 12
36 36 36 36 36
21. −24 − 9 = −24 + (−9) = −33
22. −5 − 15 = −5 + (−15) = −20
 21 20 
5  7  20  21 
6.
+−  =
+ −  = − − 
9  12  36  36 
 36 36 
1
=−
36
23. 50 − 62 = 50 + (−62) = −12
24. 38 − 46 = 38 + (−46) = −8
25. −17 − (−25) = −17 + 25 = 8
20  21 
5  7
7. − +  −  = − +  − 
36  36 
9  12 
26. −2 − (−66) = −2 + 66 = 64
 20 21 
41
= − +  = −
36
 36 36 
27. −8 − (−8) = −8 + 8 = 0
28. −14 − (−14) = −14 + 14 = 0
 2
8. 3+ (−4.2) +  −  + 5 = −1.2 + (−0.4) + 5
 5
= −1.6 + 5 = 3.4
29. 120 − (−41) = 120 + 41 = 161
30. 91 − (−62) = 91 + 62 = 153
31. −15 − 19 = −15 + (−19) = −34
 1
9.  −  + 6.5 + (−8) + 2 + (−4)
 2
32. −82 − 44 = −82 + (−44) = −126
= (−0.5) + 6.5 + (−8) + 2 + (−4)
= 6 + (−8) + 2 + (−4)
= −2 + 2 + (−4) = 0 + (−4) = −4
33. 3 − 25 = 3 + (−25) = −22
34. 6 − 33 = 6 + (−33) = −27
10. To subtract two numbers, add the opposite
of the second number to the first.
35. −13 − 13 = −13 + (−13) = −26
36. −43 − 43 = −43 + (−43) = −86
11. 2 − 9 = 2 + (−9) = –7
37. 24 − 25 = 24 + (−25) = −1
12. 5 − 11 = 5 + (−11) = −6
38. 43 − 98 = 43 + (−98) = −55
13. 4 − (−3) = 4 + 3 = 7
39. −6 − (−38) = −6 + 38 = 32
14. 12 − (−8) = 12 + 8 = 20
40. −75 − (−21) = −75 + 21 = −54
15. −3 − 15 = −3 + (−15) = −18
332
Section 10.3
41. −48 − (−33) = −48 + 33 = −15
66.
42. −29 − (−32) = −29 + 32 = 3
Subtraction of Real Numbers
5  2 5 2 7
−−  = + =
9  9 9 9 9
3  7
3 7
4 2
−−  = − + = =
10  10 
10 10 10 5
43. Minus, difference, decreased, less than,
subtract from
67. −
44. Subtraction is not commutative.
3−7≠7−3
5  3
5 3
5 6 1
68. − −  −  = − + = − + =
8  4
8 4
8 8 8
45. 14 − 23 = 14 + −23 = −9
69. 1
46. 27 − 40 = 27 + (−40) = −13
47. 5 − 12 = 5 + (−12) = −7
3
5 17  19  17  38 
− 2 = + −  = + − 
14
7 14  7  14  14 
21
1
= − = −1
14
2
48. 16 − 10 = 16 + (−10) = 6
70. 2
49. 105 − 110 = 105 + (−110) = −5
50. 70 − 98 = 70 + (−98) = −28
51. 320 − (−20) = 320 + 20 = 340
1 5
1  5
2  5
7
71. − − = − +  −  = − +  −  = −
2 4
2  4
4  4
4
52. 150 − 75 = 150 + (−75) = 75
53. −35 − 24 = −35 + (−24) = −59
72. −
54. 175 − 189 = 175 + (−189) = −14
55. −34 − 21 = −34 + (−21) = −55
56. −90 − 22 = −90 + (−22) = −112
11  9 
11 3
11  3 
− = − +−  = − +− 
15  15 
15 5
15  5 
20
4
=−
=−
15
3
73. 2 + 5 − (−3) − 10 = 2 + 5 + 3 + (−10)
= 7 + 3 + (−10)
= 10 + (−10)
=0
57. 5.2 − 13.5 = 5.2 + (−13.5) = −8.3
58. 4.4 − 10.2 = 4.4 + (−10.2) = −5.8
74. 4 − 8 + 12 − (−1) = 4 + (−8) + 12 + 1
= −4 + 12 + 1 = 8 + 1 = 9
59. −2.3 − 1.9 = −2.3 + (−1.9) = −4.2
60. −4.1 − 2.1 = −4.1 + (−2.1) = −6.2
75. −5 + 6 + (−7) − 4 − (−9)
= −5 + 6 + (−7) + (−4) + 9
= 1 + (−7) + (−4) + 9
= −6 + (−4) + 9 = −10 + 9 = −1
61. −3.6 − (−9.1) = −3.6 + 9.1 = 5.5
62. −8.9 − (−10.5) = −8.9 + 10.5 = 1.6
63. 5.5 − (−2.8) = 5.5 + 2.8 = 8.3
76. −2 − 1 + (−11) + 6 − (−8)
= −2 + (−1) + (−11) + 6 + 8
= −3 + (−11) + 6 + 8
= −14 + 6 + 8 = −8 + 8 = 0
64. 11.9 − (−4.3) = 11.9 + 4.3 = 16.2
65.
3
9 43  59  43  118 
−5 =
+ −  =
+ −

20
10 20  10  20  20 
75
15
3
=−
= −3 = −3
20
20
4
2  1 2 1 4 1 5
−−  = + = + =
3  6 3 6 6 6 6
77. [ −2 − ( −6)]2 = [ −2 + 6]2 = [4]2 = 16
333
Chapter 10
Real Numbers
88. −1+ 0 + 0 + (−1) + 0 + 1+ 0 + (−2) + 0
= −1+ 0 + (−1) + 0 + 1+ 0 + (−2) + 0
= −1+ (−1) + 0 + 1+ 0 + (−2) + 0
= −2 + 0 + 1+ 0 + (−2) + 0
= −2 + 1+ 0 + (−2) + 0
= −1+ 0 + (−2) + 0
= −1+ (−2) + 0
= −3 + 0
= −3
Ernie’s score is −3, which means 3 below
par.
78. [ −1 − ( −4)]2 = [ −1 + 4]2 = [3]2 = 9
79. [ −5 − ( −6)]3 = [ −5 + 6]3 = [1]3 = 1
80. [ −3 − ( −5)]3 = [ −3 + 5]3 = [2]3 = 8
81. 25 − 13− (−40) = 25 + (−13) + 40
= 12 + 40 = 52
82. −35 + 15 − (−28) = −35 + 15 + 28
= −20 + 28 = 8
 1  1 
 1 1
83. 5.5 −  −  = 5.5 −  +  −  
 2 5
 2  5 
 5  2 
= 5.5 −  +  −  
 10  10  
3
= 5.5 + (−0.3)
= 5.5 −
10
= 5.2
89. 0.14 − (−0.04) = 0.14 + 0.04 = 0.18
The difference is 0.18 point.
90. 0.1 − (−0.05) = 0.1 + 0.05 = 0.15
The difference is 0.15 point.
91. −$320 − $55 = −$320 + (−$55) = −$375
His new balance is −$375.
92. −$210 + 25 = −$185
His balance is −$185.
2 3
 4
3
84. −6.8 −  +  = −6.8 −  + 
 5 10 
 10 10 
93. 29, 029 − (−35, 798 ) = 29, 029 + 35, 798
= 64, 827 ft
 7
= −6.8 −  
 10 
= −6.8 + (−0.7) = −7.5
94. 4392 − (−86 ) = 4392 + 86 = 4478 m
85. 6000°F − (–423°F) = 6000°F + 423°F
= 6423°F
The temperature difference is 6423°F.
95. The range is 3° − (−8°) = 11°.
86. 214°C − (−184°C) = 214°C + 184°C
= 398°C
97. For example, 4 − 10
96. The range is −1° − (−12°) = 11°.
98. For example, 10 − 30
87. 17, 476.55 + 1786.84 − 2342.47 − 754.32
+321.63 + 1597.28
= 19, 263.39 − 2342.47 − 754.32
+321.63 + 1597.28
= 16,920.92 − 754.32 + 321.63 + 1597.28
= 16,166.60 + 321.63 + 1597.28
= 16, 488.23 + 1597.28
= 18,085.51
The balance was $18,085.51.
99. To find each number, subtract 4 from the
previous number.
−7 − 4 = −7 + (−4) = −11
−11 − 4 = −11 + (−4) = −15
−15 − 4 = −15 + (−4) = −19
100. To find each number, subtract 5 from the
previous number.
−28 − 5 = −28 + (−5) = −33
−33 − 5 = −33 + (−5) = −38
−38 − 5 = −38 + (−5) = −43
334
Section 10.3
101. To find each number, subtract
104. Negative, since b − a = b + (−a) and a is
positive so −a is negative.
1
from the
3
previous number.
2 1
2  1
3
− − = − +  −  = − = −1
3 3
3  3
3
105. Positive, since |a| and |b| are both positive
106. Positive or zero
1
3  1
4
−1 − = − +  −  = −
3
3  3
3
4 1
4  1
5
− − = − +−  = −
3 3
3  3
3
102. To find each number, subtract
Subtraction of Real Numbers
107. Negative
108. Negative
109. Negative, since a is positive
1
from the
4
110. Positive, since b is negative
111. −190 − 223 = −413
previous number.
1 1
2  1
3
− − = − +−  = −
2 4
4  4
4
112. −288 − 145 = −433
113. −23.24 − (−90.01) = 66.77
3 1
3  1
4
− − = − +  −  = − = −1
4 4
4  4
4
1
4  1
5
−1 − = − +  −  = −
4
4  4
4
114. −14.93 − (−34.99) = 20.06
115. 89.2 − (−23.6) = 112.8
116. 104.9 − (−24.8) = 129.7
103. Positive, since a − b = a + (−b) and b is
negative so −b is positive.
Problem Recognition Exercises: Addition and Subtraction of Real
Numbers
1. −7 − 5 = −7 + (−5 ) = − (7 + 5 ) = −12
10. −31.2 + (−52.6 ) = − (31.2 + 52.6 ) = −83.8
2. −7 − (−5 ) = −7 + 5 = − (7 − 5 ) = −2
11. −31.2 − 52.6 = −31.2 + (−52.6 )
= − (31.2 + 52.6 ) = −83.8
3. −7 + (−5 ) = − (7 + 5 ) = −12
12. −31.2 + 52.6 = 52.6 − 31.2 = 21.4
4. −7 + 5 = − (7 − 5 ) = −2
13. −19.5 + 21.5 = 21.5 − 19.5 = 2
5. 10 − (−45 ) = 10 + 45 = 55
14. −19.5 − 21.5 = −19.5 + (−21.5 )
= − (19.5 + 21.5 ) = −41
6. 10 − 45 = 10 + (−45 ) = − (45 − 10 ) = −35
15. −19.5 + (−21.5 ) = − (19.5 + 21.5 ) = −41
7. 10 + (−45 ) = − (45 − 10 ) = −35
16. −19.5 − (−21.5 ) = −19.5 + 21.5
= 21.5 − 19.5 = 2
8. 10 + 45 = 55
9. −31.2 − (−52.6 ) = −31.2 + 52.6
= 52.6 − 31.2 = 21.4
17.
335
− 12 + 8 = − 4 = 4
Chapter 10
Real Numbers
1  5  13  5  26  5 
+− 
30. 4 +  −  = +  −  =
3  6 3  6 6  6
26 5 21 7
1
=
− =
= or 3
6 6 6 2
2
18. 12 − 8 = 4 = 4
19.
− 12 − 8 = − 20 = 20
20. 12 − (−8 ) = 12 + 8 = 20 = 20
21.
2
1
7 31
14 31
31. −1 − 3 = − −
=− −
5
10
5 10
10 10
45
 31 14 
= − +  = −
 10 10 
10
9
1
= − or − 4
2
2
1 5 1 10
 10 1 
− = −
= − − 
 8 8
8 4 8 8
9
1
= − or − 1
8
8
1 5
1 10
 1 10 
= − + 
22. − − = − −
8 8 
8 4
8 8
11
3
=−
or − 1
8
8
5  1
5
1 17 7 24
+ =
32. 2 −  −1  = 2 + 1 =
6  6
6
6 6 6 6
=4
1  5  1  10 
 10 1 
+ −  = + −  = − − 
23.




 8 8
8
4
8
8
9
1
= − or − 1
8
8
33. −
34. −
1  5
1 10 9
1
= or 1
24. − −  −  = − +
8  4
8 8 8
8
3
3 12 12 3 9
1
+ 3= − +
=
− = or 2
4
4 4
4 4 4
4
4
4 5
 4 5
−1= − − = − + 
 5 5
5
5 5
9
4
= − or − 1
5
5
35. −2 + 0.001 = − (2 − 0.001) = −1.999
25. −
7 1
14 3
17
 14 3 
− = − − = − +  = −
 18 18 
9 6
18 18
18
26. −
7 1
14 3
11
 14 3 
+ = − + = − −  = −


9 6
18 18
18 18
18
37. −56 + 56 = 56 − 56 = 0
7  1
7 1
14 3
−−  = − + = − +
9  6
9 6
18 18
14
3
11


= − −  = −
 18 18 
18
39. −56 − 56 = − (56 + 56 ) = −112
27. −
28.
36. 4 − 5.987 = − (5.987 − 4 ) = −1.987
38. 14 + (−14 ) = 0
40. 14 − (−14 ) = 14 + 14 = 28
7 1 14 3 11
− =
−
=
9 6 18 18 18
29. 2
1
1 9 11 9 22
−5 = − = −
4
2 4 2 4 4
13
1
 22 9 
= −
−
=−
or − 3
 4 4 
4
4
336
Section 10.4
Section 10.4
Multiplication and Division of Real Numbers
Multiplication and Division of Real Numbers
Section 10.4 Practice Exercises
1. (a) positive; negative
15. 9(−8) = −72
(b) positive; negative
16. 8(−3) = −24
(c) reciprocals
17. (−1.2)(−3.2) = 3.84
1
1
18. (−3.3)(−2.5) = 8.25
2. (a) 2 ⋅ 3 = 1 = 1
3 2 1
1
19. −6(0.4) = −2.4
1
20. −8(1.3) = −10.4
1
(b) 7 ⋅ 1 = 7 ⋅ 1 = 1
14 1 14 2
21. 7(−1.1) = −7.7
2
1
22. 5(−3.4) = −17
5
23. −14 ⋅ 0 = 0
(c) 3 ÷ 6 = 3 ⋅ 25 = 5
5 25 5 6
2
1
24. −8 ⋅ 0 = 0
2
2
3. 14 − (−5) = 14 + 5 = 19
25.  − 2   − 6  = 4
 3  7  7



4. −24 − 50 = −24 + (−50) = −74
1
5. −33 + (−11) = −(33 + 11) = −44
2
1
26.  − 8  − 3  = 2
 9  4  3



6. −7 − (−23) = −7 + 23 = 16
3
7. 23 − 12 + (−4) − (−10)
= 23+ (−12) + (−4) + 10
= 11+ (−4) + 10 = 7 + 10 = 17
1
1
1
1
7
27. 3  − 5  = − 1


5  21 
7
8. 9 + (−12) − 17 − 4 − (−15)
= 9 + (−12) + (−17) + (−4) + 15
= −3 + (−17) + (−4) + 15
= −20 + (−4) + 15 = −24 + 15 = −9
28.
5
12
3
9. −3(5) = −15
1
5
 4
 − 7  = − 21


1
10. −2(13) = −26
29. 6 ⋅  − 5  = 6 ⋅  − 5  = − 5 or − 2 1




2
2
 12  1  12 
11. −12 ⋅ 4 = −48
2
12. −6 ⋅ 11 = −66
1
30. 4 ⋅  − 1  = 4 ⋅  − 1  = − 1




4
 16  1  16 
13. −15(–3) = 45
4
14. −3(–25) = 75
337
Chapter 10
Real Numbers
1
52. (3)(0)(−13)(22) = 0
31.  −2 3  −1 2  =  − 13   − 5  = 13 or 4 1


 


3
 5  3   5  3  3
53. (−1)(−1)(−1)(−1)(−1)(−1) = 1
1
54. (−1)(−1)(−1)(−1)(−1)(−1)(−1) = −1
2
32.  −3 1   −2 1  =  − 10   − 11
 3   5   3   5 


55. (−1)(−1)(−1)(−1)(−1) = −1
56. (−1)(−1)(−1)(−1) = 1
1
=
22
1
or 7
3
3
57. −102 = −(10)(10) = −100
58. −82 = −(8)(8) = −64
 8
33.  −  ⋅ 0 = 0
 9
59. (−10) 2 = ( −10)( −10) = 100
 2
34.  −  ⋅ 0 = 0
 11 
60. ( −8) 2 = ( −8)( −8) = 64
35.
(−3.5)(−1.4)= 4.9
61. −33 = −(3)(3)(3) = −27
36.
(−1.6)(−6.5)= 10.4
62. −43 = −(4)(4)(4) = −64
37. −3(−1) = 3
63. ( −3)3 = ( −3)( −3)( −3) = −27
38. −12(−4) = 48
64. ( −4)3 = ( −4)( −4)( −4) = −64
39. −5 ⋅ 3 = −15
65. −0.23 = −(0.2)(0.2)(0.2) = −0.008
40. 9(−2) = −18
66. −0.43 = −(0.4)(0.4)(0.4) = −0.064
41. 1.3(−3) = −3.9
3
8
 2   2  2  2 
67.  −  =  −  −  −  = −
27
 3   3  3  3 
42. (−2.3)(6) = −13.8
43. (3)(−12) = –36 customers
3
27
 3   3  3  3 
68.  −  =  −  −  −  = −
125
 5   5  5  5 
44. (5)(−15) = –$75
45. (4)(−8) = –32 yd
46. (11)(−20) = –220°F
47. (5)(−2)(4)(−10) = 400
48. (−3)(−5)(−2)(4) = −120
69.
(−6 )2 = (−6)(−6)= 36
70.
(−6)3 = (−6 )(−6 )(−6 )= −216
( )2 = − (−6 )(−6)= −36
71. − −6
49. (−11)(−4)(−2) = −88
50. (20)(−3)(−1) = 60
( )3 = − (−6)(−6 )(−6 )= 216
72. − −6
51. (24)(−2)(0)(−3) = 0
338
Section 10.4
73.
−15
= −3
5
74.
30
= −5
−6
75.
56
= −7
−8
76.
 1
91.  −  ÷ 0 is undefined.
 6
2
92.   ÷ 0 is undefined.
 11 
−48
= −16
3
−25 5
77.
=
−15 3
78.
82.
−41
is undefined.
0
0
=0
5
−1 1
=
−3 3
−18
3
=−
24
4
12
2
=−
−30
5
98. 46 ÷ (−23) = −2
13
is undefined.
0
84.
94.
97. −100 ÷ 20 = −5
81.
0
=0
−2
−5 5
=
−8 8
96. 12 ÷ (−30 ) =
−9 9
=
−8 8
83.
93.
95. −18 ÷ 24 =
−6 1
=
−18 3
−2 2
79.
=
−3 3
80.
Multiplication and Division of Real Numbers
99. −32 ÷ (−64) =
100. 108 ÷ (−24) =
−32 1
=
−64 2
108
9
=−
−24
2
101. −52 ÷ 13 = −4
102. −45 ÷ (−15) = 3
103. 8 + (−6) = 2
85. (−20) ÷ (−5) = 4
104. 8 − (−6) = 8 + 6 = 14
86. (−10) ÷ (−2) = 5
105. 8(−6) = −48
87. −0.91 ÷ −0.7 = 1.3
106. 8 ÷ (−6) =
88. −1.3 ÷ −0.5 = 2.6
2
107. −9 − (−12) = −9 + 12 = 3
89.  8  ÷  − 4  =  8  ⋅  − 5  = − 10
 7   5   7   4 
7
108. (−9)(−12) = 108
1
1
109. −36 ÷ (−12) = 3
3


