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Division of Whole Numbers
1.6 OBJECTIVES
1.
2.
3.
4.
Use repeated subtraction to divide whole numbers
Check the results of a division problem
Divide whole numbers using long division
Estimate a quotient
Overcoming Math Anxiety
Learn to Take Useful Notes
Although some students find it easier to be organized than do other students,
every student can become a better note taker. Below are some hints that can
help you learn to take more useful notes. Good note taking begins with your
preparation for class. Note that the first several items refer to your preparation.
1. Read the assigned material before the lecture. This helps you become
familiar with both the vocabulary and the concepts.
2. Review your notes from the previous class meeting. If you already have a
concept in your lecture notes and the idea is referred to again, you then
simply jot a note to refer back to previous material in your notes.
3. Get to class a few minutes early. Have your materials out and ready to
go when your instructor walks in the door.
4. Be ready to listen as soon as the instructor walks in the door.
5. Have more than one pencil sharpened and ready to use.
6. Have a highlighter or colored pen available to mark particularly important
segments of your notes.
7. Know how to spell. This includes both plain old English words and
technical words that have already been presented in class. If you can spell
them, you won’t have to waste time trying to figure out How to spell them.
8. Be aggressive in notetaking. Don’t wait for an idea to strike you—it’s
better to have too much material than too little.
9. If the professor repeats something, write it down.
10. Take notes, not dictation. That means being able to develop your own
form of shorthand.
11. Develop abbreviations for words that are used frequently in the course.
• Real numbers R
• Natural numbers N
© 2010 McGraw-Hill Companies
12. Use symbols when you can.
• & and
• E there exists
• B but
•
• 5 for each
•
I such that
H is an element of
• ‹ therefore
13. Skip lines. Leave visual breaks between definitions, lists, or explanations.
14. If you miss something, leave a blank in your notes. You can fill it in later.
If you try to copy it from your neighbor during the lecture, both of you will
lose more material.
15. Get together with your classmates. Do so after lecture and pool your
notes. That way, you can be sure you have everything down. It will also
help make sure you understand what you have written down.
71
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We now examine a fourth arithmetic operation, division. Just as multiplication was repeated addition, division is repeated subtraction. Division asks how many times one number is contained in another.
OBJECTIVE 1
Example 1 Dividing by Using Subtraction
Joel needs to set up 48 chairs in the student union for a concert. If there is room for 8 chairs
per row, how many rows will it take to set up all 48 chairs?
This problem can be solved by subtraction. Each row subtracts another 8 chairs.
48
8
40
8
32
8
24
8
16
8
8
8
40
32
24
16
8
0
Because 8 can be subtracted from 48 six times, there will be 6 rows.
This can also be seen as a division problem
NOTE Each of these notations
48 8 6
or
6
8B48
or
48
6
8
No matter which notation we use, we call the 48 the dividend, the 8 the divisor, and the 6
the quotient.
CHECK YOURSELF 1
Carlotta is creating a garden path made of bricks. She has 72 bricks. Each row will
have 6 bricks in it. How many rows can she make?
Units Analysis
When dividing a denominate number by an abstract number, the result has the
units of the denominate number. Here are a couple of examples.
76 trombones 4 19 trombones
$55 11 $5
When one denominate number is divided by another, the result has the units
of the dividend over the units of the divisor.
144 mi 6 gal 24 mi/gal (which we read as “miles per gallon”)
$120 8 h 15 dollars/h (“dollars per hour”)
© 2010 McGraw-Hill Companies
represent the same division
problem, “48 divided by 8 is 6.”
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73
To solve a problem that requires division, you first set up the problem as a division
statement. Example 2 illustrates this.
Example 2 Writing a Division Statement
Write a division statement that corresponds to the following situation. You need not do the
division.
The staff at the Wok Inn Restaurant splits all tips at the end of each shift. Yesterday’s
evening shift collected a total of $224. How much should each of the seven employees get
in tips?
$224 7 employees
Note that the units for the answer will be
“dollars per employee.”
CHECK YOURSELF 2
Write a division statement that corresponds to the following situation. You need
not do the division.
All nine sections of basic math skills at SCC (Sum Community College) are full.
