3.4
Adding and Subtracting
Mixed Numbers
3.4
OBJECTIVES
1.
2.
3.
4.
Add any two mixed numbers
Add any group of mixed numbers
Subtract any two mixed numbers
Solve an application that involves mixed number
addition or subtraction
Once you know how to add fractions, adding mixed numbers should be no problem if you
keep in mind that addition involves combining groups of the same kind of objects. Because
mixed numbers consist of two parts—a whole number and a fraction—we could work with
the whole numbers and the fractions separately. Generally, it is easier to rewrite mixed
numbers as improper fractions, then do the addition.
6
5
12
5
© 2001 McGraw-Hill Companies
The sum of the
whole-number
parts
18
3
or 3
5
5
The sum of the
fractional parts
This suggests the following general rule.
Step by Step: To Add Mixed Numbers
NOTE Step 2 requires that the
fractional parts have the same
denominator.
Step 1 Change the mixed numbers to improper fractions.
Step 2 Add the fractions.
Step 3 Rewrite the result as a mixed number if required.
261
262
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
Our first example illustrates the use of this rule.
Example 1
Adding Mixed Numbers
Add and write the result as a mixed numbers.
3
1
2
16
22
4 5
5
5
5
Rewrite as improper fractions.
38
5
3
7
5
Add the numerators.
Rewrite as a mixed number.
CHECK YOURSELF 1
Add 2
3
4
3 . Write the result as a mixed number.
10
10
When the fractional portions of the mixed numbers have different denominators, we
must rename these fractions as equivalent fractions with the least common denominator to
perform the addition in step 2. Consider Example 2.
Example 2
Adding Mixed Numbers with Different Denominators
Add, and write the result as a mixed number.
NOTE
1
3
19
19
2 6
8
6
8
5
24B133
120
13
76
57
24
24
133
24
5
The LCD of the fractions is 24. Rename
them with that denominator.
Then add as before.
13
24
CHECK YOURSELF 2
Add 5
7
5
3 . Write the result as a mixed number.
10
6
You follow the same procedure if more than two mixed numbers are involved in the
problem.
© 2001 McGraw-Hill Companies
3
ADDING AND SUBTRACTING MIXED NUMBERS
SECTION 3.4
263
Example 3
Adding Mixed Numbers with Different Denominators
Add.
NOTE The LCD of the three
2
fractions is 40. Convert to
equivalent fractions.
1
3
1
11
15
33
3 4 5
4
8
5
4
8
88
150
165
40
40
40
403
40
10
3
40
CHECK YOURSELF 3
Add 5
1
2
3
4 3 .
2
3
4
We can use a similar technique for subtracting mixed numbers. The rule is similar to that
stated earlier for adding mixed numbers.
Step by Step: To Subtract Mixed Numbers
Step 1 Change the mixed numbers to improper fractions.
Step 2 Subtract the fractions.
Step 3 Rewrite the result as a mixed number if required.
Example 4 illustrates the use of this rule.
Example 4
Subtracting Mixed Numbers with Like Denominators
© 2001 McGraw-Hill Companies
Subtract.
5
7
5
67
41
3
12
12
12
12
26
12
13
6
2
1
6
264
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
CHECK YOURSELF 4
Subtract 8
7
3
5 .
8
8
Again, we must rename the fractions if different denominators are involved. This
approach is shown in Example 5.
Example 5
Subtracting Mixed Numbers with Different Denominators
Subtract.
8
7
3
87
27
3 10
8
10
8
348
135
40
40
213
40
5
Write the fractions with denominator 40.
Subtract as before.
13
40
CHECK YOURSELF 5
Subtract 7
11
5
3 .
12
8
To subtract a mixed number from a whole number, we use the same techniques.
Example 6
Subtracting Mixed Numbers
Subtract.
6
6
24
1
4
Multiply the numerator and
denominator by 4 to form a
common denominator.
3
4
62
3
24
11
4
4
4
Write both the whole number and the
mixed number as improper fractions with
a common denominator.
13
4
3
1
4
CHECK YOURSELF 6
2
Subtract 7 3 .
5
© 2001 McGraw-Hill Companies
NOTE
62
ADDING AND SUBTRACTING MIXED NUMBERS
SECTION 3.4
265
Example 7
An Application of the Subtraction of Mixed Numbers
1
5
Linda was 48 inches (in.) tall on her sixth birthday. By her seventh year she was 51 in.
4
8
tall. How much did she grow during the year?
Because we want the difference in height, we must subtract.
51
5
1
413
193
48 8
4
8
4
413
386
8
8
27
in.
8
3
3 in.
8
3
Linda grew 3 in. during the year.
8
CHECK YOURSELF 7
You use 4
3
yards (yd) of fabric from a 50-yd bolt. How much fabric remains on the
4
© 2001 McGraw-Hill Companies
bolt?