90.  − 2  ÷  8  =  − 2  ⋅ 15 = − 3


 5   15   5 
4
 8 
1
8
4
=−
−6
3
110. −36 + (−12) = −48
4
111. (−5)(−4) = 20
339
Chapter 10
Real Numbers
112. −90 ÷ (−6) = 15
122. (6)11 has one more factor of 6 than (6)10 ,
so (6)11 is greater.
113. 0 + (−15) = −15
114. 0 − (−15) = 0 + 15 = 15
123. a ⋅ b is negative since signs are different.
2
124. b ÷ a is negative since signs are different.
115. 1 ÷  − 5  = 1 ⋅  − 6  = − 2


3  6  3  5 
5
125. −a ÷ (b) is positive since −a and b are both
negative.
1
116.
1  5 2  5
3
1
+−  = +−  = − = −
3  6 6  6
6
2
126. a(−b) is positive since both factors are
positive.
117.
1  5 1 5 2 5 7
−−  = + = + =
3  6 3 6 6 6 6
127. (−413)(871) = −359,723
118.
1 5
5
−  = −
3 6 
18
129.
(−52.12 )(−101.5 ) = 5290.18
130.
−576,828
= 54
−10,682
128. −6125 ⋅ (−97) = 594,125
119. ( −2)50 is positive and ( −2)51 is negative,
so ( −2)50 is greater.
131. 5,945,308 ÷ (−9452) = −629
120. ( −3) 20 is positive and ( −3) 21 is negative,
so ( −3)
20
132.
is greater.
−301, 224
= −33
9128
121. (5)41 has one more factor of 5 than (5) 40 ,
so (5)41 is greater.
Problem Recognition Exercises: Operations on Real Numbers
1. 15 − (−5 ) = 15 + 5 = 20
8. −36 + (−2 ) = −38
2. 15 (−5 ) = −75
9. 20 (−4 ) = −80
3. 15 + (−5 ) = 10
10. −20 (−4 ) = 80
4. 15 ÷ (−5 ) = −3
11. −20 (4 ) = −80
5. −36 (−2 ) = 72
12. 20 (4 ) = 80
6. −36 − (−2 ) = −36 + 2 = −34
13. −5 − 9 − 2 = −5 + (−9 ) + (−2 ) = −14 + (−2 )
= −16
7.
−36
= 18
−2
14. −4 (−9 )(−2 ) = 36 (−2 ) = −72
340
Problem Recognition Exercises: Operations on Real Numbers
15. 10 + (−3) + (−12 ) = 7 + (−12 ) = −5
9  13 
 1  4
27.  −2   1  = − ⋅  
 4  9
4  9
13
1
=−
or − 3
4
4
16. 10 − (−3) − (−12 ) = 10 + 3 + 12 = 13 + 12
= 25
17.
(−1)(−2 )(−3)(−4 ) = 2 (−3)(−4 )
= −6 (−4 ) = 24
28. −2
18.
(−1)(−2 )(3)(4 ) = 2 (3)(4 ) = 6 (4 ) = 24
29. 41.5 − (−13.6 ) = 41.5 + 13.6 = 55.1
19.
(−1)(−2 )(−3)(4 ) = 2 (−3)(4 ) = −6 (4 )
30. −13.56 + 4.12 = − (13.56 − 4.12 ) = −9.44
20.
(−1)(2 )(3)(4 ) = −2 (3)(4 ) = −6 (4 ) = −24
31. −60.41 − 33.50 = −60.41 + (−33.50 )
= −93.91
21.
27
3  10  3  9 
÷  −  = ⋅ −  = −
50
5  9  5  10 
22.
23.
24.
= −24
1
4
9 13
81 52
29
+1 = − + = −
+
=−
4
9
4 9
36 36
36
32. −0.06 − (−0.04 ) = −0.06 + 0.04 = −0.02
30
2
3  10 
=−
 −  = −
9
45
3
5
23
3  10  27  50 
+−  =
+−  = −
45
5  9  45  45 
3  10  27 50 77
−−  =
+
=
5  9  45 45 45
33.
−12 12
=
−11 11
34.
1
5
=−
6
−30
35.
0
=0
−8
36. −4.5 ÷ 0 Undefined
37. 42 ÷ (−0.002 ) = −21, 000
2  7
6  7
13
25. − +  −  = − +  −  = −




3
9
9
9
9
38. −360 ÷ (−0.009 ) = 40, 000
2  7
2  9  18 6
=
26. − ÷  −  = − ⋅  −  =
3  9
3  7  21 7
39. −44 − (−44 ) = −44 + 44 = 0
60  1 
 1
40. −60 ⋅  −  = −
⋅ −
=1
 60 
1  60 
Section 10.5
Order of Operations
Section 10.5 Practice Exercises
1
1. −28 + 72 = 44
3
3.  − 2  ÷  8  =  − 2  ⋅  27  = − 3


   
 
4
 9   27   9   8 
2. −100 – (−4) = –96
1
341
4
Chapter 10
Real Numbers
2
26. 1− (−5)(−3)2 = 1− (−5)(9) = 1− (−45)
= 1+ 45 = 46
4. 10 ⋅  − 3  = 10 ⋅  − 3  = −6




 5 1  5 
1
27. 12 + (14 − 16)2 ÷ (−4) = 12 + (−2)2 ÷ (−4)
= 12 + (4) ÷ (−4)
= 12 + (−1) = 11
5. −2.8(−1.1) = 3.08
6. 5.5 ÷ (−0.5) = −11
28. −7 + (1− 5)2 ÷ 4 = −7 + (−4)2 ÷ 4
= −7 + 16 ÷ 4 = −7 + 4
= −3
7. (−1)(−5)(−8)(3) = −120
8. −1 + ( −5) + (−8) + 3 = −(1 + 5 + 8) + 3
= −14 + 3 = −11
29. −48 ÷ 12 ÷ (−2) = −4 ÷ (−2) = 2
9. 5 + 2 (3 − 5 ) = 5 + 2 (−2 ) = 5 + (−4 ) = 1
30. −100 ÷ (−5) ÷ (−5) = 20 ÷ (−5) = −4
10. 6 − 4 (8 − 10 ) = 6 − 4 (−2 ) = 6 + 8 = 14
31. 90 ÷ (−3)(−1) ÷ (−6) = −30(−1) ÷ (−6)
= 30 ÷ (−6) = −5
11. −8 − 62 = −8 − 36 = −8 + (−36 ) = −44
32. 64 ÷ (−4)2 ÷ (−16) = −16 ⋅ 2 ÷ (−16)
= −32 ÷ (−16) = 2
12. −10 − 5 = −10 − 25 = −10 + (−25 ) = −35
2
2
2
2
2
( )2 ÷ (−4)= 81− 49 ÷ (−4)
= 32 ÷ (−4 )
33. 92 − −7
13. 4 + (3 − 8 ) = 4 + (−5 ) = 4 + 25 = 29
14. 5 + (2 − 9 ) = 5 + (−7 ) = 5 + 49 = 54
= 32 ÷ (−4) = −8
15. 120 ÷ (−4 )(5 ) = −30 (5 ) = −150
34. (−8)2 − 52 ÷ (−3) = 64 − 25 ÷ (−3)
16. 36 ÷ (−2 )(3) = −18 (3) = −54
= 39 ÷ (−3)
= 39 ÷ (−3) = −13
17. −2.1 − 6 ÷ 5 = −2.1 − 1.2 = −3.3
3
3
35. 2 + 2 − 10 − 12 = 2 + 2 − − 2
18. −8.3 − 10 ÷ 8 = −8.3 − 1.25 = −9.55
2
2
= 2 + 23 − 2 = 2 + 8 − 2
= 10 − 2 = 8
19.  5.3 − (−2.7 ) = [5.3 + 2.7 ] = 8 = 64
2
21. −2(3− 6) + 10 = −2(−3) + 10 = 6 + 10 = 16
36. 14 − 42 + 2 − 10 = 14 − 16 + 2 − 10
= −2 + 2 − 10 = 0 − 10
= −10
22. −4 ÷ (1− 3) − 8 = −4 ÷ (−2) − 8 = 2 − 8 = −6
37. −6(48 ÷12)2 = −6(4)2 = −6(16) = −96
23. −16 ÷ (−4)(−5) = 4(−5) = −20
38. −5(35 ÷ 5) 2 = −5(7) 2 = −5(49) = −245
20.
(−7.1 − 1.9 )2 = (−9 )2 = 81
24. −12(−1) ÷ 6 = 12 ÷ 6 = 2
2
1 1
1 12
 1 1 1
39.  −  ⋅ ÷ = − ÷ = − ⋅
= −2
6 12
6 1
 2  3 12
25. 8 − (−3)(−2)3 = 8 − (−3)(−8) = 8 − 24
= −16
1
342
Section 10.5
1
2
2
=−
4
2
4
46.  −2 1  ÷ 1 1  −  1  =  − 7  ÷  4  −  1 
 3  3
 2
 3  3
 2
 7   16   1 
= −  ÷   −  
 3   9   16 
2 6
2  1 6 2  3  6
40.
÷  −  ⋅ = ⋅ −  ⋅ = − ⋅
3 5
9  3 5 9  1  5
3
Order of Operations
1
4
5
3
 7  9  1
= −  ⋅  −
 3   16  16
1
1  5  4 1  5  1  10 
+ −
⋅ = + − = + −
41.
6  4  3 6  3  6  6 
1
21 1
22
− =−
16 16
16
11
3
= − = −1
8
8
=−
1
9
3
=− =−
6
2
1
1
4
1
3 5  5
3 5  6
42. − +
÷−  = − +
⋅ −
8 24  6 
8 24  5 
47. 21−  4 − (5 − 8 ) = 21 −  4 − (−3)
= 21 − [4 + 3] = 21− 7
= 14
3  1
= − +− 
8  4
3  2
5
= − +−  = −
8  8
8
48. 15 − 10 − (20 − 25 ) = 15 − 10 − (−5 )
= 15 − [10 + 5 ]
= 15 − 15 = 0
49. −17 − 2 18 ÷ (−3) = −17 − 2 [−6 ]
= −17 + 12 = −5
2
 2
 5  15 4 5 15
43.  −  −   ÷ = − ÷
 3
 21 7 9 21 7
=
=
1
1
3
3
50. −8 − 5 (−45 ÷ 15 ) = −8 − 5 (−3) = −8 + 15
=7
4 5 7
−
⋅
9 21 15
51. 4 + 2  9 + (−4 + 12 ) = 4 + 2 [9 + 8 ]
= 4 + 2 [17 ] = 4 + 34
= 38
4 1 3 1
− = =
9 9 9 3
52. −13 + 3 11 + (−15 + 10 )
= −13 + 3 11 + (−5 )
= −13 + 3[6 ] = −13 + 18
=5
1
3
44.  − 1  +  2  ⋅ 5 = − 1 + 2 ⋅ 5
 3   9  6
27 9 6
3
=−
2
3
1
5
4
+
=
27 27 27
2
1  1
5  3
 1
 1
45. 2 ⋅ 1  +  −  = ⋅   +  − 






 2
2
2
2
2 2
5 9  1
= ⋅ + − 
2 4  8
45  1 
=
+ − 
8  8
44 11
1
=
= =5
8
2
2
53. 22 − − 3 + 9 = 22 − 6 = 4 − 6 = −2
3
54. 5 2 − 10 + (−8 ) = 5 2 − 2 = 25 − 2 = 23
55.
343
3 + (−5 )
−2
2
4 − (3)(−2 ) 4 − (3)(−2 ) 4 − (3)(−2 )
2
2
2 1
=
=
=
=
4 − (−6 ) 4 + 6 10 5
=
=
Chapter 10
56.
Real Numbers
− 11 + 7
−4
66. −6 + (−2) + 5 + 1+ 0 + (−3) + 7 + 2 + (−4)
9
0
= =0
9
4
8 − 4 (−1) 8 − 4 (−1) 8 − 4 (−1)
4
4 1
=
=
=
8 + 4 12 3
=
=
57. 13 − (2 )(4 ) = 13 − (2 )(4 ) = 13 − 8 = 5
−1 − 4 −5
−1 − 4
−1− 2 2
= −1
9
( −89.6 ) + 32 = 1.8 ( −89.6 ) + 32
5
= −161.28 + 32 = −129.28º F
67. ºF =
58. 10 − (−3)(5 ) = 10 − (−3)(5 ) = 10 − (−15 )
−9 − 16
−9 − 16
−9 − 4 2
10 + 15
25
=
=
= −1
−9 − 16 −25
9
68. °F (−27.2) + 32 = 1.8(−27.2) + 32
5
= −48.96 + 32 = 16.96 °F
59. 1− 4 (3 − 5 ) = 1− 4 (−2 ) = 1− 4 (−2 )
25 − 4
52 − 22
52 − 22
1+ 8
9 3
=
=
=
25 − 4 21 7
 1
 3 8
69.   ÷ 0.05 +  − ⋅ 
 2
 4 3
1 1
2
6 − 32
=
2
(5 − 2 )
6 − 32
2
3
=
2
2
 1
=   ÷ 0.05 + (−2)
 2
1
= ÷ 0.05 + (−2)
4
= 0.25 ÷ 0.05 + (−2)
= 5 + (−2) = 3
60. −3 − (2 + 4 ) = −3 − 6 = −3 − 6 = −9
6 2 − 32 36 − 9 27
6 2 − 32
1
=−
3
61.
1
6 − 9 −3
1
=
=−
9
9
3
70. −0.8 −
2
2
62. −2 − 5 = −2 − 5 = −2 − 25 = −27 = −3
9
9
32
(6 − 3)2
63. −8° + (−11°) + (−4°) + 1° + 9° + 4° + (−5°)
7
−14°
=
= −2°
7
19  4 1 
+
÷
− (−0.15)
20  5 2 
19  4 2 
+
⋅ − (−0.15)
= −0.8 −
20  5 1 
19 8
+ − (−0.15)
= −0.8 −
20 5
= −0.8 − 0.95 + 1.6 + 0.15
=0
 7 1
 7 2
71. 2  −  − (−1.5)2 = 2  −  − (−1.5)2
 8 4
 8 8
 5
= 2   − (−1.5)2
 8
 5
= 2   − (2.25)
 8
64. 15° + 12° + 10° + 3° + 0° + (−2°) + (−3°)
7
35°
=
= 5°
7
65. −8+(−8) +(−6) +(−5) +(−2) +3+3+0+(−4)
9
−27
=
=−3
9
10
− 2.25
8
= 1.25 − 2.25 = −1
=
344
Section 10.5
Order of Operations
 7

72. −2.1+ 4  − 0.0375
 16

= −2.1+ 4(0.4375 − 0.0375)
= −2.1+ 4(0.4) = −2.1+ 1.6
1
= −0.5 or −
2
Chapter 10
Review Exercises
Section 10.1
17. |3| > −|3|
1. −10,227
18. 8 < |−9|
2. −$5 billion
19. −2.8 < −2
3. 15°
20. −
4. $10
3
1
>−
2
2
Section 10.2
5−8.
21. 6 + (−2) = 4
9. Opposite: 4
absolute value: 4
22. −3 + 6 = 3
23. −3 + −2 = −5
1
10. Opposite:
2
absolute value:
24. −3 + 0 = −3
1
2
25. To add two numbers with the same sign,
add their absolute values and apply the
common sign.
11. Opposite: −3.5
absolute value: 3.5
26. To add two numbers with different signs,
subtract the smaller absolute value from
the larger absolute value. Then apply the
sign of the number having the larger
absolute value.
12. Opposite: −6
absolute value: 6
13. (a) −(−9) = 9
27. 35 + (−22) = 35 − 22 = 13
(b) −|−9| = −9
28. −105 + 90 = −(105 − 90) = −15
14. (a) −(1.5) = −1.5
29. −29 + (−41) = −(29 + 41) = −70
(b) −|1.5| = −1.5
30. 3.22 + (−4.1) = −(4.1 − 3.22) = −0.88
5
15. − > −1
6
31. −6.5 + (−4.16) = −(6.5 + 4.16) = −10.66
16. −0.5 < 0.5
345
Chapter 10
Real Numbers
 3  1  7   5
32.  −1  +  2  =  −  +  
 4  2  4  2
 7   10 
= −  +  
 4  4 
10 7 3
= − =
4 4 4
46. −2 − (−24) = −2 + 24 = 22
 1  7   2   7 
33.  −  +  −  =  −  +  − 
 5   10   10   10 
 2 7
9
= − +  = −
10
 10 10 
 5  5   5 5
50.  −  −  −  =  −  +
 3   12   3 12
 20  5
= −  +
 12  12
15
5
=− =−
12
4
47. −289 − 130 = −289 + (−130) = −419
48. −2.9 − 4.5 = −2.9 + (−4.5) = −7.4
49. 3.8 − 4.5 = 3.8 + (−4.5) = −0.7
1
 7 6  7
7 6
34. 2 +  −  = +  −  = −  −  = −
3
 3 3  3
 3 3
35. 23 + (−35) = −(35 − 23) = −12
20 20
 20 
=
51. 0 −  −  = 0 +
21 21
 21 
36. 57 + (−10) = 57 − 10 = 47
52. For example: The difference of 4 and 6
37. −5 + (−13) + 20 = −18 + 20 = 2
53. For example: 23 minus negative 6
38. −42 + 12 = −30
54. For example: 14 subtracted from −2
39. −12 + 3 = −9
55. For example: Subtract −7 from −25.
40. −89 + (−22) = −111
56. −1°F − (−6°F) = −1°F + 6°F = 5°F
The temperature rose 5°F.
41. −3 + (−10) + 12 + 14 + (−10)
= −13 + 12 + 14 + (−10)
= −1 + 14 + (−10)
= 13 + (−10) = 3
57. −$40 + $132 = $92
Sam’s balance is now $92.
58.
42. 9 + (−15) + 2 + (−7) + (−4)
= −6 + 2 + (−7) + (−4)
= −4 + (−7) + (−4)
= −11 + (−4) = −15
Section 10.3
−3 + 4 + 0 + 9 + (−2) + (−1) + 0 + 5 + (−3)
9
9
= =1
9
The average is 1 above par.
Section 10.4
43. 1. Leave the first number (the minuend)
unchanged.
2. Change the subtraction sign to an
addition sign.
3. Add the opposite of the second
number (the subtrahend).
59. 6(−3) = −18
44. 4 − (−23) = 4 + 23 = 27
60.
−12
= −3
4
61.
−900
= 15
−60
62. (−7)(−8) = 56
45. 19 − 44 = 19 + (−44) = −25
346
Chapter 10
63. −2.8 ÷ 0.04 = −70
80. −4 ⋅ 19 = −76
64. (−62.6)(2.5) = −156.5
81. 30(−5) = −150
1
7
1
4
Review Exercises
82. −136 ÷ (−8) = 17
65.  2   21  7
 − 3  − 8  = 4