There are a total of 315 students in the classes. How many students are in each
class? What are the units for the answer?
In Section 1.5, we used a rectangular array of stars to represent multiplication. These
same arrays can represent division. Just as 3 4 12 and 4 3 12, so is it true that
12 3 4 and 12 4 3.
© 2010 McGraw-Hill Companies
4 • 3 12
or
12 3 4
3 • 4 12
or
12 4 3
This relationship allows us to check our division results by doing multiplication.
OBJECTIVE 2
NOTE For a division problem
to check, the product of the
divisor and the quotient must
equal the dividend.
Example 3 Checking Division by Using Multiplication
3
(a) 7B21
(b) 48 6 8
Check: 7 3 21
Check: 6 8 48
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CHECK YOURSELF 3
Complete the division statements and check your results.
(b) 28 7 (a) 9B45
NOTE Because 36 9 4, we
say that 36 is exactly divisible
by 9.
In our examples so far, the product of the divisor and the quotient has been equal to the
dividend. This means that the dividend is exactly divisible by the divisor. That is not always
the case. In Example 4, we are again using repeated subtraction.
Example 4 Dividing by Using Subtraction, Leaving a Remainder
How many times is 5 contained in 23?
NOTE The remainder must be
smaller than the divisor or we
could subtract again.
23
5
18
18
5
13
13
5
8
8
5
3
We see that 5 is contained 4
times in 23, but 3 is “left over.”
23 is not exactly divisible by 5. The “leftover” 3 is called the remainder in the division.
CHECK YOURSELF 4
How many times is 7 contained in 38?
Property:
Remainder
Dividend divisor quotient remainder
We can check our work in a division problem with a remainder as follows.
Example 5 Checking Division by a Single-Digit Number
Using the work of Example 4, we can write
NOTE Another way to write
the result is
4
5B23
4 r3
5B 23
To apply the remainder property, we have
with remainder 3
NOTE The multiplication is
done before the 3 is added.
Dividend
Quotient
23 5 4 3
Remainder
23 20 3
23 23
The division checks.
CHECK YOURSELF 5
Evaluate 7B38. Check your answer.
We must be careful when 0 is involved in a division problem. There are two special cases.
© 2010 McGraw-Hill Companies
Divisor
The “r” stands for remainder.
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Property:
SECTION 1.6
75
Division and Zero
1. 0 divided by any whole number (except 0) is 0.
2. Division by 0 is undefined.
The first case involving zero occurs when we are dividing into zero.
Example 6 Dividing into Zero
0 5 0 because 0 5 0.
CHECK YOURSELF 6
(a) 0 7 (b) 0 12 Our second case illustrates what happens when 0 is the divisor. Here we have a special
problem.
Example 7 Dividing by Zero
8 0 ? This means that 8 0 ?
Can 0 times some number ever be 8? From our multiplication facts, the answer is no! There
is no answer to this problem, so we say that 8 0 is undefined.
CHECK YOURSELF 7
Decide whether each problem results in 0 or is undefined.
(a) 9 0
(b) 0 9
(c) 0 15
(d) 15 0
© 2010 McGraw-Hill Companies
It is easy to divide when small whole numbers are involved, because much of the work
can be done mentally. In working with larger numbers, we turn to a process called long
division. This is a method for performing the steps of repeated subtraction.
To start, we can look at an example in which we subtract multiples of the divisor.
OBJECTIVE 3
NOTE With larger numbers,
repeated subtraction is just too
time-consuming to be practical.
Example 8 Dividing by a Single-Digit Number
Divide 176 by 8.
Because 20 eights are 160, we know that there are at least 20 eights in 176.
Step 1 Write
20 eights
20
8B176
160
16
Subtracting 160 is just a shortcut for
subtracting eight 20 times.
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After subtracting the 20 eights, or 160, we are left with 16. There are 2 eights in 16, and so
we continue.
Step 2
2 eights
2
20 22
8B176
160
16
16
0
Adding 20 and 2 gives us the quotient, 22.
Subtracting the 2 eights, we have a 0 remainder. So 176 8 22.
CHECK YOURSELF 8
Verify the results of Example 8, using multiplication.
The next step is to simplify this repeated-subtraction process one step further. The
result is the long-division method.
