3.4 adding and subtracting mixed · pdf fileadding and subtracting mixed numbers section 3.4...
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Adding and SubtractingMixed Numbers
3.4
3.4 OBJECTIVES
1. Add any two mixed numbers2. Add any group of mixed numbers3. Subtract any two mixed numbers4. Solve an application that involves mixed number
addition or subtraction
261
Once you know how to add fractions, adding mixed numbers should be no problem if youkeep in mind that addition involves combining groups of the same kind of objects. Becausemixed numbers consist of two parts—a whole number and a fraction—we could work withthe whole numbers and the fractions separately. Generally, it is easier to rewrite mixednumbers as improper fractions, then do the addition.
This suggests the following general rule.
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512
56
The sum of the The sum of thefractional parts
�
partswhole-number
�
�
or 3
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.
Step by Step: To Add Mixed Numbers
NOTE Step 2 requires that thefractional parts have the samedenominator.
262 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
Our first example illustrates the use of this rule.
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Example 1
Adding Mixed Numbers
Add and write the result as a mixed numbers.
Rewrite as a mixed number.� 7
3
5
Add the numerators.�38
5
Rewrite as improper fractions.� 16
5�
22
53
1
5� 4
2
5
Example 2
Adding Mixed Numbers with Different Denominators
Add, and write the result as a mixed number.
� 5
13
24
�133
24
Then add as before.�76
24�
57
24
The LCD of the fractions is 24. Renamethem with that denominator.
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19
83
1
6� 2
3
8
C H E C K Y O U R S E L F 1
Add � Write the result as a mixed number.3
410
.2
310
When the fractional portions of the mixed numbers have different denominators, wemust rename these fractions as equivalent fractions with the least common denominator toperform the addition in step 2. Consider Example 2.
C H E C K Y O U R S E L F 2
Add � Write the result as a mixed number.3
56
.5
710
NOTE 5
12013
24B133
You follow the same procedure if more than two mixed numbers are involved in theproblem.
ADDING AND SUBTRACTING MIXED NUMBERS SECTION 3.4 263
Adding Mixed Numbers with Different Denominators
Add.
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40
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Example 3
NOTE The LCD of the threefractions is 40. Convert toequivalent fractions.
C H E C K Y O U R S E L F 3
Add � � 3
34
.4
23
5
12
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.
Step by Step: To Subtract Mixed Numbers
We can use a similar technique for subtracting mixed numbers. The rule is similar to thatstated earlier for adding mixed numbers.
Example 4 illustrates the use of this rule.
Subtracting Mixed Numbers with Like Denominators
Subtract.
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1
6
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6
�26
12
�67
12�
41
12 5
7
12� 3
5
12
Example 4
264 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
Again, we must rename the fractions if different denominators are involved. Thisapproach is shown in Example 5.
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C H E C K Y O U R S E L F 4
Subtract � 5
38
.8
78
Example 5
Subtracting Mixed Numbers with Different Denominators
Subtract.
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13
40
�213
40
Subtract as before.�348
40�
135
40
Write the fractions with denominator 40. �87
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27
8 8
7
10� 3
3
8
C H E C K Y O U R S E L F 5
Subtract � 3
58
.7
1112
To subtract a mixed number from a whole number, we use the same techniques.
Example 6
Subtracting Mixed Numbers
Subtract.
� 3
1
4
�13
4
Write both the whole number and themixed number as improper fractions witha common denominator.
6 � 2
3
4�
24
4�
11
4
6 � 2
3
4
C H E C K Y O U R S E L F 6
Subtract 7 � 3
25
.
NOTE
Multiply the numerator anddenominator by 4 to form acommon denominator.
6 �61
�244
ADDING AND SUBTRACTING MIXED NUMBERS SECTION 3.4 265
An Application of the Subtraction of Mixed Numbers
Linda was inches (in.) tall on her sixth birthday. By her seventh year she was in.
tall. How much did she grow during the year?
Because we want the difference in height, we must subtract.
Linda grew in. during the year.3
3
8
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3
8 in.
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8 in.
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Example 7
C H E C K Y O U R S E L F 7
You use yards (yd) of fabric from a 50-yd bolt. How much fabric remains on the
bolt?
4
34
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
A rectangular poster is to have a total length of in. We want a -in. border on the top
and a 2-in. border on the bottom. What is the length of the printed part of the poster?
1
3
812
1
4
266 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
First, we will draw a sketch of the poster:
Now, we will use that sketch to find the total width of the top and bottom borders.
Now subtract that sum (the top and bottom borders) from the total length of the poster.
The length of the printed part is in.8
7
8
� 8
7
8 in.
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8 in.
83
in.1
12
2 in.
in.41 ?