Section 10.5
83. 28 ÷ (−7) ⋅ 3 − (−1) = −4 ⋅ 3 − (−1)
= −12 − (−1) = −12 + 1
= −11
 1   1   17   5 
66.  −2  ÷  1  =  −  ÷  
 8  4  8   4
1
84. (−4)3 ÷ 8 − (−6) = −64 ÷ 8 − (−6)
= −8 − (−6) = −8 + 6 = −2
 17   4 
=  −  ⋅ 
 8   5
2
85. 10 − (−3)2 ⋅ (−11) + 4 = 10 − 9 ⋅(−11) + 4
17
7
=−
or − 1
10
10
= 1 ⋅(−11) + 4
= 1⋅(−11) + 4
= −11+ 4 = −7
 1
67.  −  ÷ 0 is undefined.
 5
86. [−9 − (−7)]3 ⋅ 3 ÷ (−6) = [−9 + 7]3 ⋅3 ÷ (−6)
0
68.
=0
−5
= [−2]3 ⋅3 ÷ (−6)
= −8⋅ 3 ÷ (−6)
= −24 ÷ (−6) = 4
69. (−1)(−8)(2)(1)(−2) = −32
70.
−9 9
4
= or 1
−5 5
5
87. 18 − (−5)2 + 14 ÷ 2 = 18 − 25 + 14 ÷ 2
= 18 − 25 + 7 = 0
71. ( −6) 2 = ( −6)( −6) = 36
 1   7   3  6 
88.   ÷  −  ⋅   +  − 
 15   10   2   7 
72. −6 2 = −(6)(6) = −36
5
1
 1   10   3   6 
=
+ −
⋅
 ⋅ −
 15   7   2   7 
3
27
 3   3  3  3 
73.  −  =  −  −  −  = −
64
 4   4  4  4 
5
1
5
6
1
6
+− =− +−
35
7
7
7
7
= − = −1
7
3
27
3
 3  3  3 
74. −   = −     = −
64
4
 4  4  4 
=−
75. ( −1)10 = 1
2
3
 3
 1
9  1 9 1
89.  −  −  −  =
− −
+
=
64  8  64 8
 8
 2
9
8 17
=
+
=
64 64 64
76. ( −1) 21 = −1
77. Negative
78. Positive
79. −45 ÷ (−15) = 3
347
Chapter 10
Real Numbers
(
)
( )
90. 6 − 5 − 2 − 8  = 6 − 5 − −6 
= 6 − 5 + 6  = 6 − 11 = −5
91.
( ) = 3− − 5
3− 2 + −7
2
2
3 −5
2
=
7
=
3− 5
2
Test
1. (a) −$220
18.
(b) 26
5  7  10  7  3
+−  = +−  =
4  8 8  8 8
2. −3, 0, 4, −1
19. 6(−12) = −72
3
4
3. −3, − , 0, 4, − 1,
5
7
20. (−11)(−8) = 88
4.
21.
−24
=2
−12
22.
54
= −18
−3
23.
−44
is undefined.
0
7, − π
5. −5 < −2
6. |−5| > |−2|
7. 0 > −2.4
8.
−7°
= −1°
7
3 − 52
3 −5
3− 5
−2 1
=
=
=
9 − 25 −16 8
Chapter 10
2
92. 2° + 4° + (−6°) + (−1°) + 0° + (−4°) + (−2°)
24. (−91)(0) = 0
4
2
>−
5
3
9. −|−9| < 9
10. −|33.1| < |−33.1|
11. 9 + (−14) = −5
25.
15
3
3  4 3  5
÷  −  = ⋅ −  = −
=−
40
8
10  5  10  4 
26.
−13 13
=
−6
6
27. (a) Positive
12. −23 + (−5) = −28
(b) Negative
13. −4 − (−13) = −4 + 13 = 9
28. (a) ( −8) 2 = ( −8)( −8) = 64
14. −30 − 11 = −30 + (−11) = −41
15. −1.5 + 2.1 = 0.6
(b) −82 = −(8)(8) = −64
16. 0.5 − 2.8 = 0.5 + (−2.8) = −2.3
(c) ( −4)3 = ( −4)( −4)( −4) = −64
2 4
14 12
26
5
17. − − = − −
=−
or − 1
3 7
21 21
21
21
(d) −43 = −(4)(4)(4) = −64
348
Chapter 10
29. −3(−7) = 21
2
2
 1   5 1   1   15 2 
40.  −  ÷  −  =  −  ÷  − 
 3   6 9   3   18 18 
2
 1   13 
= −  ÷ 
 3   18 
30. −13 + 8 = −5
31. 18 − (−4) = 18 + 4 = 22
18
 2 6  3
32. 6 ÷  −  = ⋅  −  = − = −9
2
 3 1  2
2
1 13 1 18 2
=
= ÷ = ⋅
9 18 9 13 13
1
33. −8.1 + 5 = −3.1
( )
34. −3 + 15 + (−6) + (−1) = 12 + (−6) + (−1)
= 6 + (−1) = 5
= 16 − 2 8  = 16 − 16
=0
36. (−3)(−1)(−4)(−1)(−5) = −60
42.
37. −20 ÷ (−2)2 + (−14) = −20 ÷ (4) + (−14)
= −5 + (−14) = −19
1
3
1
15 − 2 3 − 9
8− 2
43.
2  20 
22
+−  = −
15  15 
15
Cumulative Review Exercises
1. 3490
+ 123
3613
2.
=
2901
− 332
2569
349
15 − 2 − 6
2
8− 2
15 − 2 6
()
8− 4
=
()
15 − 2 6
8 − 22
15 − 12 3
=
=
8− 4
4
4° + (−3°) + (−1°) + 5° + (−2°) + 0° + 4°
7
7°
= = 1°
7
39. − 2 +  − 20 ⋅ 7  = − 2 +  − 4 


15  21 5 
15  3 
=−
2
=
38. 12 ⋅(−6) + [20 − (−12)] − 15
= 12 ⋅ (−6) + (20 + 12) − 15
= 12 ⋅ (−6) + 32 − 15
= −72 + 32 − 15
= −40 + (−15) = −55
4
( )
41. 16 − 2 5 − 1− 4  = 16 − 2 5 − −3 
= 16 − 2 5 + 3
35. −14 + 22 − (−5) + (−10) = 8 + 5 + (−10)
= 13 + (−10) = 3
Chapters 1–10
Test
3.
23
34
98
+ 22
177
4.
790
× 24
3 160
15 800
18,960
Chapter 10
Real Numbers
5. 720 = 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 3 ⋅ 3 ⋅ 5 or 24 ⋅ 32 ⋅ 5
6.
115
35 4025
−35
52
−35
175
−175
0
16. 3.5 402.5
5
8
7. Harold answered 14 − 3 = 11 questions
11
of the quiz correct.
correctly. He got
14
8.
20 oz
=8
2 12 oz
17.
Amy will have 8 packages.
9. 16 = 2 ⋅ 2 ⋅ 2 ⋅ 2
40 = 2 ⋅ 2 ⋅ 2 ⋅ 5
10 = 2 ⋅ 5
LCM: 2 ⋅ 2 ⋅ 2 ⋅ 2 ⋅ 5 = 80
10.
=
1
2
7 35 7 6
2
= ÷ = ⋅
=
3 6 3 35 5
1
5
9964
= 2491 gal/hr.
4
1
7 6 42
19. 3 ⋅ 6 = ⋅ =
= 21 mi
2
2 1 2
3 33 7 15 66 56 25 5
+
− =
+
−
=
=
16 40 10 80 80 80 80 16
20.
1
7
21
= 16
= 15
2
14
14
13
13
13
− 12
− 12
− 12
14
14
14
4
8
=3
3
7
14
16
x 2.4
=
9
8
8 x = (9)(2.4)
8 x 21.6
=
8
8
x = 2.7 cm
y
8
=
2.1 2.4
2.4 y = (2.1)(8)
2.4 y 16.8
=
2.4
2.4
y = 7 cm
21. (0.32)(600) = 192
22.
23.
13. (a) 34.230
(b) 9.0
336
= 1.12 or 112%
300
15 = (0.06) x
15
0.06 x
=
0.06 0.06
250 = x
24. (0.20)($86) = $17.20
$86 − $17.20 = $68.80
The sale price is $68.80.
14. 209¢ = $2.09
15.
5 56 m
7
3
35
6
18. The aircraft used
13 18 43 54 43
3
11. 3 + 2 = +
=
+
15 5 15 15 15
5
97
7
=
or 6
15
15
12.
2 13 m
204.55
× 2.4
81 820
409 100
490.920
2 ft 12 in.
⋅
= 24 in.
1
1 ft
2 ft 4 in. = 24 in. + 4 in. = 28 in.
25. 2 ft =
26. 20 qt =
20 qt 1 gal
⋅
= 5 gal
1
4 qt
27. 60 mL = 0.06 L
350
Chapters 1–10
28. 30 oz =
30 oz 1 lb
7
⋅
= 1.875 lb or 1 lb
1 16 oz
8
Cumulative Review Exercises
34.
29. 62 + 82 = c 2
36 + 64 = c 2
100 = c 2
100 = c
10 = c
The distance is 10 mi.
4 + 4 + 4 + 3+ 6 + 4 + 6 + 5+ 3+ 4
10
43
=
= 4.3 mi
10
35. Mean =
30. A = bh = (5 yd)(3.3 yd) = 16.5 yd 2
2
2
 1 
9 
81 2
31. A = s 2 =  2 m  =  m  =
m
16
 4 
4 
1
=5
m2
16
36. 18% will be spent on postage. Find 18%
of $1200.
(0.18)($1200) = $216
32. A = πr 2
38. −12 + (−5) − 3− (−8) = −17 − 3+ 8
= −20 + 8 = −12
37. 43 − (−12) = 43 + 12 = 55
≈ (3.14)(1.5 km) 2
= (3.14)(2.25 km 2 )
39. ( −4) 2 − 6 2 = 16 − 36 = −20
= 7.065 km 2
3
8  1 5  2
40. ⋅  −  ÷  − 
9  3 6   3
C = 2πr
≈ 2(3.14)(1.5 km)
= 9.42 km
33.
3
8  2 5  2
= ⋅ −  ÷  − 
9  6 6   3
Number
of miles
Tally
Frequency
3
||
2
4
||||
5
5
|
1
6
||
2
3
8  3  2 
= ⋅ −  ÷  − 
9  6  3
3
8  1  2
= ⋅ −  ÷  − 
9  2   3
8  1  2 
= ⋅ −  ÷  − 
9  8  3
8  1  3
= ⋅ −  ⋅ − 
9  8  2
24 1
=
=
144 6
351
Chapter 11
Solving Equations
Chapter Opener Puzzle
Section 11.1
Properties of Real Numbers
Section 11.1 Practice Exercises
11. 2g
1. (a) variable
(b) commutative; x + 6
(c) b • a; 6x
(d) associative; ( −5 + 7) + x
12. 2t
13. (a) −6x = −6( ) = −6(2) = −12
(b) −6x = −6( ) = −6(−5) = 30
(e) (a • b • c); (−5  7) x
(f) a  b + a  c ; 4x + 20
14. (a) −2 y 2 = −2( )2 = −2(3)2 = −2(9) = −18
2. w + 54
(b) −2 y 2 = −2( )2 = −2(−3)2 = −2(9)
= −18
 1
15. (a) 3p + 5q = 3( ) + 5( ) = 3(2) + 5  − 
 5
3. 8p
4. 5r
= 6 + (−1) = 5
(b) 3p + 5q = 3( ) + 5( ) = 3(−5) + 5(0)
= −15 + 0 = −15
5. t + 4
6. l +
1
2
 1
16. (a) 9c − 2d = 9( ) − 2( ) = 9(−1) − 2  
 2
= −9 − 1 = −10
(b) 9c − 2 d = 9( ) − 2( ) = 9(3) − 2(−2)
= 27 − (−4) = 27 + 4 = 31
7. v − 6
8. A − 30
9.
10.
4
n
17. (a) − a2 = −( )2 = −(−7)2 = −49
p
20
(b) − a2 = −( )2 = −(7)2 = −49
352
Section 11.1
26. 5b − c = 5( ) − ( ) = 5(−3) − (−2)
= −15 + 2 = −13
18. (a) −b3 = −( )3 = −(−3)3 = −(−27) = 27
(b) −b3 = −( )3 = −(3)3 = −27
27. b2 − c 2 = ( )2 − ( )2 = (−3)2 − (−2)2
= 9−4=5
19. (a) −4( r − s)2 = −4[( ) − ( )]2
= −4[(8) − (6)]2 = −4(2)2
= −4(4) = −16
28. b2 + c 2 = ( )2 + ( )2 = (−3)2 + (−2)2
= 9 + 4 = 13
(b) −4( r − s)2 = −4[( ) − ( )]2
29. P = 2l + 2 w = 2( ) + 2( )
= 2(6 in.) + 2(2.3 in.)
= 12 in. + 4.6 in. = 16.6 in.
2
= −4[(3) − (−1)]
= −4(3 + 1)2 = −4(4)2
= −4(16) = −64
3 
30. A = lw = ( )( ) =  ft  (4 ft) = 6 ft 2
2 
20. (a) −5(u + v )2 = −5[( ) + ( )]2
= −5[(10) + (−7)]2 = −5(3)2
= −5(9) = −45
31. A = πr 2 = ( )( )2 =  22   7 m 
 7   2 
(b) −5(u + v )2 = −5[( ) + ( )]2
11
2
7
 22   49 2  77 2
=
m =
m

 7  4
 2
= −5[(0) + (−2)]2
= −5(−2)2 = −5(4) = −20
1
21. y(x − 4) = ( )[( ) − 4]
 2
2
=   [(−2) − 4] = (−6) = −4
3
 3
2
32. A = 1 bh = 1 ( )( )
2
2
2
2
 4
14
  10
yd  =
yd 2
=  yd  
25
11
11


22. w(− x − 4) = ( )[−( ) − 4]
 1
=  −  [−(−2) − 4]
 2
1
1
33. 5 + w = w + 5
1
1
= − (2 − 4) = − (−2) = 1
2
2
34. t + 2 = 2 + t
23. z 2 − x + 6 = ( )2 − ( ) + 6 = (4)2 − (−2) + 6
= 16 + 2 + 6 = 24
24. x 3 − w −
Properties of Real Numbers
1
1
 1
35. − + b = b +  −  or b −
3
3
 3
3
3
= ( )3 − ( ) −
2
2


1
3
1 3
= (−2)3 −  −  − = −8 + −
2 2
 2 2
1
1
 1
36. − + c = c +  −  or c −
2
2
2


37. r(2) = 2r
16 1 3 −18
=− + − =
= −9
2 2 2
2
38. a(−4) = −4a
39. t(−s) = −st
25. bc ÷ a = ( )( ) ÷ ( ) = (−3)(−2) ÷ (12)
1
= 6 ÷ 12 =
2
40. d(−c) = −cd
353
Chapter 11
Solving Equations
41. xy = yx
62. −7( p − 4) = −7( p ) + (−7)(−4)
= −7 p + 28
42. ab = ba
63. −5(−2 + x ) = −5(−2) + (−5)( x )
= 10 + −5x = 10 − 5x
( )
43. 7 − p = 7 + (−p) = −p + 7
44. 8 − q = 8 + (−q) = −q + 8
64. −8(−3+ y ) = −8(−3) + (−8)( y )
= 24 + −8 y = 24 − 8 y
( )
45. −2(6b) = (−2 ⋅ 6)b = −12b
46. −3(2c) = (−3 ⋅ 2)c = −6c
65. 4( a + 4b − c) = 4[ a + 4b + (− c)]
= 4( a ) + 4(4b) + 4(− c)
= 4 a + 16b + (−4c)
= 4 a + 16b − 4c
47. 3 + (8 + t) = (3 + 8) + t = 11 + t
48. 7 + (5 + p) = (7 + 5) + p = 12 + p
49. −4.2 + (2.5 + r ) = (−4.2 + 2.5 ) + r
= −1.7 + r
66. 2(3q − r + s) = 2[3q + (−r ) + s]
= 2(3q) + 2(−r ) + 2( s )
= 6q + (−2r ) + 2s
= 6q − 2r + 2 s
50. 1.1+ (−0.8 + w) = [1.1+ (−0.8)] + w
= 0.3 + w
8
2

2
67. 4  + g  = 4   + 4( g ) = + 4 g
3
3

3
51. 3(6x) = (3 ⋅ 6)x = 18x
52. 9(5k) = (9 ⋅ 5)k = 45k
 5
5

40
68. 8  + m = 8   + 8( m) =
+ 8m
6
 6
6

20
=
+ 8m
3
4  7   4  7 
53. −  − d  =  − ⋅  −   d = 1d = d
7  4   7  4 
54.
56  5 6
 m  =  ⋅  m = 1m = m
65  6 5
69. −(3 − n) = −[3 + (− n)] = −1[3 + (− n)]
= −1(3) + (−1)(− n) = −3 + n
55. −9 + (−12 + h ) = (−9 + (−12 ))+ h
= −21 + h
70. −(13 − t ) = −[13 + (− t )] = −1[13 + (− t )]
= −1(13) + (−1)(− t ) = −13 + t
56. −11+ (−4 + s ) = (−11+ (−4 ))+ s = −15 + s
71. −(− a − 8) = −[− a + (−8)] = −1[− a + (−8)]
= −1(− a ) + (−1)(−8) = a + 8
57. 4(x + 8) = 4(x) + 4(8) = 4x + 32
58. 5(3 + w) = 5(3) + 5(w) = 15 + 5w
72. −(− d − 10) = −[− d + (−10)]
= −1[− d + (−10)]
= −1(− d ) + (−1)(−10) = d + 10
59. −2( p + 4) = −2( p ) + (−2)(4)
= −2 p + (−8) = −2 p − 8
73. −(3x + 9 − 5 y )
= −[3x + 9 + (−5 y )]
= −1[3x + 9 + (−5 y )]
= −1(3x ) + (−1)(9) + (−1)(−5 y )
= −3x + (−9) + 5 y = −3x − 9 + 5 y
60. −6( k + 2) = −6( k ) + (−6)(2)
= −6 k + (−12) = −6 k − 12
61. −10(t − 3) = −10(t) + (−10)(−3)
= −10t + 30
354
Section 11.1
74. −( a − 8b + 4c)
= −[ a + (−8b) + 4c]
= −1[ a + (−8b) + 4c]
= −1( a ) + (−1)(−8b) + (−1)(4c)
= − a + 8b + (−4c) = − a + 8b − 4c
86. 3+ (25 − m) = 3 + [25 + (− m)]
= (3+ 25) + (− m)
= 28 + (− m) = 28 − m
87. −8(4 − p ) = −8[4 + (− p )]
= −8(4) + (−8)(− p ) = −32 + 8 p
75. −(−5q − 2s − 3t )
= −[−5q + (−2s) + (−3t )]
= −1[−5q + (−2s) + (−3t )]
= −1(−5q ) + (−1)(−2s) + (−1)(−3t )
= 5q + 2s + 3t
88. 3(25 − m) = 3[25 + (− m)] = 3(25) + 3(− m)
= 75 + (−3m) = 75 − 3m
 1   1
89. 8  a  =  8⋅  a = 4 a
 2   2
76. −(−10 p − 12q + 3)
= −[−10 p + (−12q ) + 3]
= −1[−10 p + (−12q ) + 3]
= −1(−10 p ) + (−1)(−12q ) + (−1)(3)
= 10 p + 12q + (−3) = 10 p + 12q − 3
1  
1
90. −20  b =  −20 ⋅  b = −4b
5
5  
1