Example 9 Dividing by a Single-Digit Number
Divide 358 by 6.
The dividend is 358. We look at the first digit, 3. We cannot divide 6 into 3, so we look
at the first two digits, 35. There are 5 sixes in 35, and so we write 5 above the tens digit of
the dividend.
5
6B358
When we place 5 as the tens digit,
we really mean 5 tens, or 50.
Now multiply 5 6, place the product below 35, and subtract.
5
6B358
300
5
We have actually subtracted 50 sixes
(300) from 358.
Because the remainder, 5, is smaller than the divisor, 6, we bring down 8, the ones digit of
the dividend.
Now divide 6 into 58. There are 9 sixes in 58, and so 9 is the ones digit of the quotient. Multiply 9 6 and subtract to complete the process.
NOTE Because the 4 is smaller
than the divisor, we have a
remainder of 4.
NOTE Verify that this is true
and that the division checks.
59
6B358
30
58
54
4
We now have:
358 6 59 r4
To check: 358 6 59 4.
© 2010 McGraw-Hill Companies
5
6B358
300
58
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77
CHECK YOURSELF 9
Divide 7B 453
Long division becomes a bit more complicated when we have a two-digit divisor. It becomes, in part, a matter of trial and error. We round the divisor and dividend to form a trial
divisor and a trial dividend. We then estimate the proper quotient and must determine
whether our estimate is correct.
Example 10 Dividing by a Two-Digit Number
Divide
38B293
7
NOTE Think: 4B 29.
Round the divisor and dividend to the nearest ten. So 38 is rounded to 40, and 293 is
rounded to 290. The trial divisor is then 40, and the trial dividend is 290.
Now look at the nonzero digits in the trial divisor and dividend. They are 4 and 29. We
know that there are 7 fours in 29, and so 7 is our first estimate of the quotient. Now we will
see if 7 works.
7
38B293
266
27
Your estimate
Multiply 7 38. The product, 266, is
less than 293, and so we can subtract.
The remainder, 27, is less than the divisor, 38, and so the process is complete.
293 38 7 r27
Check: 293 38 7 27.
You should verify that this statement is true.
CHECK YOURSELF 10
Divide.
57B 482
Because this process is based on estimation, our first guess will sometimes be wrong.
Example 11 Dividing by a Two-Digit Number
© 2010 McGraw-Hill Companies
Divide
54B428
8
NOTE Think: 5B 43.
Rounding to the nearest ten, we have a trial divisor of 50 and a trial dividend of 430.
Looking at the nonzero digits, how many fives are in 43? There are 8. This is our first
estimate.
8
54B428
432
Too large
We multiply 8 54. Do you see what’s wrong? The
product, 432, is too large. We cannot subtract. Our
estimate of the quotient must be adjusted downward.
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NOTE If we tried 6 as the
quotient
We adjust the quotient downward to 7. We can now complete the division.
6
54B428
324
104
7
54B428
378
50
We have 104, which is too large
to be a remainder.
We have
428 54 7 r50
Check: 428 54 7 50.
CHECK YOURSELF 11
Divide.
63B 557
We have to be careful when a 0 appears as a digit in the quotient. Next, we look at an
example in which this happens with a two-digit divisor.
Example 12 Dividing with Large Dividends
Divide
32B9871
Rounding to the nearest ten, we have a trial divisor of 30 and a trial dividend of 100.
Think, “How many threes are in 10?” There are 3, and this is our first estimate of the
quotient.
3
32B9871
96
2
Everything seems fine so far!
Bring down 7, the next digit of the dividend.
30
32B9871
96
27
Now do you see the difficulty? We cannot divide
32 into 27, and so we place 0 in the tens place
of the quotient to indicate this fact.
We continue by multiplying by 0. After subtraction, we bring down 1, the last digit of the
dividend.
30
32B9871
96
27
00
271
© 2010 McGraw-Hill Companies
NOTE Our divisor, 32, divides
into 98, the first two digits of
the dividend.
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Another problem develops here. We round 32 to 30 for our trial divisor, and we round 271
to 270, which is the trial dividend at this point. Our estimate of the last digit of the quotient
must be 9.