Often we will have to use more than one operation to find the solution to a problem.
Consider Example 8.
Example 8
An Application Involving Mixed Numbers
1
3
A rectangular poster is to have a total length of 12 in. We want a 1 -in. border on the top
4
8
and a 2-in. border on the bottom. What is the length of the printed part of the poster?
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
First, we will draw a sketch of the poster:
3
1 8 in.
12
1
4
in.
?
2 in.
Now, we will use that sketch to find the total width of the top and bottom borders.
1
3
11
16
27
2
in.
8
8
8
8
Now subtract that sum (the top and bottom borders) from the total length of the poster.
1
27
49
27
98
27
12 4
8
4
8
8
8
71
8
7
8 in.
8
7
The length of the printed part is 8 in.
8
CHECK YOURSELF 8
You cut one shelf 3
3
1
feet (ft) long and one 4 ft long from a 12-ft piece of lumber.
4
2
Can you cut another shelf 4 ft long?
CHECK YOURSELF ANSWERS
7
7
5
57
23
171
115
286
16
8
2. 5
3 9
9
10
10
6
10
6
30
30
30
30
15
11
1
11
5
95
29
190
87
103
7
3. 13
4. 3
5. 7
3 4
12
2
12
8
12
8
24
24
24
24
3
3
1
6. 3
7. 45 yd
8. No, only 3 ft is “left over.”
5
4
4
1. 5
© 2001 McGraw-Hill Companies
266
Name
3.4
Exercises
Section
Date
Do the indicated operations.
ANSWERS
2
5
1. 2 3
9
9
3. 2
1
5
5
9
9
2
4
2. 5 6
9
9
4. 1
1
5
5
6
6
1.
2.
3.
5
7
5. 6 4
9
9
8
4
6. 5 4
9
9
1
1
7. 1 2
3
5
1
1
8. 2 1
4
6
4.
5.
9. 2
1
5
1
3 1
4
8
6
10. 3
1
1
1
2 5
5
2
4
6.
7.
8.
11. 3
3
1
3
4 5
5
4
10
12. 4
5
2
5
3 7
6
3
9
13. 7
7
3
3
8
8
14. 3
5
1
1
6
6
9.
10.
11.
15. 3
2
4
1
5
5
16. 5
3
1
2
7
7
12.
13.
2
1
17. 3 2
3
4
© 2001 McGraw-Hill Companies
19. 7
5
11
3
12
18
21. 5 2
23. 3
25. 2
1
4
3
1
3
5 2
4
2
8
3
1
5
2 1
8
4
6
4
1
18. 5 1
5
6
20. 9
3
13
2
7
21
22. 4 1
24. 1
26. 1
2
3
5
5
1
3
2
6
12
4
1
3
4
3
2
15
10
5
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
267
ANSWERS
Solve the following applications.
27.
27. Plumbing. A plumber needs pieces of pipe 15
28.
the total length of the pipe that is needed?
5
3
and 25 inches (in.) long. What is
8
4
29.
28. Postage. Marcus has to figure the postage for sending two packages. One weighs
7
3
3 pounds (lb), and the other weighs 2 lb. What is the total weight?
8
4
30.
1
3
hours (h) on Monday, 5 h on Wednesday and
4
4
1
4 h Friday. What was the total number of hours that he worked?
2
31.
29. Working hours. Franklin worked 2
32.
1
1
1
mi on Sunday, 2 mi on Tuesday, and 3 mi on Friday. How
3
4
2
far did she run during the week?
30. Distance. Robin ran 5
33.
34.
31. Perimeter. Find the perimeter of the figure below.
35.
3
1
1 8 in.
1 4 in.
36.
5
1 8 in.
32. Perimeter. Find the perimeter of the figure below.
5
1
1 8 in.
1 2 in.
1
2 4 in.
33. Consumer purchases. Senta is working on a project that uses three pieces of fabric
1
3 1
5
with lengths , 1 , and yd. She needs to allow for yd of waste. How much fabric
4 4
8
8
should she buy?
1
5
in. thick. We apply in. wallboard and
2
8
1
3
-in. paneling to the inside. Siding that is in. thick is applied to the outside. What
4
4
is the finished thickness of the wall?
3
points on Monday. By closing time Friday, it was
8
3
at 28 . How much did it drop during the week?
4
35. Stocks. A stock was listed at 34
1
3
lb before cooking and 3 lb after cooking. How many
4
8
pounds were lost during the cooking?
36. Cooking. A roast weighed 4
268
© 2001 McGraw-Hill Companies
34. Construction. The framework of a wall is 3
ANSWERS
37. Quantity of material. A roll of paper contains 30
roll, how much paper remains?
1
7
yd. If 16 yd is cut from the
4
8
37.
38.
38. Geometry. Find the missing dimension in the figure below.
39.