 1
91. 8  + a  = 8   + 8 a = 4 + 8a
2

 2
()
77. 6(2x) = (6 ⋅ 2)x = 12x
78. −3(12k) = (−3 ⋅ 12)k = −36k
1 
 1
92. −20  + b = −20   + −20 b
5 
 5
= −4 + −20b = −4 − 20b
( )( )
( )
79. 6(2 + x) = 6(2) + 6(x) = 12 + 6x
80. −3(12 + k ) = −3(12) + (−3)( k )
= −36 + (−3k ) = −36 − 3k
(
93.
( ( ))
)
81. −6 −1− k = −6 −1+ − k
( ) ( )( )
= −6 −1 + −6 − k = 6 + 6k
(
)
( ) ( )( )
( )
= 32 + −4 h = 32 − 4 h
( ( ))
= (−6 + (−1))+ (− k )
)
83. −6 + −1− k = −6 + −1+ − k
95.
( )
= −7 + − k = −7 − k
(
) (
5
5
5
5
(9 + y ) = (9) + ( y ) = 5 + y
9
9
9
9
3
3
94. − (8 − b) = − [8 + (− b)]
4
4
 3
3
= − (8) +  −  (− b)
4
 4
3
= −6 + b
4
82. −4 −8 + h = −4 −8 + −4 h
(
Properties of Real Numbers
( ))
5
5 
(9 y ) =  ⋅ 9  y = 5 y
9
9 
3
 3 
96. − (8b) =  − ⋅ 8  b = −6b
4
 4 
84. −4 + −8 + h = −4 + −8 + h = −12 + h
85. −8 + (4 − p ) = −8 + [4 + (− p )]
= (−8 + 4) + (− p )
= −4 + (− p ) = −4 − p
355
Chapter 11
Solving Equations
Section 11.2
Simplifying Expressions
Section 11.2 Practice Exercises
14. variable term
1. (a) term
15. variable term
(b) variable; constant
16. constant term
(c) coefficient
17. 6, −4
(d) like
18. −5, −2
2. (a) 6( a + 3b) = 6a + 18b
= 6(4) + 18(−5)
= 24 − 90
= −66
19. −14, 12
20. 8, 9
(b) 6a + 18b
= 6(4) + 18(−5)
= 24 − 90
= −66
21. 1, −1
22. 1, −1
23. 5, −8, −3
3. 6(p + 3) = 6(p) + 6(3) = 6p + 18
4.
24. 6, −16, −2
(−7 p + 2 ) + 10 = −7 p + (2 + 10 )
25. Like terms
= −7 p + 12
5. 4(−6q) = [4 ⋅ (−6)]q = −24q
26. Like terms
6. −3(t − 2) = −3[t + (−2)] = −3(t ) + (−3)(−2)
= −3t + 6
27. Unlike terms
28. Unlike terms
7. 13 + (−4 − h) = 13+ [−4 + (− h)]
= [13 + (−4)] + (− h)
= 9 + (− h) = 9 − h
29. Like terms
30. Like terms
31. Unlike terms
8. −( x − 20 y − 14 z )
= −[ x + (−20 y ) + (−14 z )]
= −1[ x + (−20 y ) + (−14 z )]
= −1( x ) + (−1)(−20 y ) + (−1)(−14 z )
= − x + 20 y + 14 z
32. Unlike terms
33. Unlike terms
34. Unlike terms
9. variable term
35. Like terms
10. constant term
36. Like terms
11. constant term
37. 6rs + 8rs = (6 + 8)rs = 14rs
12. variable term
38. 4x + 21x = (4 + 21)x = 25x
13. variable term
39. −4h + 12h = (−4 + 12)h = 8h
356
Section 11.2
40. 9p − 13p = (9 − 13)p = −4p
57.
41. 4 x 2 + 9 − x 2 = 4 x 2 − x 2 + 9
= (4 − 1) x 2 + 9 = 3x 2 + 9
42. 13t 2 − t 2 + 4 = (13 − 1)t 2 + 4 = 12t 2 + 4
43. 10 x − 12 y − 4 x − 3 y = 10 x − 4 x − 12 y − 3 y
= 6 x − 15 y
58.
44. 14a − 5b + 3a − b = 14a + 3a − 5b − b
= 17a − 6b
45. −6k − 9k + 12k = −3k
1
2
1
2
b−4+ b−4= b+ b−4−4
3
9
3
9
3
2
= b+ b−4−4
9
9
5
= b −8
9
= 2.3x 2 − 5.3x 2 + 4.1x − 6 x
47. −8uv + 6u + 12uv = −8uv + 12uv + 6u
= 4uv + 6u
= −3x 2 − 1.9 x
60. 1.2 y − 0.4 y 2 − 0.3 y − 1.5 y 2
48. 9 pq − 9 p + 13 pq = 9 pq + 13 pq − 9 p
= 22 pq − 9 p
= −0.4 y 2 − 1.5 y 2 + 1.2 y − 0.3 y
= −1.9 y 2 + 0.9 y
49. 6 − 14m − 15 − 2m = −14m − 2m + 6 − 15
= −16m − 9
61. 4.4 − 0.9a + 3.2 = −0.9a + 4.4 + 3.2
= −0.9a + 7.6
50. 1− 8n + 5 − 3n = −8n − 3n + 1+ 5 = −11n + 6
62. 9.7 − 8.8b − 3.2 = −8.8b + 9.7 − 3.2
= −8.8b + 6.5
51. 18 − 3a + 5b − 6 a + 2
= −3a − 6 a + 5b + 18 + 2
= −9 a + 5b + 20
63. 5(t − 6) + 2 = 5[t + (−6)] + 2
= 5(t ) + 5(−6) + 2
= 5t + (−30) + 2
= 5t + (−28)
= 5t − 28
52. 13 + w − 5 z − 4 + 7 w = w + 7 w − 5 z + 13 − 4
= 8w − 5 z + 9
53. −5 p 2 + 6 p − p 2 + 7 − 8 p
= −5 p 2 − p 2 + 6 p − 8 p + 7
64. 7(a − 4) + 8 = 7[a + (−4)] + 8
= 7(a) + 7(−4) + 8
= 7a + (−28) + 8
= 7a + (−20)
= 7a − 20
= −6 p 2 − 2 p + 7
54. −3q 2 − 10q + q 2 − 15 + 5q
= −3q 2 + q 2 − 10q + 5q − 15
65. −3(2 x + 1) − 13 = −3(2 x) + (−3)(1) − 13
= −6 x + (−3) − 13
= −6 x + (−16)
= −6 x − 16
= −2q 2 − 5q − 15
1
3
5 4
5
5
y + y − = y − = 2y −
2
2
6 2
6
6
56. −
3
1
3
1
a +3− a + 6 = a − a +3+ 6
4
8
4
8
6
1
= a − a +3+ 6
8
8
5
= a+9
8
59. 2.3x 2 + 4.1x − 5.3x 2 − 6 x
46. −11p + 23p − p = 11p
55.
Simplifying Expressions
4
2
4
2
4
p+ p+ =− p+
5
5
7
5
7
357
Chapter 11
Solving Equations
66. −2(4b + 3) − 10 = −2(4b) + (−2)(3) − 10
= −8b + ( −6) − 10
= −8b + ( −16)
= −8b − 16
73. −2( a + 3b) − (4 a − 5b)
= −2( a + 3b) + (−1)[4 a + (−5b)]
= −2( a ) + (−2)(3b) + (−1)(4 a ) + (−1)(−5b)
= −2 a + (−6b) + (−4 a ) + (5b)
= −2 a + (−4 a ) + (−6b) + (5b)
= −6 a + (− b)
= −6 a − b
67. 4 + 6( y − 3) = 4 + 6[ y + (−3)]
= 4 + 6( y ) + 6(−3)
= 4 + 6 y + ( −18)
= 6 y + 4 + ( −18)
= 6 y + (−14)
= 6 y − 14
74. −(2m − 7n) − 3(6m − n)
= −1[2m + (−7n)]+ (−3)[6m + (−n)]
= −1(2m) + (−1)(−7n) + (−3)(6m) + (−3)(−n)
= −2m + 7n + (−18m) + 3n
= −2m + (−18m) + 7n + 3n
= −20m +10n
68. 11 + 2( p − 8) = 11 + 2[ p + (−8)]
= 11 + 2( p) + 2(−8)
= 11 + 2 p + (−16)
= 2 p + 11 + (−16)
= 2 p + (−5)
= 2p −5
75. 10( x + 5) − 3(2 x + 9)
= 10( x + 5) + (−3)(2 x + 9)
= 10( x ) + 10(5) + (−3)(2 x ) + (−3)(9)
= 10 x + 50 + (−6 x ) + (−27)
= 10 x + (−6 x ) + 50 + (−27)
= 4 x + 23
69. 21 − 7(3 − q) = 21 + (−7)[3 + (− q)]
= 21 + (−7)(3) + (−7)( −q )
= 21 + (−21) + (7 q )
= 0 + 7q
= 7q
76. 6( y − 9) − 5(2 y − 5)
= 6[ y + (−9)] + (−5)[2 y + (−5)]
= 6( y ) + 6(−9) + (−5)(2 y ) + (−5)(−5)
= 6 y + (−54) + (−10 y ) + 25
= 6 y + (−10 y ) + (−54) + 25
= −4 y + (−29)
= −4 y − 29
70. 10 − 5(2 − 5m) = 10 + (−5)[2 + (−5m)]
= 10 + (−5)(2) + (−5)(−5m)
= 10 + (−10) + 25m
= 0 + 25m
= 25m
77. −(12z + 1) + 2(7z − 5)
= −1(12z + 1) + 2[7z + (−5)]
= −1(12z) + (−1)(1) + 2(7z) + 2(−5)
= −12z + (−1) + 14z + (−10)
= −12z + 14z + (−1) + (−10)
= 2z + (−11)
= 2z − 11
71. −3 − (2n + 1) = −3 + (−1)(2n + 1)
= −3 + (−1)(2n) + (−1)(1)
= −3 + (−2n) + (−1)
= −2n + (−3) + (−1)
= −2n + (−4)
= −2n − 4
72. −13 − (6s + 5) = −13 + (−1)(6 s + 5)
= −13 + (−1)(6 s) + (−1)(5)
= −13 + (−6 s) + (−5)
= −6s + (−13) + (−5)
= −6s + (−18)
= −6s − 18
78. −(8w + 5) + 3(w − 15)
= −1(8w + 5) + 3[w + (−15)]
= −1(8w) + (−1)(5) + 3(w) + 3(−15)
= −8w + (−5) + 3w + (−45)
= −8w + 3w + (−5) + (−45)
= −5w + (−50)
= −5w − 50
358
Section 11.2
Simplifying Expressions
79. 3( w + 3) − (4 w + y ) − 3 y
= 3( w + 3) + (−1)(4 w + y ) + (−3 y )
= 3( w) + 3(3) + (−1)(4 w) + (−1)( y ) + (−3 y )
= 3w + 9 + (−4 w) + (− y ) + (−3 y )
= 3w + (−4 w) + (− y ) + (−3 y ) + 9
= − w + (−4 y ) + 9
= −w − 4 y + 9
85. 15+2(w−4)−(2w−5z)+7z
=15+2[w+(−4)]+(−1)[2w+(−5z)]+7z
=15+2(w)+2(−4)+(−1)(2w)+(−1)(−5z)+7z
=15+2w+(−8)+(−2w)+5z +7z
= 2w+(−2w)+5z +7z +15+(−8)
= 0+12z +7
=12z +7
80. 2( s + 6) − (8s − t ) + 6t
= 2( s + 6) + (−1)[8s + (−t )] + 6t
= 2( s) + 2(6) + (−1)(8s) + (−1)(−t ) + 6t
= 2s + 12 + (−8s) + t + 6t
= 2s + (−8s) + t + 6t + 12
= −6s + 7t + 12
86. 7+3(2a−5)−(6a−8b)−2b
=7+3[2a+(−5)]+(−1)[6a+(−8b)]+(−2b)
=7+3(2a)+3(−5)+(−1)(6a)+(−1)(−8b)+(−2b)
=7+6a+(−15)+(−6a)+8b+(−2b)
=6a+(−6a)+8b+(−2b)+7+(−15)
=0+6b+(−8)
=6b−8
81. 20a − 4(b + 3a) − 5b
= 20a + (−4)(b + 3a) + (−5b)
= 20a + (−4)(b) + (−4)(3a) + (−5b)
= 20a + (−4b) + (−12a) + (−5b)
= 20a + (−12a) + (−4b) + (−5b)
= 8a + (−9b)
= 8a − 9b
1
2  5
3
87. 6  x −  − 4  x + 
3  2
4
2
 1  2 
5
3
= 6  x +  −   + (−4)  x + 
4
2
 2  3 
 1   2
5 
 3
= 6  x + 6  −  + (−4)  x + (−4)  
 2   3
2 
 4
82. 16 p − 3(2 p − q ) + 7 q
= 16 p + (−3)[2 p + (− q )] + 7 q
= 16 p + (−3)(2 p ) + (−3)(− q ) + 7 q
= 16 p + (−6 p ) + (3q ) + 7 q
= 10 p + 10q
= 3x + (−4) + (−10x) + (−3)
= 3x + (−10x) + (−4) + (−3)
= −7 x + (−7)
= −7 x − 7
5
1  2
1
88. −12  p +  + 9  p − 
4  9
3
6