309
32B9871
96
27
00
271
288
Too large
We cannot subtract. The trial quotient must be adjusted downward to 8. We can now
complete the division.
308
32B9871
96
27
00
271
256
15
9,871 32 308 r15
Check: 9,871 32 308 15.
CHECK YOURSELF 12
Divide.
43B 8857
© 2010 McGraw-Hill Companies
Because of the availability of the handheld calculator, it is rarely necessary that people
find the exact answer when performing long division. On the other hand, it is frequently
important that one be able to either estimate the result of long division or confirm that a
given answer (particularly from a calculator) is reasonable. As a result, the emphasis in this
section will be to improve your estimation skills in division.
In Example 13, we divide a four-digit number by a two-digit number. Generally, we
round the divisor to the nearest ten and the dividend to the nearest hundred.
OBJECTIVE 4
Example 13 Estimating the Result of a Division Application
The Ramirez family took a trip of 2,394 mi in their new car, using 77 gal of gas. Estimate
their gas mileage (mi/gal).
Our estimate will be based on dividing 2,400 by 80.
30
80B2400
They got approximately 30 mi/gal.
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CHECK YOURSELF 13
Troy flew a light plane on a trip of 2,844 mi that took 21 h. What was his
approximate speed in miles per hour?
As before, we may have to combine operations to solve an application of the mathematics you have learned.
Example 14 Estimating the Result of a Division Application
Charles purchases a used car for $8,574. He agrees to make payments for 4 years. Interest
charges will be $978. Approximately what should his monthly payments be?
First, we find the amount that Charles owes:
$8,574 $978 $9,552
Now, to find the monthly payment, we divide that amount by 48 (months). To estimate the
payment, we divide $9,600 by 50 months.
192
50B9600
The payments will be approximately $192 per month.
CHECK YOURSELF 14
One $10 bag of fertilizer will cover 310 ft 2. Approximately what would it cost to
cover 2,200 ft 2?
Using a Scientific Calculator
to Divide
Of course, division is easily done using your calculator. However, as we will see, some
special things come up when we use a calculator to divide. First we outline the steps of
division as it is done on a calculator.
RECALL A graphing calculator
uses the Enter key rather
than .
Step 1 Enter the dividend.
2380
Step 2 Press the divide key.
Step 3 Enter the divisor.
35
Step 4 Press the equals key.
The desired quotient is now in your display.
The display shows 68.
We have already mentioned some of the difficulties related to division with 0. We will
experiment on the calculator.
© 2010 McGraw-Hill Companies
Divide 35B2380.
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81
Example 15 Using a Scientific Calculator to Divide
To find 0 5, we use this sequence:
0 5 Display
0
There is no problem with this. Zero divided by any whole number other than 0 is just 0.
CHECK YOURSELF 15
What is the result when you use your calculator to perform the given operation?
0 17
We see what happens when dividing zero by another number, but what happens when
we try to divide by zero? More importantly to this section, how does the calculator handle
division by zero? Example 16 illustrates this concept.
Example 16 Using a Scientific Calculator to Divide
To find 5 0, we use this sequence:
5 0 Display
NOTE You may find that you
must “clear” your calculator
after trying this.
Error
If we try this sequence, the calculator gives us an error! Do you see why? Division by 0 is
not allowed. Try this on your calculator to see how this error is indicated.
CHECK YOURSELF 16
What is the result when you use your calculator to perform the given operation?
17 0
Example 17 Using a Scientific Calculator to Divide
Dividing 293 by 38 gives 7 with remainder 27.
NOTE Be aware that the
calculator will not give you a
remainder in the form we have
been using in this chapter.
293 38 7.7105263
© 2010 McGraw-Hill Companies
Another special problem comes up when a remainder is involved in a division problem.
Quotient
Remainder
7 is the whole-number part of the
quotient as before.
0.7105263 is the decimal form of the
remainder, 27, as a fraction of 38.
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CHECK YOURSELF 17
What is the result when you use your calculator to perform the given operation?
458 36
The calculator can also help you combine division with other operations.
Example 18 Using a Scientific Calculator to Divide
To find 18 2 3, use this sequence:
18 2 3 Display 12
Do you see that the calculator has
done the division as the first step?