3
3 8 in.
40.
?
41.
1
5 4 in.
42.
1
1
in. bolt is placed through a board that is 3 in. thick. How far
4
2
does the bolt extend beyond the board?
39. Carpentry. A 4
43.
1
h
2
3
on Monday and 3 h on Tuesday. How many more hours can he work during the week?
4
40. Working hours. Ben can work 20 h per week on a part-time job. He works 5
41. Geometry. Find the missing dimension in the figure below.
5
8
in.
?
1
5 4 in.
3
square yards (yd2) of carpet for their living room,
4
1
1
15 yd2 for the dining room, and 6 yd2 for a hallway. How much will remain if a
2
4
50 yd2 roll of carpeting is used?
42. Carpeting. The Whites used 20
1 3
2 4
1
3 miles (mi) for the month of July. With their present equipment, they can pave 8 mi
3
in 1 month. How much more work can they take on in July?
© 2001 McGraw-Hill Companies
43. Construction. A construction company has bids for paving roads of 1 , , and
269
ANSWERS
44.
44. Travel. On an 8 h trip, Jack drives 2
left to drive?
3
1
h and Pat drives 2 h. How many hours are
4
2
45.
45. Distance. A runner has told herself that she will run 20 mi each week. She runs
46.
1
1
3
1
5 mi on Sunday, 4 mi on Tuesday, 4 mi on Wednesday, and 2 mi on Friday.
2
4
4
8
How far must she run on Saturday to meet her goal?
47.
48.
1
1
of the space in a landfill and plastic takes up
of
2
10
the space, how much of the landfill is used for other materials?
46. Environment. If paper takes up
1
of the space in a landfill and organic waste takes
2
1
up of the space, how much of the landfill is used for other materials?
8
47. Environment. If paper takes up
3
8
1
was up to 14 %. How much did the interest rate increase over the period?
4
48. Interest. The interest rate on an auto loan in May was 12 %. By September the rate
Answers
1. 5
7
9
3. 7
2
3
5. 11
1
3
7. 3
8
15
9. 7
1
24
11. 13
3
20
1
2
4
7
4
3
5
29
15. 3 1 2 1 1
17. 1
19. 3
2
5
5
5
5
5
12
36
3
7
19
3
1
1
21. 2
23. 6
25. 2
27. 41 in.
29. 12 h
31. 4 in.
4
8
24
8
2
4
3
5
3
3
in.
35. 5 points
37. 13 yd
39.
41. 4 in.
33. 2 yd
4
8
8
4
5
3
3
mi
43. 2
45. 3 mi
47.
12
8
8
270
© 2001 McGraw-Hill Companies
13. 4
Using Your Calculator to Add
and Subtract Mixed Numbers
We have already seen how to add, multiply, and divide fractions using our calculators. Now
we will use our calculators to add and subtract mixed numbers.
Scientific Calculator
To enter a mixed number on a scientific calculator, press the fraction key between both the
7
whole number and the numerator and denominator. For example, to enter 3 , press
12
3 a b/c 7 a b/c 12
Example 1
Adding Mixed Numbers
Add.
3
7
11
2
12
16
The keystroke sequence is
3 a b/c 7 a b/c 12 2 a b/c 11 a b/c 16 The result is 6
13
.
48
Graphing Calculator
As with multiplying and dividing fractions, when using a graphing calculator, you must
choose the fraction option from the math menu before pressing Enter .
7
11
For the problem in Example 1, 3
2 , the keystroke sequence is
12
16
3 7 12 2 11 16
The display will read
Frac
Enter
301
.
48
CHECK YOURSELF 1
© 2001 McGraw-Hill Companies
Find the sum.
4
3
5
4
7
6
CHECK YOURSELF ANSWERS
1. 9
11
42
271
Name
Section
Calculator Exercises
Date
Add or subtract the following.
ANSWERS
1. 3
2
1
2
3
4
2. 6
1
2
8
6
3
3. 5
4
2
2
9
3
4. 2
3
9
4
7
14
3
5
14
8
12
1.
2.
3.
4.
5. 11
2
1
5
3
4
6. 6
7. 14
13
23
22
18
27
8. 82
5.
41
25
97
45
27
6.
7.
9. 4
7
11
2
9
18
10. 7
8
13
4
11
22
11. 5
11
5
2
16
12
12. 18
13. 6
2
5
1
3
6
14. 131
8.
9.
10.
5
3
11
24
40
43
27
99
45
60
11.
15. 10
12.
2
1
2
4 7
3
5
15
16. 7
1
2
1
3 1
5
3
5
13.
14.
Answers
15.
1. 5
11
12
5
13. 4
6
1
9
5. 16
11
12
7. 37
31
54
9. 2
1
6
11. 3
13
48
15. 22
© 2001 McGraw-Hill Companies
16.
3. 8
272