5
 1 
1
2
= −12  p +  + 9  p +  −  
4  9
6
 3 
5 
 1   2   1
= −12  p + (−12)   + 9  p + 9  − 
6 
 4   9   3
83. 6−(3m−n)−2(m+8)+5n
=6+(−1)[3m+(−n)]+(−2)(m+8)+5n
=6+(−1)(3m)+(−1)(−n)+(−2)(m)+(−2)(8)+5n
=6+(−3m)+n+(−2m)+(−16)+5n
= −3m+(−2m)+n+5n+6+(−16)
= −5m+6n+(−10)
= −5m+6n−10
= −10 p + (−3) + 2 p + (−3)
= −10 p + 2 p + (−3) + (−3)
= −8 p + (−6)
= −8 p − 6
84. 12−(5u + v)−4(u −6)+2v
=12+(−1)(5u + v)+(−4)[u +(−6)]+2v
=12+(−1)(5u)+(−1)(v)+(−4)(u)+(−4)(−6)+2v
=12+(−5u)+(−v)+(−4u)+24+2v
=−5u +(−4u)+(−v)+2v +12+24
=−9u + v +36
359
Chapter 11
89.
Solving Equations
92. 100(0.14b+0.2)−10(1.3b−4)
=100(0.14b+0.2)+(−10)[1.3b+(−4)]
=100(0.14b)+100(0.2)+(−10)(1.3b)+(−10)(−4)
=14b+20+(−13b)+40
=14b+(−13b)+20+40
=b+60
3
2
(9y + 6) − (18y −16)
2
3
 3
2
= (9y + 6) +  −  [18y + (−16)]
3
 2
 3
 3
2
2
= (9y) + (6) +  −  (18y) +  −  (−16)
3
3
 2
 2
= 6y + 4 + (−27y) + 24
= 6y + (−27y) + 4 + 24
= −21y + 28
93. 100(1.04a − 2.1b) − 10(21.1a + 0.3b)
= 100[1.04a + (−2.1b)] + (−10)(21.1a + 0.3b)
= 100(1.04a) + 100(−2.1b) + (−10)(21.1a)
+(−10)(0.3b)
= 104a + (−210b) + (−211a) + (−3b)
= 104a + (−211a) + (−210b) + (−3b)
= −107a + (−213b)
= −107a − 213b
1
1
90. − (4w − 8) + (4w + 10)
4
2
1
1
= − [4w + (−8)] + (4w + 10)
2
4
 1
1
1
1
= − (4w) +  −  (−8) + (4w) + (10)
2
2
4
 4
= −w + 2 + 2w + 5
= −w + 2w + 2 + 5
= w+7
94. 10(−7.2x − y) + 1000(0.023x + 0.004 y)
= 10[−7.2x + (− y)] + 1000(0.023x + 0.004 y)
= 10(−7.2x) + 10(− y) + 1000(0.023x)
+1000(0.004 y)
= −72x + (−10 y) + 23x + 4 y
= −72x + 23x + (−10 y) + 4 y
= −49x + (−6 y)
= −49x − 6 y
91. 10(0.2q−3)−100(0.04q−0.5)
=10[0.2q+(−3)]+(−100)[0.04q+(−0.5)]
=10(0.2q)+10(−3)+(−100)(0.04q)+(−100)(−0.5)
=2q+(−30)+(−4q)+50
=2q+(−4q)+(−30)+50
=−2q+20
Section 11.3
Addition and Subtraction Properties of Equality
Section 11.3 Practice Exercises
3. −10 a + 3b − 3a + 13b
= −10 a − 3a + 3b + 13b
= −13a + 16b
1. (a) linear
(b) solution
4. 4 − 23 y + 11 − 16 y = −23 y − 16 y + 4 + 11
= −39 y + 15
(c) equivalent
(d) addition
5. −(−8h + 2 k − 13)
= −1[−8h + 2 k + (−13)]
= −1(−8h) + (−1)(2 k ) + (−1)(−13)
= 8h + (−2 k ) + 13
= 8h − 2 k + 13
(e) subtraction
2. The exponents on the variable x are
different. Therefore, the terms are not like
terms and cannot be combined.
360
Section 11.3
Addition and Subtraction Properties of Equality
6. 3( −4m + 3) − 12 = 3( −4m ) + 3(3) − 12
= −12m + 9 − 12
= −12m − 3
?
−7 − (−3) = − 10
?
−7 + 3 = − 10
− 4 ≠ −10
−3 is not a solution.
7. 5 z − 8( z − 3) − 20
= 5 z + (−8)[ z + (−3)] + (−20)
= 5 z + (−8)( z ) + (−8)(−3) + (−20)
= 5 z + (−8 z ) + 24 + (−20)
= −3z + 4
15.
8. −(7 p − 12) − 10(1 − p ) + 6
= −1[7 p + ( −12)] + (−10)[1 + (− p )] + 6
= −1(7 p ) + (−1)(−12) + ( −10)(1)
+ (−10)( − p ) + 6
= −7 p + 12 + (−10) + 10 p + 6
= −7 p + 10 p + 12 + (−10) + 6
= 3p + 8
9.
−7 − w = −10
14.
6m − 3 = −6
?
 1
6 −  − 3 = − 6
 2
?
−3−3 = −6
−6 = −6
−
16.
5 x + 3 = −2
?
5(−1) + 3 = − 2
1
is a solution.
2
−12n + 2 = −1
?
1
−12   + 2 = − 1
4
?
− 3 + 2 = −1
−1 = −1
?
−5+3 = −2
−2 = −2
−1 is a solution.
10.
1
is a solution.
4
3y − 2 = 4
17. 13 = 13 + 6t
?
?
3(2) − 2 = 4
13 = 13 + 6(0)
?
?
6−2 = 4
4=4
2 is a solution.
13 = 13 + 0
13 = 13
0 is a solution.
11. 10 = p − 16
1
1
18. − = r −
5
5
1 ?
1
− = 0−
5
5
1
1
− =−
5
5
0 is a solution.
?
10 = 26 − 16
10 = 10
26 is a solution.
12. −14 = q − 1
?
−14 = − 13 − 1
−14 = −14
−13 is a solution.
13.
19.
25 = −5q − 5
?
25 = − 5(4) − 5
− z + 8 = 20
?
25 = − 20 − 5
25 ≠ −25
4 is not a solution.
?
−12 + 8 = 20
−4 ≠ 20
12 is not a solution.
361
Chapter 11
20.
Solving Equations
39 = −7 p − 4
33.
?
39 = − 7(5) − 4
?
39 = − 35 − 4
39 ≠ −39
5 is not a solution.
21. 13 + (−13) = 0
22. 6 + (−6) = 0
34.
23. 7 + (−7) = 0
24. 1 + (−1) = 0
25. 3.2 + (−3.2) = 0
26. 0.3 + (−0.3) = 0
5
1
− + p=
6
3
5 5
2 5
− + + p= +
6 6
6 6
7
0+ p =
6
7
p=
6
3
3
− +q=
4
2
3 3
6 3
− + +q= +
4 4
4 4
9
0+q =
4
9
q=
4
27.
g − 23 = 14
g − 23 + 23 = 14 + 23
g + 0 = 37
g = 37
35.
k − 4.3 = −1.2
k − 4.3 + 4.3 = −1.2 + 4.3
k + 0 = 3.1
k = 3.1
28.
h − 12 = 30
h − 12 + 12 = 30 + 12
h + 0 = 42
h = 42
36.
a − 0.04 = −2.04
a − 0.04 + 0.04 = −2.04 + 0.04
a + 0 = −2
a = −2
29.
−4 + k = 12
−4 + 4 + k = 12 + 4
0 + k = 16
k = 16
37.
13 = −21+ w
13+ 21 = −21+ 21+ w
34 = 0 + w
34 = w
30.
−16 + m = 4
−16 + 16 + m = 4 + 16
0 + m = 20
m = 20
38.
2 = −17 + w
2 + 17 = −17 + 17 + w
19 = 0 + w
19 = w
31.
−18 = n − 3
−18 + 3 = n − 3 + 3
−15 = n + 0
−15 = n
39. 52 − 52 = 0
32.
40. 2 − 2 = 0
41. 18 − 18 = 0
−9 = t − 6
−9 + 6 = t − 6 + 6
−3 = t + 0
−3 = t
42. 15 − 15 = 0
43. 100 − 100 = 0
44. 21 − 21 = 0
362
Section 11.3
45.
x + 34 = 6
x + 34 − 34 = 6 − 34
x + 0 = −28
x = −28
46.
y + 12 = 4
y + 12 − 12 = 4 − 12
y + 0 = −8
y = −8
47.
17 + b = 20
17 − 17 + b = 20 − 17
0+b =3
b=3
48.
5 + c = 14
5 − 5 + c = 14 − 5
0+c =9
c=9
49.
−32 = t + 14
−32 − 14 = t + 14 − 14
−46 = t + 0
−46 = t
50.
−23 = k + 11
−23 − 11 = k + 11 − 11
−34 = k + 0
−34 = k
51.
8.2 = 21.8 + m
8.2 − 21.8 = 21.8 − 21.8 + m
−13.6 = 0 + m
−13.6 = m
52.
16.01 = 20.88 + n
16.01− 20.88 = 20.88 − 20.88 + n
−4.87 = 0 + n
−4.87 = n
53.
Addition and Subtraction Properties of Equality
54.
3
7
=−
5
10
3 3
7 3
a+ − =− −
5 5
10 5
7 6
a+0=− −
10 10
13
a=−
10
55.
21 = 14 + w
21− 14 = 14 − 14 + w
7 = 0+ w
7=w
56.
9 = 8+u
9−8 = 8−8+u
1= 0 + u
1= u
57.
1+ p = 0
1 −1 + p = 0 −1
0 + p = −1
p = −1
58.
r − 12 = 13
r − 12 + 12 = 13 + 12
r + 0 = 25
r = 25
59.
−34 + t = −40
−34 + 34 + t = −40 + 34
0 + t = −6
t = −6
60.
7+q=4
7−7+q =4−7
0 + q = −3
q = −3
61.
a+
363
1
3
=−
4
8
1 1
3 1
b+ − =− −
4 4
8 4
3 2
b+0=− −
8 8
5
b=−
8
b+
2
=
3
2 5
+ =
3 12
8 5
+ =
12 12
13
=
12
5
12
5 5
y− +
12 12
y−
y+0
y
Chapter 11
62.
63.
Solving Equations
7
3
=z+
11
11
7 3
3 3
− =z+ −
11 11
11 11
4
= z+0
11
4
=z
11
−2.5 = −1.1 + m
−2.5 + 1.1 = −1.1 + 1.1 + m
−1.4 = 0 + m
−1.4 = m
64.
−4.1 = −3.5 + n
−4.1 + 3.5 = −3.5 + 3.5 + n
−0.6 = 0 + n
−0.6 = n
65.
w − 23 = −11
w − 23 + 23 = −11 + 23
w + 0 = 12
w = 12
71.
4.01 + p = 3.22
4.01 − 4.01 + p = 3.22 − 4.01
0 + p = −0.79
p = −0.79
72.
2.8 + q = 6.1
2.8 − 2.8 + q = 6.1 − 2.8
0 + q = 3.3
q = 3.3
73.
74.
3
=2
8
3 3
3
t + − =2−
8 8
8
16 3
t+0= −
8 8
13
5
or 1
t=
8
8
t+
4
= −1
7
4 4
4
r − + = −1 +
7 7
7
7 4
r+0=− +
7 7
3
r=−
7
r−
66.
p − 10 = −9
p − 10 + 10 = −9 + 10
p + 0 =1
p =1
67.
x + 21 = 16
x + 21 − 21 = 16 − 21
x + 0 = −5
x = −5
75.
27 = z − 22
27 + 22 = z − 22 + 22
49 = z + 0
49 = z
68.
y + 18 = −4
y + 18 − 18 = −4 − 18
y + 0 = −22
76.
109 = x + 49
109 − 49 = x + 49 − 49
60 = x + 0
60 = x
69.
−2 = a − 15
−2 + 15 = a − 15 + 15
13 = a + 0
13 = a
77.
−70 = −55 + w
−70 + 55 = −55 + 55 + w
−15 = 0 + w
−15 = w
70.
−1 = b − 49
−1 + 49 = b − 49 + 49
48 = b + 0
48 = b
78. 5h − 4h + 4 = 3
h+4=3
h+ 4−4 =3−4
h + 0 = −1
h = −1
364
Section 11.3
Addition and Subtraction Properties of Equality
79. 10 x − 9 x − 11 = 15
x − 11 = 15
x − 11 + 11 = 15 + 11
x + 0 = 26
x = 26
80. 9 + (−2) = 4 + t
7 =4+t
7−4=4−4+t
3=0+t
3=t
81. −13 + 15 =
2=
2−5=
−3 =
−3 =
82.
3(r − 2) − 2r = 6 + (−2)
3[r + (−2)] − 2r = 6 + (−2)
3(r ) + 3(−2) − 2r = 6 + (−2)
3r + (−6) − 2r = 6 + (−2)
3r − 2r + (−6) = 6 + (−2)
r + (−6) = 4
r + ( −6) + 6 = 4 + 6
r + 0 = 10
r = 10
83.
4(k + 2) − 3k = −6 + 9
4(k ) + 4(2) − 3k = −6 + 9
4k + 8 − 3k = −6 + 9
4k − 3k + 8 = −6 + 9
k +8=3
k +8−8 = 3−8
k + 0 = −5
k = −5
p+5
p+5
p +5−5
p+0
p
Section 11.4
Multiplication and Division Properties of Equality
Section 11.4 Practice Exercises
1. (a) multiplication
(b) division
2 x + 3 = 13
2(5) + 3 = 13?
2. (a)
; Yes
10 + 3 = 13?
13 = 13
2 x = 10
(b) 2(5) = 10? ; Yes
10 = 10
3.
p − 12 = 33
p − 12 + 12 = 33 + 12
p + 0 = 45
p = 45
4.
−8 = 10 + k
−8 − 10 = 10 − 10 + k
−18 = 0 + k
−18 = k
5.
6.
−4 + w = 22
−4 + 4 + w = 22 + 4
0 + w = 26
w = 26
7.
p − 6 = −19
p − 6 + 6 = −19 + 6
p + 0 = −13
p = −13
8.
9.
16 = h − 5
16 + 5 = h − 5 + 5
21 = h + 0
21 = h
365
11
1
=− +m
6
6
11 11
1 11
+ =− + +m
6 6
6 6
12
= 0+m
6
2=m
1
2
=−
2
3
1 1
2 1
n+ − =− −
2 2
3 2
4 3
n+0=− −
6 6
7
n=−
6
n+
Chapter 11
10.
Solving Equations
2.4 + z = −12
2.4 − 2.4 + z = −12 − 2.4
0 + z = −14.4
z = −14.4
24.
1
11. 3 ⋅ = 1
3
25.
 1
12. −6 ⋅  −  = 1
 6
4  7
13. − ⋅  −  = 1
7  4
14.
26.
3 10
⋅ =1
10 3
15. −7 ÷ (−7) = 1
27.
16. 2 ÷ 2 = 1
−h = −17
−1h = −17
−1h −17
=
−1
−1
h = 17
2
m = 14
3
3 2
3  14 
⋅ m=  
2 3
2 1 
m = 21
5
n = 40
9
9 5
9  40 
⋅ n=  
5 9
5 1 
n = 72
b
= −3
7
1
b = −3
7
1 
7  b  = 7(−3)
7 
b = −21
17. 5.1 ÷ 5.1 = 1
18. −6.8 ÷ (−6.8) = 1
19. 14b = −42
14b −42
=
14
14
b = −3
28.
a
= −12
4
1
a = −12
4
1 
4  a  = 4( −12)
4 
a = −48
20. −6 p = 12
−6 p 12
=
−6 −6
p = −2
21. −8k = 56
−8k 56
=
−8 −8
k = −7
29. −2.8 = −0.7t
−2.8 −0.7t
=
−0.7 −0.7
4=t
22. 5 y = −25
5 y −25
=
5
5
y = −5
30. −3.3 = −3r
−3.3 −3r
=
−3
−3
1.1 = r
23.
−t = −13
−1t = −13
−1t −13
=
−1
−1
t = 13
366
Section 11.4
31.
−
Multiplication and Division Properties of Equality
u
= −15
2
38.
1
− u = −15
2
 1 
−2  − u  = −2(−15)
 2 
u = 30
39.
v
32.
− = −4
10
1
− v = −4
10
 1 
−10  − v  = −10(−4)
 10 
v = 40
33.
34.
40.
6 = −18w
6
−18w
=
−18 −18
1
− =w
3
41.
4 = −32 g
−32 g
4
=
−32 −32
1
− =g
8
42.
35. 1.3 x = 5.33
1.3x 5.33
=
1.3
1.3
x = 4.1
36. 8.1y = 17.82
8.1y 17.82
=
8.1
8.1
y = 2.2
37.
11
1
h=−
12
6
12  11 
12  1 
− − h = − − 
11  12 
11  6 
2
h=
11
−
3
0= m
8
8
83 
(0) =  m 
3
3 8 
0=m
1
n
10
1 
10(0) = 10  n 
 10 
0=n
0=
9
3
− x=−
4
5
4 9 
4 3
− − x = − − 
9 4 
9 5
4
x=
15
15
1
y=
14
2
14  15 
14  1 
− − y = −  
15  14 
15  2 
7
y=−
15
−
43. 100 = 5k
100 5k
=
5
5
20 = k
5
1
k =−
4
2
4 5  4 1
 k = − 
5 4  5 2
2
k =−
5
44. 95 = 19h
95 19h
=
19 19
5=h
45.
367
31 = − p
31 −1 p
=
−1 −1
−31 = p
Chapter 11
Solving Equations
46. −6 = − z
−6 − z
=
−1 −1
6=z
5
47.
3p =
2
1
1 5
(3 p ) =  
3
3 2 
5
p=
6
48.
55.
q − 4 = −12
q − 4 + 4 = −12 + 4
q = −8
56.
p − 6 = −18
p − 6 + 6 = −18 + 6
p = −12
57.
1
h = −12
4
1 
4  h  = 4(−12)
4 
h = −48
7
5
1
17
(2q ) =  
2
2 5
7
q=
10
2q =
58.
50. −7b = 0
−7b 0
=
−7 −7
b=0
4 + x = −12
4 − 4 + x = −12 − 4
x = −16
52.
6 + z = −18
6 − 6 + z = −18 − 6
z = −24
w
= −18
6
1
w = −18
6
1 
6  w  = 6(−18)
6 
w = −108
49. −4a = 0
−4a 0
=
−4 −4
a=0
51.
h
= −12
4
59.
60.
53. 4 y = −12
4 y −12
=
4
4
y = −3
54. 6 p = −18
6 p −18
=
6
6
p = −3
2
+ t =1
3
2 2
2
− + t =1−
3 3
3
3 2
t= −
3 3
1
t=
3
3
+ q =1
4
3 3
3
− + q =1−
4 4
4
4 3
q= −
4 4
1
q=
4
61. −9a = −12
−9a −12
=
−9
−9
4
a=
3
368
Section 11.4
Multiplication and Division Properties of Equality
62. −8b = −44
−8b −44
=
−8
−8
11
b=
2
63.
69.
7 = r − 23
7 + 23 = r − 23 + 23
30 = r
64.
11 = s − 4
11 + 4 = s − 4 + 4
15 = s
65.
−
70.
y
=5
3
t − 12.9 = 15
t − 12.9 + 12.9 = 15 + 12.9
t = 27.9
72.
c − 4.11 = 1.2
c − 4.11 + 4.11 = 1.2 + 4.11
c = 5.31
73.
5 + u = 3.2
5 − 5 + u = 3.2 − 5
u = −1.8
74.
3 + v = 1.7
3 − 3 + v = 1.7 − 3
v = −1.3
5
6
1
15
(2 p ) =  
2
26
5
p=
12
75.
50 = a + 72
50 − 72 = a + 72 − 72
−22 = a
76.
23 = w + 41
23 − 41 = w + 41− 41
−18 = w
3
5
1
13
(4q ) =  
4
45
3
q=
20
77.
−1 = b − 16
−1+ 16 = b − 16 + 16
15 = b
78.
−5 = y − 8
−5 + 8 = y − 8 + 8
3= y
−
h
=1
5
1
− h =1
5
1