CHECK YOURSELF 18
Use your calculator to compute.
15 5 7
Example 19 Using a Scientific Calculator to Divide
To find 6 3 2, use this sequence:
6 3 2 Display 4
CHECK YOURSELF 19
Use your calculator to compute.
18 6 5
© 2010 McGraw-Hill Companies
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SECTION 1.6
83
READING YOUR TEXT
The following fill-in-the-blank exercises are designed to assure that you understand
the key vocabulary used in this section. Each sentence comes directly from the
section. You will find the correct answers in Appendix C.
Section 1.6
(a) If you read the assigned material before the lecture, you will become familiar
with both the
and the concepts.
(b) Just as multiplying was repeated addition, division is repeated
(c) You can check division by using
(d) Division by
.
is undefined.
CHECK YOURSELF ANSWERS
© 2010 McGraw-Hill Companies
1. 12
2. 315 students 9 classes; students per class
3. (a) 5; 9 5 45;
(b) 4; 7 4 28
4. 5
5. 5 with remainder 3
6. (a) 0; (b) 0
7. (a) undefined; (b) 0; (c) 0; (d) undefined
8. 8 22 176
9. 64 with remainder 5
10. 8 with remainder 26
11. 8 with remainder 53
12. 205 with remainder 42
13. 140 mi/h
14. $70
15. 0
16. Error message
17. 12.72222
18. 10
19. 15
.
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Page 84
1.6
Exercises
1. Given 48 8 6, 8 is the
, 48 is the
, and 6 is the
.
9
2. In the statement 5B45 , 9 is the
, 5 is the
, and 45 is the
.
Name
Section
Date
3. Find 36 9 by repeated subtraction.
4. Find 40 8 by repeated subtraction.
ANSWERS
Stefanie is planting rows of tomato plants. She wants to plant
63 plants with 9 plants per row. How many rows will she have?
5. Problem Solving
1.
Nick is designing a parking lot for a small office building. He must
make room for 42 cars with 7 cars per row. How many rows should he plan for?
6. Construction
2.
Divide and identify the correct units for the quotient.
3.
4.
5.
6.
7. 36 pages 4
7.
8.
9. 4,900 km 7
9.
10.
11. 160 mi 4 h
12. 264 ft 3 s
11.
12.
13. 3,720 h 5 months
14. 560 cal 7 g
8. $96 8
10. 360 gal 18
13.
15.
16.
17.
18.
19.
20.
15. 5B 43
16. 40 9
17. 9B65
18. 6B 51
19. 57 8
20. 74 8
21. 0 6
22. 18 0
21.
Divide.
22.
23. 5B 83
24. 9B 78
25. 8B293
26. 7B 346
27. 8B 3136
28. 9B3527
29. 8B 22,153
30. 5B 43,287
31. 48B 892
32. 54B372
33. 45B2367
34. 53B 3480
36. 871B 4321
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
35. 763B 3071
33.
34.
Solve the applications.
35.
36.
37. Statistics
37.
38.
84 SECTION 1.6
Ramon bought 56 bags of candy. There were 8 bags in each box. How
many boxes were there?
There are 32 students who are taking a field trip. If each car can hold
4 students, how many cars are needed for the field trip?
38. Statistics
© 2010 McGraw-Hill Companies
Divide using long division and check your work.
14.
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ANSWERS
Ticket receipts for a play were $552. If the tickets were
$4 each, how many tickets were purchased?
39. Business and Finance
39.
40.
41.
42.
43.
Construction of a fence section requires 8 boards. If you have 256
boards, how many sections can you build?
40. Construction
The homeowners along a street must share the $2,030 cost of new
street lighting. If there are 14 homes, what amount will each owner pay?
41. Construction
44.
A bookstore ordered 325 copies of a textbook at a cost of
$7,800. What was the cost to the store for an individual textbook?
42. Business and Finance
A company distributes $16,488 in year-end bonuses. If
each of the 36 employees receives the same amount, what bonus will each receive?