−5  − h  = −5(1)
5


h = −5
67.
68.
2
4
y=
11
15
11  2 
11  4 
− − y = −  
2  11 
2  15 
22
y=−
15
−
71.
1
− y =5
3
 1 
−3  − y  = −3(5)
 3 
y = −15
66.
3
9
− x=
7
10
7 3 
7 9 
− − x = −  
3 7 
3  10 
21
x=−
10
2p =
4q =
369
Chapter 11
79.
Solving Equations
−12 = 30x
1
1
−12 =
30x
30
30
2
− =x
5
85. 3 p + 4 p = 25 − 4
7 p = 21
7 p 21
=
7
7
p=3
−10 = 12h
1
1
−10 =
12h
12
12
5
− =h
6
86. 2q + 3q = 54 − 9
5q = 45
5q 45
=
5
5
q=9
1
−6 = − q
2
 1 
−2 −6 = −2  − q 
 2 
87.
−2(a + 3) − 6a + 6 = 8
−2(a) + (−2)(3) + (−6a ) + 6 = 8
−2a + (−6) + (−6a ) + 6 = 8
−2a + (−6a) + (−6) + 6 = 8
−8a + 0 = 8
−8a 8
=
−8 −8
a = −1
88.
−(b − 11) − 3b − 11 = −16
−1[b + (−11)] + (−3b) + (−11) = −16
−1(b) + ( −1)(−11) + (−3b) + (−11) = −16
−b + 11 + (−3b) + (−11) = −16
−b + (−3b) + 11 + (−11) = −16
−4b + 0 = −16
−4b −16
=
−4
−4
b=4
( )
80.
( )
81.
( )
( )
( )
12 = q
82.
1
4=− k
6
 1 
−6 4 = −6  − k 
 6 
()
−24 = k
83. 5 x − 2 x = −15
3 x = −15
3 x −15
=
3
3
x = −5
84. 13 y − 10 y = −18
3 y = −18
3 y −18
=
3
3
y = −6
370
Section 11.5
Section 11.5
Solving Equations with Multiple Steps
Solving Equations with Multiple Steps
Section 11.5 Practice Exercises
1. Add 6 to both sides first.
2.
3.
4.
5.
6.
1
3
1
1 1
(4c) =  − 
4
4 3
1
c=−
12
8.
5+q =0
5−5+ q = 0−5
q = −5
9.
3m + 2 = 14
3m + 2 − 2 = 14 − 2
3m = 12
3m 12
=
3
3
m=4
4c = −
1
b = −4
3
1 
3  b  = 3(−4)
3 
b = −12
1
6
− +t =
5
5
1 1
6 1
− + +t = +
5 5
5 5
7
t=
5
3
1
− = w+
8
4
3 1
1 1
− − = w+ −
8 4
4 4
3 2
− − =w
8 8
5
− =w
8
7
10
7
−1 p = −
10
7
−
−1 p
= 10
−1
−1
7
p=
10
−p = −
7. −8h = 0
−8h 0
=
−8 −8
h=0
371
10.
−2n + 5 = −15
−2n + 5 − 5 = −15 − 5
−2n = −20
−2n −20
=
−2
−2
n = 10
11.
−8c − 12 = 36
−8c − 12 + 12 = 36 + 12
−8c = 48
−8c 48
=
−8 −8
c = −6
12.
5t − 1 = −11
5t − 1 + 1 = −11 + 1
5t = −10
5t −10
=
5
5
t = −2
13.
1 = −4 z + 21
1 − 21 = −4 z + 21 − 21
−20 = −4 z
−20 −4 z
=
−4
−4
5= z
14.
−4 = −3 p + 14
−4 − 14 = −3 p + 14 − 14
−18 = −3 p
−18 −3 p
=
−3
−3
6= p
Chapter 11
15.
16.
Solving Equations
9 = 12 x − 7
9 + 7 = 12 x − 7 + 7
16 = 12 x
16 12 x
=
12 12
4
=x
3
3.4 − 2d
3.4 − 3.4 − 2d
−2d
−2d
−2
d
18.
2.9 − 4 g = 23.3
2.9 − 2.9 − 4 g = 23.3 − 2.9
−4 g = 20.4
−4 g 20.4
=
−4
−4
g = −5.1
20.
b
− 12 = −9
3
b
− 12 + 12 = −9 + 12
3
b
=3
3
1
b=3
3
1 
3  b  = 3(3)
3 
b=9
−7 = 5 y − 8
−7 + 8 = 5 y − 8 + 8
1 = 5y
1 5y
=
5 5
1
=y
5
17.
19.
21.
22.
c
+2=4
5
c
+2−2=4−2
5
c
=2
5
1
c=2
5
1 
5  c  = 5(2)
5 
c = 10
= 8.2
= 8.2 − 3.4
= 4.8
4.8
=
−2
= −2.4
23.
−0.57 = 15h + 16.23
−0.57 − 16.23 = 15h + 16.23 − 16.23
−16.8 = 15h
−16.8 15h
=
15
15
−1.12 = h
24.
1.9 = 8k + 4.06
1.9 − 4.06 = 8k + 4.06 − 4.06
−2.16 = 8k
−2.16 8k
=
8
8
−0.27 = k
372
w
−3
2
w
−9 + 3 = − 3 + 3
2
w
−6 =
2
1
−6 = w
2
1 
2(−6) = 2  w 
2 
−12 = w
−9 =
t
− 14
4
t
−16 + 14 = − 14 + 14
4
t
−2 =
4
1
−2 = t
4
1 
4(−2) = 4  t 
4 
−8 = t
−16 =
Section 11.5
25.
26.
27.
5
4
5 1
−
4 2
5 2
3x = −
4 4
3
3x =
4
1
1  3
(3x) =  
3
3 4
1
x=
4
1
=
2
1 1
3x + − =
2 2
3x +
3 9
=
8 16
3 3 9 3
9z − + = +
8 8 16 8
9 6
9z = +
16 16
15
9z =
16
1
1  15 
(9 z ) =  
9
9  16 
5
z=
48
( )( )
28.
( )
25 − c = −3
25 − 25 − c = −3− 25
−c = −28
−1 −c = −1 −28
c = 28
( )
29.
30.
2 w + 10 = 5w − 5
2 w − 5w + 10 = 5w − 5w − 5
−3w + 10 = −5
−3w + 10 − 10 = −5 − 10
−3w = −15
−3w −15
=
−3
−3
w=5
31.
7 − 5t = 3t − 2
7 − 5t − 3t = 3t − 3t − 2
7 − 8t = −2
7 − 7 − 8t = −2 − 7
−8t = −9
−8t −9
=
−8 −8
9
t=
8
32.
4 − 2p = 8+ 5p
4 − 2p − 5p = 8 + 5p − 5p
4−7p =8
4−4−7p =8−4
−7 p = 4
−7 p 4
=
−7
−7
4
p=−
7
33.
4 − 3d
4 − 3d − 5d
4 − 8d
4 − 4 − 8d
−8d
−8d
−8
d
34.
−3k + 14 = −4 + 3k
−3k − 3k + 14 = −4 + 3k − 3k
−6k + 14 = −4
−6k + 14 − 14 = −4 − 14
−6k = −18
−6k −18
=
−6
−6
k =3
9z −
10 − y = 37
10 − 10 − y = 37 − 10
− y = 27
−1 − y = −1 27
y = −27
( )
8 + 4b = 2 + 2b
8 + 4 b − 2b = 2 + 2b − 2 b
8 + 2b = 2
8 − 8 + 2b = 2 − 8
2b = −6
2b −6
=
2
2
b = −3
373
Solving Equations with Multiple Steps
= 5d − 4
= 5d − 5d − 4
= −4
= −4 − 4
= −8
−8
=
−8
=1
Chapter 11
Solving Equations
35.
12 p = 3 p + 21
12 p − 3 p = 3 p − 3 p + 21
9 p = 21
9 p 21
=
9
9
7
p=
3
36.
2 x + 10 = 4 x
2 x − 2 x + 10 = 4 x − 2 x
10 = 2 x
10 2 x
=
2
2
5= x
37.
− z − 2 = −2 z
− z + z − 2 = −2 z + z
−2 = − z
−2 −1z
=
−1 −1
2=z
38.
39.
40.
41.
9 y = − y + 25
9 y + y = − y + y + 25
10 y = 25
10 y 25
=
10 10
5
y=
2
4 2
5 1
+ q= − − q−4
3 3
3 3
1
5 12
4 2
+ q=− q− −
3
3 3
3 3
4 2
1
17
+ q=− q−
3 3
3
3
4 2
1
1
1
17
+ q+ q= − q+ q−
3 3
3
3
3
3
4
17
+q=−
3
3
4 4
17 4
− +q=− −
3 3
3 3
21
q=−
3
q = −7
4 + 2a − 7 = 3a + a + 3
4 − 7 + 2a = 3a + a + 3
−3 + 2a = 4a + 3
−3 + 2a − 4a = 4a − 4a + 3
−3 − 2a = 3
−3 + 3 − 2a = 3 + 3
−2a = 6
−2a 6
=
−2 −2
a = −3
42. 4b + 2b − 7 = 2 + 4b + 5
4b + 2b − 7 = 4b + 2 + 5
6b − 7 = 4b + 7
6b − 4b − 7 = 4b − 4b + 7
2b − 7 = 7
2b − 7 + 7 = 7 + 7
2b = 14
2b 14
=
2
2
b=7
1
3
p = 2+ p
4
4
1
1
3
1
1+ p − p = 2 + p − p
4
4
4
4
1
1= 2+ p
2
1
1− 2 = 2 − 2 + p
2
1
−1 = p
2
1 
2 −1 = 2  p 
2 
−2 = p
1+
43. −8w + 8 + 3w = 2 − 6w + 2
−8w + 3w + 8 = −6w + 2 + 2
−5w + 8 = −6w + 4
−5w + 6w + 8 = −6w + 6 w + 4
w+8= 4
w+8−8 = 4−8
w = −4
( )
374
Section 11.5
Solving Equations with Multiple Steps
49. 9q − 5(q − 3) = 5q
9q − 5q + 15 = 5q
4q + 15 = 5q
4q − 4q + 15 = 5q − 4q
15 = q
44. −12 + 5m + 10 = −2m − 10 − m
5m − 12 + 10 = −2m − m − 10
5m − 2 = −3m − 10
5m + 3m − 2 = −3m + 3m − 10
8m − 2 = −10
8m − 2 + 2 = −10 + 2
8m = −8
8m −8
=
8
8
m = −1
50. 6h − 2(h + 6) = 10h
6h − 2h − 12 = 10h
4h − 12 = 10h
4h − 4h − 12 = 10h − 4h
−12 = 6h
12 6h
− =
6
6
−2 = h
45. 6 y + 2 y − 2 = 14 + 3 y − 12
6 y + 2 y − 2 = 3 y + 14 − 12
8y − 2 = 3y + 2
8y − 3y − 2 = 3y − 3y + 2
5y − 2 = 2
5y − 2 + 2 = 2 + 2
5y = 4
5y 4
=
5 5
4
y=
5
46. −7t − 20 + 7 = −7 − 3t
−7t − 13 = −7 − 3t
−7t + 3t − 13 = −7 − 3t + 3t
−4t − 13 = −7
−4t − 13 + 13 = −7 + 13
−4t = 6
−4t 6
=
−4 −4
3
t=−
2
51.
2(1 − m) = 5 − 3m
2 − 2m = 5 − 3m
2 − 2m + 3m = 5 − 3m + 3m
2+m=5
2−2+m =5−2
m=3
52.
3(2 − g ) = 12 − g
6 − 3 g = 12 − g
6 − 3 g + g = 12 − g + g
6 − 2 g = 12
6 − 6 − 2 g = 12 − 6
−2 g = 6
−2 g 6
=
−2 −2
g = −3
53. −4(k − 2) + 14 = 3k − 20
−4k + 8 + 14 = 3k − 20
−4k + 22 = 3k − 20
−4k − 3k + 22 = 3k − 3k − 20
−7 k + 22 = −20
−7 k + 22 − 22 = −20 − 22
−7 k = −42
−7 k −42
=
−7
−7
k =6
47. 3n − 4(n − 1) = 16
3n − 4n + 4 = 16
− n + 4 = 16
−n + 4 − 4 = 16 − 4
−n = 12
−1n 12
=
−1 −1
n = −12
48. 4 p − 3( p + 2) = 18
4 p − 3 p − 6 = 18
p − 6 = 18
p − 6 + 6 = 18 + 6
p = 24
375
Chapter 11
Solving Equations
54. −3( x + 4) − 9 = −2 x + 12
−3 x − 12 − 9 = −2 x + 12
−3x − 21 = −2 x + 12
−3x + 3x − 21 = −2 x + 3 x + 12
−21 = x + 12
−21 − 12 = x + 12 − 12
−33 = x
55.
56.
58. −3t − 3(t − 4) = 2 − (2t − 1)
−3t − 3t + 12 = 2 − 2t + 1
−3t − 3t + 12 = 2 + 1 − 2t
−6t + 12 = 3 − 2t
−6t + 2t + 12 = 3 − 2t + 2t
−4t + 12 = 3
−4t + 12 − 12 = 3 − 12
−4t = −9
−4t −9
=
−4 −4
9
t=
4
3 z − 9 = 3(5 z − 1)
3 z − 9 = 15 z − 3
3 z − 3 z − 9 = 15 z − 3 z − 3
−9 = 12 z − 3
−9 + 3 = 12 z − 3 + 3
−6 = 12 z
−6 12 z
=
12 12
1
− =z
2
59. 6(u − 1) + 5u + 1 = 5(u + 6) − u
6u − 6 + 5u + 1 = 5u + 30 − u
6u + 5u − 6 + 1 = 5u − u + 30
11u − 5 = 4u + 30
11u − 4u − 5 = 4u − 4u + 30
7u − 5 = 30
7u − 5 + 5 = 30 + 5
7u = 35
7u 35
=
7
7
u =5
4 y − 9 = 8( y − 2)
4 y − 9 = 8 y − 16
4 y − 8 y − 9 = 8 y − 8 y − 16
−4 y − 9 = −16
−4 y − 9 + 9 = −16 + 9
−4 y = −7
−4 y −7
=
−4 −4
7
y=
4
60. 2(2v + 3) + 8v = 6(v − 1) + 3v
4v + 6 + 8v = 6v − 6 + 3v
4v + 8v + 6 = 6v + 3v − 6
12v + 6 = 9v − 6
12v − 9v + 6 = 9v − 9v − 6
3v + 6 = −6
3v + 6 − 6 = −6 − 6
3v = −12
3v −12
=
3
3
v = −4
57. 6 w + 2( w − 1) = 14 − (3w + 1)
6 w + 2 w − 2 = 14 − 3w − 1
6 w + 2 w − 2 = 14 − 1 − 3w
8w − 2 = 13 − 3w
8w + 3w − 2 = 13 − 3w + 3w
11w − 2 = 13
11w − 2 + 2 = 13 + 2
11w = 15
11w 15
=
11 11
15
w=
11
376
Problem Recognition Exercises: Equations versus Expressions
Problem Recognition Exercises: Equations versus Expressions
1. Equation
14. equation;
2. Expression
3. Expression
4. Equation
5. Equation
6. Expression
7. equation;
5t = 20
1
1
(5t ) = (20 )
5
5
t=4
8. equation;
6x − 2 = 36
6x − 2 + 2 = 36 + 2
6x = 38
1
1
(6x ) = (38 )
6
6
19
x=
3
2
1
(9 x − 5) +
9
9
10 1
15. expression; 2 x − 9 + 9
9
2x −
9
2x − 1
5 − 3(2t + 7)
16. expression; 5 − 6t − 21
−6t − 16
17. equation;
5 x − 3 = 20
5 x − 3 + 3 = 20 + 3
5 x = 23
1
1
(5 x ) = (23)
5
5
23
x=
5
18. equation;
6 x = 36
1
1
(6 x ) = (36 )
6
6
x=6
9. expression; 4( x − 5) + 12
4 x − 20 + 12
4x − 8
10. expression; 16 − 2k + 2 + k
− k + 18
11. equation;
5 + t = 20
5 − 5 + t = 20 − 5
t = 15
12. equation;
0 = 7 y − 3y + 8
0 = 4y + 8
0 − 8 = 4y + 8 − 8
−8 = 4 y
−8 4 y
=
4
4
−2 = y
13. equation;
s
+3
4
s
11 − 3 = + 3 − 3
4
s
8=
4
 s
4 (8 ) = 4  
 4
32 = s
11 =
19. equation;
5 (t − 3) = 20
5t − 15 = 20
5t − 15 + 15 = 20 + 15
5t = 35
1
1
(5t ) = (35 )
5
5
t=7
377
r
− 12
6
r
−14 + 12 = − 12 + 12
6
r
−2 =
6
 r
6 (−2 ) = 6  
 6
−12 = r
−14 =
Chapter 11
Solving Equations
16 − 2 k + 2 = 0
−2 k + 16 + 2 = 0
−2 k + 18 = 0
−2 k + 18 − 18 = 0 − 18
−2 k = −18
−2 k −18
=
−2
−2
k=9
21. expression; 4 − 2(4 x + 5)
4 − 8 x − 10
−8 x − 6
1
6
22. expression; ( y + 21) + y
7
7
1
6
y +3+ y
7
7
7
y+3
7
y+3
27. equation;
23. equation;
2.3u + 0.2 = −1.2u + 7.2
2.3u + 1.2u + 0.2 = −1.2u + 1.2u + 7.2
3.5u + 0.2 = 7.2
3.5u + 0.2 − 0.2 = 7.2 − 0.2
3.5u = 7
7
3.5u
=
3.5 3.5
u=2
24. equation:
6 + x = 36
6 − 6 + x = 36 − 6
x = 30
25. equation; 5 + 3 p − 2 = 0
3p + 5 − 2 = 0
3p + 3 = 0
3p + 3 − 3 = 0 − 3
3 p = −3
3 p −3
=
3
3
p = −1
29. expression;
20. equation;
26. equation;
0 = 2x + 5x + 1
0 = 7x + 1
0 − 1 = 7x + 1 − 1
−1 = 7x
−1 7x
=
7
7
1
− =x
7
28. equation;
7.5w − 2.7 = 1.4w + 15.6
7.5w − 1.4w − 2.7 = 1.4w − 1.4w + 15.6
6.1w − 2.7 = 15.6
6.1w − 2.7 + 2.7 = 15.6 + 2.7
6.1w = 18.3
6.1w 18.3
=
6.1
6.1
w=3
5+ 7p − 2p +9
5 p + 14
3( y + 2) − 21
30. expression; 3 y + 6 − 21
3 y − 15
31. equation;
32. equation;
7 3
1
b+
=
10 5
5
7
7
6
7
1
b+ −
=
−
10 10 10 10
5
1
1
b=−
10
5
 1
1 
5  b = 5  − 
 10 
5 
1
b=−
2
378
2
1
2
p− = −
3
6
3
2
1 1
2 1
− p− + = − +
3
6 6
3 6
2
4 1
− p=− +
3
6 6
2
3
− p=−
3
6
 3  2   3  1 
 −   − p  =  −   − 
3
2
2
2
3
p=
4
−
6 (x − 2 ) = 36
6x − 12 = 36
6x − 12 + 12 = 36 + 12
6x = 48
1
1
(6x ) = (48 )
6
6
x=8
Section 11.6
Section 11.6
Applications and Problem Solving
Applications and Problem Solving
Section 11.6 Practice Exercises
1.
4 x + 1 = 11
4(3) + 1 = 11?
; no
12 + 1 = 11?
13 ≠ 11
2.
3t − 15 = −22
3t − 15 + 15 = −22 + 15
3t = −7
3t −7
=
3
3
7
t=−
3
3.
6. −( y − 9) + 5( y + 3) = −2(3 y + 7)
− y + 9 + 5 y + 15 = −6 y − 14
− y + 5 y + 9 + 15 = −6 y − 14
4 y + 24 = −6 y − 14
4 y + 6 y + 24 = −6 y + 6 y − 14
10 y + 24 = −14
10 y + 24 − 24 = −14 − 24
10 y = −38
10 y −38
=
10
10
19
y=−
5
b
− 5 = −14
5
7.
b
− 5 + 5 = −14 + 5
5
1
b = −9
5
1 
5  b  = 5(−9)
5 
b = −45
4.
8.
2 x + 22 = 6 x − 2
2 x − 6 x + 22 = 6 x − 6 x − 2
−4 x + 22 = −2
−4 x + 22 − 22 = −2 − 22
−4 x = −24
−4 x −24
=
−4
−4
x=6
5. 4(r + 4) − 12 = 18 − r
4r + 16 − 12 = 18 − r
4r + 4 = 18 − r
4r + r + 4 = 18 − r + r
5r + 4 = 18
5r + 4 − 4 = 18 − 4
5r = 14
5r 14
=
5
5
14
r=
5
379
4.4 p − 2.6 = 1.2 p − 5
4.4 p − 1.2 p − 2.6 = 1.2 p − 1.2 p − 5
3.2 p − 2.6 = −5
3.2 p − 2.6 + 2.6 = −5 + 2.6
3.2 p = −2.4
3.2 p −2.4
=
3.2
3.2
p = −0.75
2
1 1
5
w− = w+
3
6 3
6
2
1
1 1
1
5
w− w− = w− w+
3
3
6 3
3
6
1
1 5
w− =
3
6 6
1
1 1 5 1
w− + = +
3
6 6 6 6
1
w =1
3
1 
3  w  = 3(1)
3 
w=3
Chapter 11
Solving Equations
9. (a) Let x represent the number. The
quotient of a number and 3 is −8.
x
= −8
3
(b)
14. (a) Let x represent the number. Sixty is −5
times a number.
60 = −5x
(b)
x
= −8
3
 x
3   = 3(−8)
3
x = −24
The number is −24.
15. (a) Let x represent the number. Five less
than the quotient of a number and 4 is
equal to −12.
x
− 5 = −12
4
10. (a) Let x represent the number. The
difference of −2 and a number is −14.
−2 − x = −14
(b)
−2 − x = −14
−2 + 2 − x = −14 + 2
− x = −12
−1(− x) = −1(−12)
x = 12
The number is 12.
(b)
−30 − x = 42
−30 + 30 − x = 42 + 30
− x = 72
−1( − x) = −1(72)
x = −72
The number is −72.
16. (a) Let x represent the number. Eight
decreased by the product of a number
and 3 is equal to 5.
8 − 3x = 5
(b)
12. (a) Let x represent the number. A number
increased by 13 results in −100.
x + 13 = −100
(b)
x + 13 = −100
x + 13 − 13 = −100 − 13
x = −113
The number is −113.
8 − 3x = 5
8 − 8 − 3x = 5 − 8
−3x = −3
−3 x −3
=
−3 −3
x =1
The number is 1.
17. (a) Let x represent the number. One-half
increased by a number is 4.
1
+x=4
2
13. (a) Let x represent the number. A total of
30 and a number is 13.
30 + x = 13
(b)
x
− 5 = −12
4
x
− 5 + 5 = −12 + 5
4
1
x = −7
4
1 
4  x  = 4( −7)
4 
x = −28
The number is −28.
11. (a) Let x represent the number. A number
subtracted from −30 results in 42.
−30 − x = 42
(b)
60 = −5 x
60 −5 x
=
−5 −5
−12 = x
The number is −12.
30 + x = 13
30 − 30 + x = 13 − 30
x = −17
The number is −17.
380
Section 11.6
(b)
−3.
x − 16 = −3x
1
+x=4
2
1 1
1
− + x =4−
2 2
2
8 1
x= −
2 2
7
1
x = or 3
2
2
7
1
The number is
or 3 .
2
2
(b)
(b)
5
− x =1
3
5 5
5
− − x =1−
3 3
3
3 5
−x = −
3 3
2
−x = −
3
 2
−1(− x) = −1 − 
 3
2
x=
3
2
The number is .
3
10( x + 5.1) = 56
10 x + 51 = 56
10 x + 51 − 51 = 56 − 51
10 x = 5
10 x 5
=
10 10
x = 0.5
The number is 0.5.
22. (a) Let x represent the number. Three
times the difference of a number and 5
is 15.
3(x − 5) = 15
(b)
19. (a) Let x represent the number. The
product of −12 and a number is the
same as the sum of the number and
26.
−12x = x + 26
(b)
x − 16 = −3x
x − x − 16 = −3x − x
−16 = −4 x
−16 −4 x
=
−4
−4
4=x
The number is 4.
21. (a) Let x represent the number. Ten times
the total of a number and 5.1 is 56.
10(x + 5.1) = 56
18. (a) Let x represent the number. Fivethirds decreased by a number is 1.
5
− x =1
3
(b)
Applications and Problem Solving
−12 x = x + 26
−12 x − x = x − x + 26
−13 x = 26
−13 x 26
=
−13 −13
x = −2
The number is −2.
3( x − 5) = 15
3x − 15 = 5
3x − 15 + 15 = 15 + 15
3 x = 30
3x 30
=
3
3
x = 10
The number is 10.
23. (a) Let x represent the number. The
product of 3 and a number is the same
as 10 less than twice the number.
3x = 2x − 10
(b)
20. (a) Let x represent the number. The
difference of a number and 16 is the
same as the product of the number and
381
3 x = 2 x − 10
3 x − 2 x = 2 x − 2 x − 10
x = −10
The number is −10.
Chapter 11
Solving Equations
24. (a) Let x represent the number. Six less
than a number is the same as 3 more
than twice the number.
x − 6 = 2x + 3
(b)
 Total 
 Boyz II Men   Metallica  
+
=

  hits   number 
hits
 of hits 
x + x − 6 = 26
2x − 6 = 26
2x − 6 + 6 = 26 + 6
2x = 32
2x 32
=
2
2
x = 16
x − 6 = 16 − 6 = 10
Metallica had 10 hits. while Boyz II Men
had 16 hits.
x − 6 = 2x + 3
x − x − 6 = 2x − x + 3
−6 = x + 3
−6 − 3 = x + 3 − 3
−9 = x
The number is −9.
25. Let x represent the length of the shorter
piece. Then 3x represents the length of the
other piece.
 Total 
 Length of   Length of  
 one piece  +  other piece  =  length 
 of wire
x + 3x = 12
4x = 12
4x 12
=
4
4
x=3
3x = 3 3 = 9
28. Let x represent the cost of the small bag.
Then x + 40 represents the cost of the
large bag.
 Cost of   Cost of   Cost of 
 small bag  +  large bag  =  set 
x + x + 40 = 150
2x + 40 = 150
2x + 40 − 40 = 150 − 40
2x = 110
2x 110
=
2
2
x = 55
x + 40 = 55 + 40 = 95
The large bag costs $95 and the small bag
costs $55.
()
The pieces are 3 m and 9 m long.
26. Let x represent the length of the shorter
piece. Then 2x represents the length of the
other piece.
 Total 
 Length of   Length of  
 one piece  +  other piece  =  length 
 of strip
x + 2x = 9
3x = 9
3x 9
=
3 3
x=3
2x = 2 3 = 6
29. Let w represent the width. Then w + 30
represents the length.
P = 2l + 2w
460 = 2(w + 30) + 2w
460 = 2w + 60 + 2w
460 = 4w + 60
460 − 60 = 4w + 60 − 60
400 = 4w
400 4w
=
4
4
100 = w
x + 30 = 100 + 30 = 130
The soccer field is 100 yd by 130 yd.
()
The pieces are 3 ft and 6 ft long.
27. Let x represent the number of hits for Boyz
II Men. Then x − 6 represents the number
of hits for Metallica.
382
Section 11.6
30. Let l represent the length. Then
1
l
2
Applications and Problem Solving
 Total 
 Length of   Length of  

+
=
 one piece   other piece   length 

 
 of ribbon


x + 2x = 4
3x = 4
3x 4
=
3 3
1
x =1
3
1
The pieces are 1 ft and
3
2
 1
4 8
2 1  = 2   = = 2 ft long.
3
3
3
3
 
 
represents the width.
P = 2l + 2 w
1 
150 = 2l + 2  l 
2 
150 = 2l + l
150 = 3l
150 3l
=
3
3
50 = l
The length is 50 m, and the width is
1
(50 m) = 25 m.
2
34. Let x represent the length of the shorter
section. Then 3x represents the length of
the other section.
 Total 
 Length of   Length of  

+
=
 one section   other section   length 

 
 of pipe


x + 3x = 6
4x = 6
4x 6
=
4 4
3
x=
2
3
1
The sections should be = 1 m and
2
2
1
3 9
3   = = 4 m long.
2
2 2
31. Let x represent the number of minutes
over 500. Then 0.25x represents the cost
for
x minutes over 500.
 Cost of 
 Monthly  
  Total 
 fee  +  additional  =  cost 




 minutes 
49.95 + 0.25 x = 62.45
49.95 − 49.95 + 0.25 x = 62.45 − 49.95
0.25 x = 12.50
0.25 x 12.50
=
0.25
0.25
x = 50
Jim used 50 min over the 500 min.
32. Let x represent the number of miles over
200. Then 0.30x represents the cost for
x miles over 200.
 cost of 
 Daily  
  Total 
+
 charge   additional  =  cost 




 miles 
19 + 0.30 x = 62.50
19 − 19 + 0.30 x = 62.50 − 19
0.30 x = 43.50
0.30 x 43.50
=
0.30
0.30
x = 145
Mr. Cain traveled 145 mi over 200 mi.
35. Let x represent the number of points for
Oakland. Then 2x + 6 represents the
number of points for Tampa Bay.
 Total 
 Oakland's   Tampa Bay's  

+
=
 points   points   points 

 
 scored


x + 2 x + 6 = 69
3 x + 6 = 69
3 x + 6 − 6 = 69 − 6
3 x = 63
3 x 63
=
3
3
x = 21
Oakland scored 21 points and Tampa Bay
scored 2(21) + 6 = 42 + 6 = 48 points.
33. Let x represent the length of the shorter
piece. Then 2x represents the length of the
other piece.
383
Chapter 11
Solving Equations
36. Let x represent the number of points for
Kansas City. Then 3x + 5 represents the
number of points for Green Bay.
 Total 
 Kansas City's   Green Bay's  

+
=
 points   points   points 

 
 scored


x + 3x + 5 = 45
4x + 5 = 45
4x + 5 − 5 = 45 − 5
4x = 40
4x 40
=
4
4
x = 10
Kansas City scored 10 points and Green
Bay scored 3(10) + 5 = 30 + 5 = 35 points.
 paycheck 
(Salary) + (overtime pay) = 
 amount 
480 + 18 x = 588
480 − 480 + 18 x = 588 − 480
18 x = 108
18 x 108
=
18
18
x=6
Stefan worked 6 hr of overtime.
40. Let x represent the overtime pay per hour.
Then 8x represents the overtime earnings.
 paycheck 
(salary) + (overtime pay) = 

 amount 
480 + 8 x = 672
480 − 480 + 8 x = 672 − 480
8 x = 192
8 x 192
=
8
8
x = 24
Victoria makes $24 an hour for overtime.
37. Let x represent the rent. Then x − 350
represents the security deposit.
 security  1st month's 
(rent) + 
=

 deposit   payment 
x + x − 350 = 950
2 x − 350 = 950
2 x − 350 + 350 = 950 + 350
2 x = 1300
2 x 1300
=
2
2
x = 650
Charlene’s rent is $650 a month with a
security deposit of $650 − $350 = $300.
41. Let x represent the hours for the fall. Then
x + 4 represents the hours for the spring.
 fall   spring   total 
 hours  +  hours  =  hours 

 
 

x + x + 4 = 28
2 x + 4 = 28
2 x + 4 − 4 = 28 − 4
2 x = 24
2 x 24
=
2
2
x = 12
Raul took 12 hr in the fall and signed up
for 12 + 4 = 16 hr in the spring.
38. Let x represent the cost of the pants. Then
x − 20 represents the cost of the shirt.
 Cost for   Cost for   Total 
 pants  +  shirt  =  paid 

 
 

x + x − 20 = 71.90
2 x − 20 = 71.90
2 x − 20 + 20 = 71.90 + 20
2 x = 91.90
2 x 91.90
=
2
2
x = 45.95
The pants cost $45.95 and the shirt cost
$45.95 − $20 = $25.95.
39. Let x represent the number of hours of
overtime. Then 18x represents the
earnings for working overtime.
384
Section 11.6
42. Let x represent the price of the monitor.
Then x + 241 represents the price of the
computer.
 Price of   Price of   Total 
 monitor  +  computer  =  cost 

 
 

x + x + 241 = 899
2 x + 241 = 899
2 x + 241 − 241 = 899 − 241
2 x = 658
2 x 658
=
2
2
x = 329
The monitor costs $329 and the computer
costs $329 + $241 = $570.
Ann-Marie must sell $1250 of
merchandise each week.
44.
43. Let x represent the amount of merchandise
to sell. Then 0.24x represents the
commission earned.
 Earnings from 
 commission  = (Salary)


0.24 x = 300
0.24 x 300
=
0.24 0.24
x = 1250
Chapter 11
Applications and Problem Solving
Let x represent the cost per minute for
local calls. Then x + 0.05 represents the
cost per minute for long-distance calls.
 cost per 
 cost per   total value


35 min for +52 min for  =  of calling 
 local calls
 long distance  card 



 

35(x)+52(x +0.05)=20
35x +52x +2.6=20
87x +2.6=20
87x +2.6−2.6=20−2.6
87x =17.4
87x 17.4
=
87 87
x =0.20
Local calls are $0.20 per minute, and
long-distance calls are $0.20 + $0.05 =
$0.25 per minute.
Review Exercises
Section 11.1
1. (a) a + 8
2
(b) a + 8 = ( ) + 8 = 35 + 8 = 43 years old
2
2
2
 4 
 4 
16
4
2. (a) −   = −   = − 
 =−
9
 x
 (3) 
( )
2
2
2
2
2
2
 4
 4 
 4 
(b) −   = −   = − 
 x
 ( )
 (4) 
16
= − = −1
16
 4
 4 
 4 
(c) −   = −   = − 
 x
 ( )
 (2) 
16
= − = −4
4
2
 4
 4 
 4 
(d) −   = −   = − 
 x
 ( )
 (−2) 
16
= − = −4
4
2
3. −2(x + y)2 = −2[( ) + ( )]2
= −2[(6) + (−9)]2 = −2(−3)2
= −2(9) = −18
385
Chapter 11
Solving Equations
4
4  22   3 
4. V = πr 3 ≈    m 
3
3 7   2 
1
9
1
2
18. −5(p + 4) + 6(p + 1) − 2
= −5p + (−5)(4) + 6p + 6(1) − 2
= −5p + (−20) + 6p + 6 − 2
= −5p + 6p + (−20) + 6 − 2
= p + (−16) = p − 16
3
4  22   27 3  198 3
=  
m =
m
3  7  8
 14
=
Section 11.3
99 3
m
7
19.
5 x + 10 = −5
?
5(−3) + 10 = − 5
5. (a) t − 5 = t + (−5) = −5 + t
?
− 15 + 10 = − 5
−5 = −5
−3 is a solution.
(b) h ⋅ 3 = 3h
6. (a) −4(2 ⋅ p) = (−4 ⋅ 2)p = −8p
20.
(b) ( m + 10) − 12 = m + (10 − 12)
= m + (−2) = m − 2
−3( x − 1) = −9 + x
?
−3[(−3) − 1] = − 9 + ( −3)
?
7. 3(2b + 5) = 3(2b) + 3(5) = 6b + 15
−3( −4) = − 9 + (−3)
12 ≠ −12
−3 is not a solution.
8. −(−4k + 8m − 12)
= −1[−4k + 8m + (−12)]
= −1(−4k ) + (−1)(8m) + (−1)(−12)
= 4k + (−8m) + 12 = 4k − 8m + 12
21. If a constant is being added to the variable
term, use the subtraction property. If a
constant is being subtracted from a
variable term, use the addition property.
Section 11.2
9. 3, −5, 12
22.
r + 23 = −12
r + 23 − 23 = −12 − 23
r = −35
23.
k − 3 = −15
k − 3 + 3 = −15 + 3
k = −12
24.
10 = p − 4
10 + 4 = p − 4 + 4
14 = p
25.
21 = q − 3
21 + 3 = q − 3 + 3
24 = q
26.
4.1 + m = 5.2
4.1 − 4.1 + m = 5.2 − 4.1
m = 1.1
10. −6, −1, 2, 1
11. Unlike terms
12. Like terms
13. Like terms
14. Unlike terms
15. 6y + 8x − 2y − 2x + 10
= 8x − 2x + 6y − 2y + 10
= 6x + 4y + 10
16. 12a − 5 + 9b − 5a + 14
= 12a − 5a + 9b − 5 + 14
= 7 a + 9b + 9
17. 4(u − 3v) − 5u + v = 4u − 4(3v) − 5u + v
= 4u − 12v − 5u + v
= 4u − 5u − 12v + v
= −u − 11v
386
Chapter 11
27.
28.
29.
30.
−3.1 + n = 1.9
−3.1 + 3.1 + n = 1.9 + 3.1
n=5
33.
2 5
=
3 6
2 2 5 2
a− + = +
3 3 6 3
5 4
a= +
6 6
9
a=
6
3
a=
2
a−
34.
Review Exercises
t
= −13
2
1
t = −13
2
1 
2  t  = 2(−13)
2 
t = −26
p
=7
5
1
p=7
5
1 
5  p  = 5(7)
5 
p = 35
1
9
=−
5
10
1 1
9 1
b+ − =− −
5 5
10 5
9 2
b=− −
10 10
11
b=−
10
b+
35.
36.
3
+h=2
4
3 3
3
− +h = 2−
4 4
4
8 3
h= −
4 4
5
1
h = or 1
4
4
37.
Section 11.4
4
− y = −16
5
5 4 
5  16 
− − y = − − 
4 5 
4 1 
y = 20
2
x = −14
3
3  2  3  14 
 x = − 
2 3  2 1 
x = −21
1.4 = −0.7 m
1.4 −0.7 m
=
−0.7
−0.7
−2 = m
38. −3.6 = 0.9n
−3.6 0.9n
=
0.9
0.9
−4 = n
31. 4d = −28
4d −28
=
4
4
d = −7
39.
32. −3c = −12
−3c −12
=
−3
−3
c=4
387
1
3
w=
3
7
1 
3
3 w  = 3 
3 
7
9
w=
7
Chapter 11
40.
Solving Equations
1
2
− s=
4
3
 1 
2
−4  − s  = −4  
 4 
3
8
s=−
3
47.
41. −42 = −7 p
−42 −7 p
=
−7
−7
6= p
42.
51 = −3b
51 −3b
=
−3 −3
−17 = b
43.
9 x + 7 = −2
9 x + 7 − 7 = −2 − 7
9 x = −9
9 x −9
=
9
9
x = −1
44.
45.
46.
48.
3
4= m−2
5
3
4+2= m−2+2
5
3
6= m
5
5 6 5 3 
  =  m
3 1  3 5 
10 = m
5
p − 1 = 14
8
5
p − 1 + 1 = 14 + 1
8
5
p = 15
8
8  5  8  15 
 p =  
58  5 1 
p = 24
8 y − 3 = 13
8 y − 3 + 3 = 13 + 3
8 y = 16
8 y 16
=
8
8
y=2
45 = 6m − 3
45 + 3 = 6m − 3 + 3
48 = 6m
48 6m
=
6
6
8=m
49.
5 x + 12 = 4 x − 16
5 x − 4 x + 12 = 4 x − 4 x − 16
x + 12 = −16
x + 12 − 12 = −16 − 12
x = −28
50.
−4t − 2 = −3t + 5
−4t + 4t − 2 = −3t + 4t + 5
−2 = t + 5
−2 − 5 = t + 5 − 5
−7 = t
51. 6( w − 2) + 15 = 3( w + 3) − 2
6w − 12 + 15 = 3w + 9 − 2
6 w + 3 = 3w + 7
6 w − 3w + 3 = 3w − 3w + 7
3w + 3 = 7
3w + 3 − 3 = 7 − 3
3w = 4
3w 4
=
3 3
4
w=
3
−25 = 2n − 1
−25 + 1 = 2n − 1 + 1
−24 = 2n
−24 2n
=
2
2
−12 = n
388
Chapter 11
52.
−4(h − 5) + h = 7( h + 1) − 5
−4h + 20 + h = 7 h + 7 − 5
−4h + h + 20 = 7 h + 7 − 5
−3h + 20 = 7 h + 2
−3h − 7 h + 20 = 7 h − 7 h + 2
−10h + 20 = 2
−10h + 20 − 20 = 2 − 20
−10h = −18
−10h −18
=
−10 −10
9
h=
5
(b)
−6 x = x + 2
−6 x − x = x − x + 2
−7 x = 2
−7 x 2
=
−7 −7
2
x=−
7
2
The number is − .
7
56. (a) Let x represent the number.
−9 + 3 + 2x = −2
(b) −9 + 3 + 2 x = −2
−6 + 2 x = −2
−6 + 6 + 2 x = −2 + 6
2x = 4
2x 4
=
2 2
x=2
The number is 2.
53. −(5a + 3) − 3(a − 2) = 24 − a
−5a − 3 − 3a + 6 = 24 − a
−5a − 3a − 3 + 6 = 24 − a
−8a + 3 = 24 − a
−8a + a + 3 = 24 − a + a
−7 a + 3 = 24
−7 a + 3 − 3 = 24 − 3
−7 a = 21
−7 a 21
=
−7 −7
a = −3
54.
Review Exercises
57. (a) Let x represent the number.
1
−x=2
3
−(4b − 7) = 2(b + 3) − 4b + 13
−4b + 7 = 2b + 6 − 4b + 13
−4b + 7 = 2b − 4b + 6 + 13
−4b + 7 = −2b + 19
−4b + 2b + 7 = −2b + 2b + 19
−2b + 7 = 19
−2b + 7 − 7 = 19 − 7
−2b = 12
−2b 12
=
−2 −2
b = −6
(b)
Section 11.6
55. (a) Let x represent the number.
−6x = x + 2
389
1
−x=2
3
1 1
1
− − x =2−
3 3
3
6 1
−x = −
3 3
5
−1x =
3
5
−1(−1x) = −1 
3
5
x=−
3
5
The number is − .
3
Chapter 11
Solving Equations
58. (a) Let x represent the number.
x
1
−2=
8
4
Tom Hanks has starred in 66 films and
has starred in 50 films.
60. Let x represent the number of text
messages over 400. Then 0.10x represents
the additional charges.
 Monthly  additional  total
 fee  +  charges  =  bill 
44.98 + 0.10x = 47.48
44.98 − 44.98 + 0.10x = 47.48 − 44.98
0.10x = 2.50
0.10x 2.50
=
0.10 0.10
x = 25
Marty made 25 text messages over the
allotted 400.
x
1
−2=
(b)
8
4
1
1
x−2+2= +2
8
4
1
1 8
x= +
8
4 4
1
9
x=
8
4
1  9
8 x  = 8 
8  4
x = 18
The number is 18.
59. Let x represent the number of films Tom
Hanks has starred in. Then x −
16represents the number of films Johnny
Depp has starred in.
 Tom Hanks   Johnny Depp   Total 
 films  + 
 =  films 
films

 
 

x + x − 16 = 116
2 x − 16 = 116
2 x − 16 + 16 = 116 + 16
2 x = 132
2 x 132
=
2
2
x = 66
x − 16 = 50
Chapter 11
61. Let w represent the width. Then w + 32
represents the length.
P = 2l + 2w
224 = 2(w + 32) + 2w
224 = 2w + 64 + 2w
224 = 2w + 2w + 64
224 = 4w + 64
224 − 64 = 4w + 64 − 64
160 = 4w
160 4w
=
4
4
40 = w
w + 32 = 40 + 32 = 72
The width is 40 in. and the length is 72 in.
Test
1. $19.95m
7. Distributive property of multiplication
over addition
2. −x 2 + y 2 = −( )2 + ( )2 = −(4)2 + (−1)2
= −16 + 1 = −15
8. Commutative property of multiplication
9. 4(a + 9) − 12 = 4a + 36 − 12 = 4a + 24
3. A = 2lw + 2lh + 2wh
= 2(3)(2.5) + 2(3)(1.75) + 2(2.5)(1.75)
10. −3 (6b ) + 5b + 8 = −18b + 5b + 8
= −13b + 8
= 15 + 10.5 + 8.75 = 34.25 ft 2
4. Associative property of multiplication
11. 14 y + 2( y − 9) + 21 = 14 y + 2 y − 18 + 21
= 16 y + 3
5. Commutative property of addition
12. 2 + (5 − w) + 3(−2w) = 2 + 5 − w + (−6w)
= 7 − 7w
6. Associative property of addition
390
Chapter 11
(
)
21.
13. − 3x − 5 + 2x − 1 = −3x + 5 + 2x − 1
= −3x + 2x + 5 − 1
= −x + 4
(
)
14. 6 y + 2 − 5 y − 4 = 6 y + 2 − 5y + 20
= 6 y − 5y + 2 + 20
= y + 22
12 = −3 p + 9
12 − 9 = −3 p + 9 − 9
3 = −3 p
3 −3 p
=
−3 −3
−1 = p
15. An expression is a collection of terms. An
equation has an equal sign that indicates
that two expressions are equal.
22. 1.8m = 0.36
1.8m 0.36
=
1.8
1.8
m = 0.2
16. (a) Expression
23.
(b) Equation
(c) Expression
(d) Equation
(e) Expression
(f) Expression
17. −6 x = 12
−6 x 12
=
−6 −6
x = −2
18.
19.
24.
−6 + x = 12
−6 + 6 + x = 12 + 6
x = 18
25.
x
= 12
−6
1
− x = 12
6
1


−6  − x  = −6(12)
6


x = −72
20. 12 x = −6
12 x −6
=
12 12
1
x=−
2
26.
1
3
=−
16
4
1 1
3 1
p+ − =− −
16 16
4 16
12 1
p=− −
16 16
13
p=−
16
p+
5
5
=− n
12
6
6 5 
6 5 
− −  = − − n
5  12 
5 6 
1
=n
2
−
5h − 2 = − h + 16
5h + h − 2 = − h + h + 16
6h − 2 = 16
6h − 2 + 2 = 16 + 2
6h = 18
6h 18
=
6
6
h=3
x
= −12
7
1
x = −12
7
1 
7  x  = 7(−12)
7 
x = −84
391
Test
Chapter 11
27.
Solving Equations
−2( q − 5) = 6q + 10
−2q + 10 = 6q + 10
−2q + 2q + 10 = 6q + 2q + 10
10 = 8q + 10
10 − 10 = 8q + 10 − 10
0 = 8q
0 8q
=
8 8
0=q
 Budget   Budget   Total 
 "Twilight"  +  "Harry"  =  Budget 