43. Business and Finance
© 2010 McGraw-Hill Companies
44. Complete the following number cross.
Across
1. 48 4
3. 1,296 8
6. 2,025 5
8. 4 5
9. 11 11
12. 15 3 111
14. 144 (2 6)
16. 1,404 6
18. 2,500 5
19. 3 5
Down
1. (12 16) 2
2. 67 3
4. 744 12
5. 2,600 13
7. 6,300 12
10. 304 2
11. 5 (161 7)
13. 9,027 17
15. 400 20
17. 9 5
1
2
3
7
6
14
18
8
13
12
15
5
10
9
11
4
17
16
19
SECTION 1.6
85
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ANSWERS
45.
46.
Estimate the result in the division problems. (Remember to round divisors to the nearest ten
and dividends to the nearest hundred.)
47.
45. 810 divided by 38
46. 458 divided by 18
49.
47. 4,967 divided by 96
48. 3,971 divided by 39
50.
49. 3,812 divided by 188
50. 5,245 divided by 255
48.
51.
In exercises 51 to 54, solve the applications.
52.
Jose drove 279 mi on 18 gal of gas. Estimate his mileage. (Hint:
Find the number of miles per gallon.)
51. Technology
53.
A contractor can build a house in 27 days. Estimate how many
houses can be built in 265 days.
52. Construction
54.
You are going to recarpet your living room. You have budgeted
$1,500 for the carpet and installation.
53. Construction
55.
(a) Determine how much carpet you will need to do the job. Draw a sketch to support your measurements.
(b) What is the highest price per square yard you can pay and still stay within
budget?
(c) Go to a local store and determine the total cost of doing the job for three different grades of carpet. Be sure to include padding, labor costs, and any other
expenses.
(d) What considerations (other than cost) would affect your decision about what
type of carpet to install?
(e) Write a brief paragraph indicating your final decision and give supporting
reasons.
54. Division is the inverse operation of multiplication. Many daily activities have in-
(a)
(b)
(c)
(d)
Spending money
Going to sleep
Turning down the volume on your CD player
Getting dressed
55. Division is not associative. For example, 8 4 2 will produce different results
if 8 is divided by 4 and then divided by 2 or if 8 is divided by the result of 4 2.
Place parentheses in the proper place so that each expression is true.
(a) 16 8 2 4
(b) 16 8 2 1
(c) 125 25 5 1
(d) 125 25 5 25
(e) Is there any situation in which the order of how the operation of division is performed produces the same result? Give an example.
86 SECTION 1.6
© 2010 McGraw-Hill Companies
verses. For each of the following activities, state the inverse activity:
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ANSWERS
56. Division is not commutative. For example, 15 5 5 15. What must be true
of the numbers a and b if a b b a?
56.
57.
Solve each chapter-activity application.
58.
A set of same-sized packages are each 13 in. wide and have a height of
3 in. Find the girth of each package (twice the width plus twice the height). You
need a length of wrapping paper equal to one more inch than the girth of the package. What length of wrapping paper do you need for each package? How many
packages can you wrap if your wrapping paper is 300 in. (25 ft) long? How long a
piece of scrap will you be left with?
57. Crafts
1
59.
60.
61.
62.
63.
If a business needs to wrap 50 packages, each as described in exercise 57, how many rolls of 300-in.-long wrapping paper will they
need? What if they needed to wrap 200 such packages?
58. Business and Finance
1
64.
65.
66.
Calculator Exercises
Use your calculator to perform the indicated operations.
59. 36,182 79
60. 464,184 189
61. 6 9 3
62. 18 6 3
63. 24 6 4
64. 1,176 42 1,572 524
65. 3 8 8 8 12
66. (89 14) 25
© 2010 McGraw-Hill Companies
Answers
1. Divisor, dividend, quotient
3. 4
5. 7
7. 9 pages
9. 700 km
11. 40 mi/h
13. 744 h/month
15. 8 r3
17. 7 r2
19. 7 r1
21. 0
23. 16 r3
25. 36 r5
27. 392
29. 2,769 r1
31. 18 r28
33. 52 r27
35. 4 r19
37. 7 boxes
39. 138 tickets
41. $145
43. $458
45. 20
47. 50
49. 20
51. 15 mi/gal
53.
55.
57. 32 in.; 33 in.; 9 packages; 3 in.
59. 458
61. 9
63. 16
65. 128
SECTION 1.6
87