 
 

x + 5 x = 300,000,000
6 x = 300,000,000
6 x 300,000,000
=
6
6
x = 500,000,000
The budget for New Moon was $50
million.
28. −(4k − 2) − k = 2( k − 6)
−4k + 2 − k = 2k − 12
−4k − k + 2 = 2k − 12
−5k + 2 = 2k − 12
−5k − 2k + 2 = 2k − 2k − 12
−7 k + 2 = −12
−7 k + 2 − 2 = −12 − 2
−7 k = −14
−7 k −14
=
−7
−7
k =2
31. Let x represent the length of the shorter
piece. Then 4x represents the length of the
other piece.
 Total 
 Length of   Length of  
+
=
 one piece   other piece  length 
 of pipe
x + 4x = 300
5x = 300
5x 300
=
5
5
x = 60
4x = 4 60 = 240
( )
29. Let x represent the number.
−2 x = 15 + x
−2 x − x = 15 + x − x
−3x = 15
−3 x 15
=
−3 −3
x = −5
The number is −5.
The lengths are 60 cm and 240 cm.
32. Let x represent the number of kWh used.
Then 0.104x represents the charge for
kWh used.
 Monthly  kWh   Total
 fee  +  charges =  bill 
9.95 + 0.104 x = 90.03
9.95 − 9.95 + 0.104 x = 90.03 − 9.95
0.104 x = 80.08
0.104 x 80.08
=
0.104 0.104
x = 770
Sela used 770 kWh.
30. Let x represent the budget for The Twilight
Saga: New Moon. Then 5x represents the
budget for Harry Potter: The Half-Blood
Prince.
Chapters 1–11
Cumulative Review Exercises
3. 1,285,000 ≈ 1,290,000
1. (a) Hundreds
4. 25,449 ≈ 25,400
(b) Ten-thousands
(c) Hundred-thousands
2. 45,921 ≈ 46,000
392
Chapters 1–11
Cumulative Review Exercises
851
5. 46 39,190
−36 8
2 39
−2 30
90
−46
44
dividend: 39,190
divisor: 46
quotient: 851
remainder: 44
1
5 1 9 35 11
15. 2 + 5 ⋅1 = + ⋅
4
6 10 4 6 10
9 385 135 385
= +
=
+
4 60
60 60
520
2
=
=8
60
3
6. (a) Prime
17. 0.16 0.08
16.
(b) Composite: 91 = 7 ⋅ 13
0.5
16 8.0
−8 0
0
(c) Composite: 39 = 3 ⋅ 13
7.
23
8
18.
23.991
3.200
+ 4.030
31.221
19.
78.002
− 34.250
43.752
3
8.
2 39 6
⋅
=
5
13 5
1
7
4
5
25
9. 21 ÷ 75 = 21 ⋅ 8 = 28
10 8 10 75 125
10.
11.
20. 4.2 − (2.0 − 1.2)2 = 4.2 − (0.8)2
= 4.2 − 0.64 = 3.56
1300
13
=
10,000 100
21. Total rooms = 5 + 6 + 4 = 15
$300
= $20
15
Sarah makes $20 per room.
9
1
4
54 15
40
79
− + =
−
+
=
25 10 15 150 150 150 150
1
1  14 
ft  (3 ft) = 7 ft 2
12. A = bh = 
2
2 3 
22.
7 3 ⋅ 8 + 7 31
13. 3 =
=
8
8
8
14.
3.1
× 4.5
1 55
12 40
13.95
28
3
=5
5
5
393
2 1.8
=
p
9
2 p = (9)(1.8)
2 p = 16.2
2 p 16.2
=
2
2
p = 8.1
Chapter 11
23.
Solving Equations
n
8
=
1
1 2 15
1 gal 4 qt 2 pt 2 c 8 oz
⋅
⋅
⋅
⋅
1 1 gal 1 qt 1 pt 1 c
= 128 oz
128 oz
1
= 21
6 oz
3
21 cups can be filled.
31. 1 gal =
 1
15n = 1  (8)
 2
15n = 12
15n 12
=
15 15
4
n=
5
24.
32. 1:41:45 = 1 hr 41 min 45 sec
= 60 min + 41 min + 0.75 min
= 101.75 min
24 m
=
15 35
15m = (24)(35)
15m = 840
15m 840
=
15
15
m = 56
33. 680 cc = 680 mL = 0.68 L
34. 3.2 km= 3200 m
35. 45 g = 4500 cg
36. r =
A = πr 2
$1.84
= $0.92/oz
2 oz
$2.25
Cat Goodies:
= $0.90/oz
2.5 oz
Kitty Treats costs $0.92 per oz. Cat
Goodies costs $0.90 per oz. Cat Goodies
is the better buy.
25. Kitty Treats:
≈ (3.14)(20 in.)2
= (3.14)(400 in.2 )
= 1256 in.2
1
1
1
1

in. − 2  in.  = 8 in. − 1 in. = 7 in.
2
2
2
2 
1

10 in. − 2  in.  = 10 in. − 1 in. = 9 in.
2 
 1

A = lw =  7 in.  (9 in.) = 67.5 in.2
2


37. 8
26. 15% = 15 × 0.01 = 0.15
15
3
15% =
=
100 20
27.
1
= 0.125
8
1 1
100
= × 100% =
% = 12.5%
8 8
8
The area is 67.5 in.2 .
38. V = πr 2 h
≈ (3.14)(6.5 in.) 2 (7.5 in.)
11
28. 1.1 =
10
1.1 = 1.1 × 100% = 110%
29.
d 40 in.
=
= 20 in.
2
2
= (3.14)(42.25 in.2 )(7.5 in.)
≈ 995 in.3
39. C = πd ≈ (3.14)(6 yd) = 18.84 yd
2
= 0.2
9
2 2
= × 100% = 22.2%
9 9
40.
x 2 + (10) 2 = (26) 2
x 2 + 100 = 676
x 2 + 100 − 100 = 676 − 100
30. 0.2% = 0.2 × 0.01 = 0.002
0.2
2
1
0.2% =
=
=
100 1000 500
x 2 = 576
x = 576
x = 24 mm
394
Chapters 1–11
41.
15
12
8
5
6
12
10
20
7
5
8
9
11
+ 12
140
Cumulative Review Exercises
48. 4(5 − 11) + ( −1) = 4( −6) + ( −1)
= −24 + ( −1)
= −25
49. 5 − 23 + 12 − 3 = −18 + 12 − 3 = −6 − 3 = −9
50. −2( x + 14) + 15 − 3x
= −2 x + (−28) + 15 − 3x
= −2 x − 3x + (−28) + 15
= −5x + (−13) = −5x − 13
51. 3 y − (5 y + 6) − 12 = 3 y − 5 y − 6 − 12
= −2 y − 18
140
= 10
14
52.
42. Arrange the numbers in order from
smallest to largest.
10 11 12 12 12 15
5 5 6 7 8 8 9
20
9 + 10 19
=
= 9.5
2
2
4 p + 5 = −11
4 p + 5 − 5 = −11 − 5
4 p = −16
4 p −16
=
4
4
p = −4
53. 9(t − 1) − 7t + 2 = t − 15
9t − 9 − 7t + 2 = t − 15
9t − 7t − 9 + 2 = t − 15
2t − 7 = t − 15
2t − t − 7 = t − t − 15
t − 7 = −15
t − 7 + 7 = −15 + 7
t = −8
43. 12
44.
45. (a) 32% are enrolled in sports. Find 32%
of 520.
(0.32)(520) = 166.4
Approximately 166 students enrolled
in sports.
(b) 20% do not participate in activities
Find 20% of 650.
(0.20)(650) = 130 students
46. −129 − (−132) = −129 + 132 = 3
47. 16 ÷ (−4) ⋅ 3 = −4 ⋅ 3 = −12
395
(
)
54.
2x = −4 x − 3
2x = −4x + 12
2x + 4x = −4x + 4x + 12
6x = 12
6x 12
=
6
6
x=2
55.
16 = 7 − 3x − 1
16 = 7 − 3x + 1
16 = 7 + 1− 3x
16 = 8 − 3x
16 − 8 = 8 − 8 − 3x
8 = −3x
8 −3x
=
−3 −3
8
− =x
3
(
)
Additional Topics Appendix
Section A.1
Energy and Power
Section A.1 Practice Exercises
1. Energy
53,000 ft ⋅ lb 1 Btu
⋅
1
778 ft ⋅ lb
≈ 68 Btu
13. 53,000 ft ⋅ lb ≈
2. Energy = (6 ft)(3800 lb) = 22,800 ft ⋅ lb
3. Energy = (5 ft)(3000 lb) = 15,000 ft ⋅ lb
14. 96,472,000 ft ⋅ lb 
2 yd 3 ft
⋅
= 6 ft
1 1 yd
Energy = (6 ft)(200 lb) = 1200 ft ⋅ lb
4. 2 yd =
= 124,000 Btu
15.
4,512, 400,000 ft ⋅ lb 1 Btu
⋅
1
778 ft ⋅ lb
= 5,800,000 Btu
1.5 yd 3 ft
⋅
= 4.5 ft
1
1 yd
Energy = (4.5 ft)(50 lb) = 225 ft ⋅ lb
16. 90,000 Btu ≈
2.5 tons 2000 lb
⋅
= 5000 lb
1
1 ton
Energy = (3 ft)(5000 lb) = 15,000 ft ⋅ lb
17. 1026 Btu ≈
5. 1.5 yd =
90,000 Btu 778 ft ⋅ lb
⋅
1
1 Btu
= 70,020,000 ft ⋅ lb
6. 2.5 tons =
1026 Btu 778 ft ⋅ lb
⋅
1
1 Btu
≈ 798, 228 ft ⋅ lb
1.5 tons 2000 lb
⋅
= 3000 lb
1
1 ton
Energy = (4 ft)(3000 lb) = 12,000 ft ⋅ lb
7. 1.5 tons =
40 min 1 hr
⋅
1
60 min
2
= hr or 0.67 hr
3
18. 40 min =
40,000 Btu 778 ft ⋅ lb
⋅
1
1 Btu
= 31,120,000 ft ⋅ lb
8. 40,000 Btu ≈
45 min 1 hr
⋅
1
60 min
3
= hr or 0.75 hr
4
19. 45 min =
3000 Btu 778 ft ⋅ lb
9. 3000 Btu ≈
⋅
1
1Btu
= 2,334,000 ft ⋅ lb
14,000 Btu 778 ft ⋅ lb
⋅
1
1 Btu
= 10,892,000 ft ⋅ lb
10. 14,000 Btu ≈
15 min 1 hr
1
⋅
= hr or 0.25 hr
1
60 min 4
1
5
1 hr 15 min = 1 + hr = hr or 1.25 hr
4
4
20. 15 min =
8000 Btu 778 ft ⋅ lb
⋅
1
1 Btu
= 6, 224,000 ft ⋅ lb
11. 8000 Btu ≈
12. 4000 ft ⋅ lb ≈
1 Btu
778 ft ⋅ lb
30 min 1 hr
1
⋅
= hr or 0.5 hr
1
60 min 2
1
5
2 hr 30 min = 2 + hr = hr or 2.5 hr
2
2
21. 30 min =
4000 ft ⋅ lb 1 Btu
⋅
≈ 5 Btu
778 ft ⋅ lb
1
396
Section A.1
24 min 1 hr
2
⋅
= hr or 0.4 hr
1
60 min 5
2
12
2 hr 24 min = 2 + hr =
hr or 2.4 hr
5
5
29. 1 hr 40 min =
22. 24 min =
5
x = (560)
3
≈ 933 Cal
4
hr
3
700 x
=4
1
3
30. Power =
(5 ft)(25 lb)
ft ⋅ lb
= 25
5 sec
sec
31. Power =
(3 ft)(40 lb)
ft ⋅ lb
= 60
2 sec
sec
32. 1 yd = 3 ft
(3 ft)(200 lb)
ft ⋅ lb
Power =
= 100
6 sec
sec
4
x = 700  
3
≈ 933 Cal
1.5 yd 3 ft
⋅
= 4.5 ft
1
1 yd
(4.5 ft)(300 lb)
ft ⋅ lb
Power =
= 135
10 sec
sec
33. 1.5 yd =
25. 45 min = 0.75 hr
590
x
=
1
0.75
x = 590(0.75)
≈ 443 Cal
34. 1 ton = 2000 lb
(3 ft)(2000 lb)
ft ⋅ lb
Power =
= 400
15 sec
sec
26. 2 hr 45 min = 2.75 hr
280
x
=
1
2.75
x = 280(2.75)
= 770 Cal
35. 1 ton = 2000 lb
(3 ft)(2000 lb)
ft ⋅ lb
Power =
= 200
30 sec
sec
27. 2 hr 30 min = 2.5 hr
500
x
=
1
2.5
x = 500(2.5)
= 1250 Cal
28. 48 min =
5
hr
3
560 x
=5
1
3
6 min 1 hr
1
23. 6 min =
hr or 0.1 hr
⋅
=
1
60 min 10
1
11
1 hr 6 min = 1 +
hr =
hr or 1.1 hr
10
10
24. 1 hr 20 min =
Energy and Power
36. 550
ft ⋅ lb
= 1 hp
sec
ft ⋅lb
1 hp
ft ⋅ lb 1100 sec
=
⋅
= 2 hp
37. 1100
1
sec
550 ft ⋅lb
48 min 1 hr
⋅
= 0.8 hr
1
60 min
sec
ft⋅lb
380
x
=
1
0.8
x = 380(0.8)
= 304 Cal
38. 4950
ft ⋅ lb 4950 sec
1 hp
=
⋅
= 9 hp
⋅lb
sec
1
550 ftsec
ft⋅lb
ft ⋅ lb 6050 sec
1 hp
=
⋅
= 11 hp
39. 6050
⋅lb
sec
1
550 ftsec
397
Additional Topics Appendix
44. (a) (250 W)(3 hr/day)(30 days)
= 22,500 Wh
ft⋅lb
400 hp 550 sec
⋅
40. 400 hp =
1 hp
1
ft ⋅ lb
= 220,000
sec
(b) 22,500 Wh = 22.5 kWh
(c) (22.5 kWh)($0.11) ≈ $2.48
45. (a) (3500 W)(6 hr/day)(30 days)
= 630,000 Wh
ft⋅lb
315 hp 550 sec
ft ⋅ lb
⋅
= 173, 250
1
1 hp
sec
41. 315 hp =
ft⋅lb
hp 550 sec
⋅
550
1
43. 215 hp =
215 hp 550 sec
ft ⋅ lb
⋅
= 118, 250
1
1 hp
sec
1 hp
(c) (630 kWh)($0.082) = $51.66
ft ⋅ lb
sec
42. 550 hp =
= 302,500
(b) 630,000 Wh = 630 kWh
ft⋅lb
Section A.2
Scientific Notation
Section A.2 Practice Exercises
1. scientific notation
15. Yes
2. 10,000 = 104
16. No; 0.02 < 1
17. No; 0.052 < 1
3. 100,000 = 105
18. 230,000,000,000 stars = 2.3 × 1011 stars
3
4. 1000 = 10
19. 25,000,000,000,000 mi. = 2.5 × 1013 mi.
5. 1,000,000 = 106
20. 0.0000002 mm = 2 × 10−7 mm
6. 0.001 = 10−3
7. 0.01 = 10
8. 0.0001 = 10
9. 0.1 = 10
21. 0.0625 in. = 6.25 × 10−2 in.
−2
22. 20,000,000 = 2 × 107
−4
23. 5000 = 5 × 103
−1
10. No; 43 > 10
24. 8,100,000 = 8.1 × 106
11. No; 82 > 10
25. 62,000 = 6.2 × 104
12. Yes
26. 0.003 = 3 × 10−3
13. Yes
27. 0.0009 = 9 × 10−4
14. Yes
28. 0.025 = 2.5 × 10−2
398
Section A.2
Scientific Notation
29. 0.58 = 5.8 × 10−1
46. 3.26 × 102 = 326
30. 142,000 = 1.42 × 105
47. 6.13 × 107 = 61,300,000
31. 25,500,000 = 2.55 × 107
48. 1.29 × 10−2 = 0.0129
32. 0.0000491 = 4.91 × 10−5
49. 4.04 × 10−4 = 0.000404
33. 0.000116 = 1.16 × 10−4
50. 2.003 × 10−6 = 0.000002003
34. 0.082 = 8.2 × 10−2
51. 5.02 × 10−5 = 0.0000502
35. 0.15 = 1.5 × 10−1
52. 9.001 × 108 = 900,100,000
36. 4920 = 4.92 × 103
53. 7.07 × 106 = 7,070,000
37. 13, 400 = 1.34 × 104
54. 0.33× 1024 = 3.3× 1023
38. 6 × 103 = 6000
55. Already in scientific notation
39. 3 × 104 = 30,000
56. Already in scientific notation
57. 0.64 × 1024 = 6.4 × 1023
40. 8 × 10−2 = 0.08
58. 189.9 × 1024 = 1.899 × 1027
41. 2 × 10−5 = 0.00002
59. 586.5 × 1024 = 5.865 × 1026
42. 4.4 × 10−1 = 0.44
43. 2.1 × 10−3 = 0.0021
60. 86.8 × 1024 = 8.68 × 1025
44. 3.7 × 104 = 37,000
61. 102.4 × 1024 = 1.024 × 1026
45. 5.5 × 103 = 5500
Section A.3
Rectangular Coordinate System
Section A.3 Practice Exercises
1. (a) x; y-axis
(b) ordered
(c) origin; (0, 0)
(d) quadrants
(e) negative
(f) III
399
Additional Topics Appendix
2−8.
29−34.
35. Quadrant IV
9−14.
36. Quadrant II
37. Quadrant III
38. Quadrant I
39. x-axis
15. First move to the left 1.8 units from the
origin. Then go up 3.1 units. Place a dot at
the final location. The point is in Quadrant
II.
40. y-axis
16. Write the fractions as mixed numbers
1
 1 1
 7 , 2  . Move 7 units to the right of
2
 2 7
1
the origin and 2 units up. Place a dot at
7
the final location. The point is in Quadrant
I.
43. Quadrant II
17−22.
48. (−1, 0)
41. y-axis
42. x-axis
44. Quadrant IV
45. Quadrant I
46. Quadrant III
47. (0, 3)
49. (2, 3)
50. (−4, 2)
51. (−5, −2)
52. (1, −1)
23−28.
53. (4, −2)
 1
54.  5, 
 2
55. (–2, −5)
400
Section A.3
Rectangular Coordinate System
59. The ordered pairs are given by (1, 13,200),
(2, 11,352), (3, 9649), (4, 8201), (5, 6971),
and (6, 5925).
56. The ordered pairs are given by (1, −7),
(2, −3), (3, 1), (4, 6), (5, 12), (6, 17),
(7, 18), (8, 18), (9, 14), (10, 6), (11, −1),
and (12, −7).
60. The ordered pairs are given by (3504, 18),
(2833, 22), (4060, 17), (2050, 38),
(2205, 36), and (2625, 28)
57. The ordered pairs are given by (1, −6),
(2, −5), (3, 1), (4, 11), (5, 18), (6, 22),
(7, 24), (8, 22), (9, 15), (10, 8), (11, 1),
and (12, −4).
61. The ordered pairs are given by (30, 33),
(40, 35), (50, 40), (60, 37), and (70, 30).
58. The ordered pairs are given by (1, 15000),
(2, 13,200), (3, 11,616), (4, 9900), (5,
8491), and (6, 7218).
401