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C H E C K Y O U R S E L F 8
You cut one shelf feet (ft) long and one ft long from a 12-ft piece of lumber.
Can you cut another shelf 4 ft long?
4
12
3
34
C H E C K Y O U R S E L F A N S W E R S
1. 2.
3. 4. 5.
6. 7. 8. No, only ft is “left over.”33
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24�
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30�
115
30�
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30� 9
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8
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Exercises
Do the indicated operations.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26. 1
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3.4
Name
Section Date
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15. 16.
17. 18.
19. 20.
21. 22.
23. 24.
25. 26.
267
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Solve the following applications.
27. Plumbing. A plumber needs pieces of pipe and inches (in.) long. What is
the total length of the pipe that is needed?
28. Postage. Marcus has to figure the postage for sending two packages. One weighs
pounds (lb), and the other weighs lb. What is the total weight?
29. Working hours. Franklin worked hours (h) on Monday, h on Wednesday and
h Friday. What was the total number of hours that he worked?
30. Distance. Robin ran mi on Sunday, mi on Tuesday, and mi on Friday. How
far did she run during the week?
31. Perimeter. Find the perimeter of the figure below.
32. Perimeter. Find the perimeter of the figure below.
33. Consumer purchases. Senta is working on a project that uses three pieces of fabric
with lengths and yd. She needs to allow for yd of waste. How much fabric
should she buy?
34. Construction. The framework of a wall is in. thick. We apply in. wallboard and
-in. paneling to the inside. Siding that is in. thick is applied to the outside. What
is the finished thickness of the wall?
35. Stocks. A stock was listed at points on Monday. By closing time Friday, it was
at . How much did it drop during the week?
36. Cooking. A roast weighed lb before cooking and lb after cooking. How many
pounds were lost during the cooking?
3
3
84
1
4
28
3
4
34
3
8
3
4
1
4
5
83
1
2
1
8
5
8
3
4, 1
1
4,
21
41
in.
in.
1
2
85
in.1
41
85
in.
in.
1
1
83 in.1
3
1
22
1
45
1
3
4
1
2
5
3
42
1
4
2
3
43
7
8
25
3
415
5
8
ANSWERS
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
268
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37. Quantity of material. A roll of paper contains yd. If yd is cut from theroll, how much paper remains?
38. Geometry. Find the missing dimension in the figure below.
39. Carpentry. A in. bolt is placed through a board that is in. thick. How far
does the bolt extend beyond the board?
40. Working hours. Ben can work 20 h per week on a part-time job. He works h
on Monday and h on Tuesday. How many more hours can he work during the week?
41. Geometry. Find the missing dimension in the figure below.
42. Carpeting. The Whites used square yards (yd2) of carpet for their living room,
yd2 for the dining room, and yd2 for a hallway. How much will remain if a
50 yd2 roll of carpeting is used?
43. Construction. A construction company has bids for paving roads of and
miles (mi) for the month of July. With their present equipment, they can pave 8 mi
in 1 month. How much more work can they take on in July?
3
1
3
1
1
2,
3
4,
6
1
415
1
2
20
3
4
?
in.85
in.415
3
3
4
5
1
2
3
1
24
1
4
3
in.415
?
in.83
16
7
830
1
4
ANSWERS
37.
38.
39.
40.
41.
42.
43.
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44. Travel. On an 8 h trip, Jack drives h and Pat drives h. How many hours areleft to drive?
45. Distance. A runner has told herself that she will run 20 mi each week. She runs
mi on Sunday, mi on Tuesday, mi on Wednesday, and mi on Friday.
How far must she run on Saturday to meet her goal?
46. Environment. If paper takes up of the space in a landfill and plastic takes up of
the space, how much of the landfill is used for other materials?
47. Environment. If paper takes up of the space in a landfill and organic waste takes
up of the space, how much of the landfill is used for other materials?
48. Interest. The interest rate on an auto loan in May was %. By September the rate
was up to %. How much did the interest rate increase over the period?
Answers
1. 3. 5. 7. 9. 11.
13. 15. 17. 19.
21. 23. 25. 27. 29. 31.
33. yd 35. points 37. yd 39. 41. 4 in.
43. 45. 47. 3
83
3
8 mi2
5
12 mi
3
4 in.13
3
85
5
82
3
4
4
1
4 in.12
1
2 h41
3
8 in.2
19
246
7
82
3
4
3
29
361
5
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2
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4
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7
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8
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37
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14
1
4
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8
1
8
1
2
1
10
1
2
2
1
84
3
44
1
45
1
2
2
1
22
3
4
ANSWERS
44.
45.
46.
47.
48.
270
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Using Your Calculator to Addand Subtract Mixed Numbers
We have already seen how to add, multiply, and divide fractions using our calculators. Nowwe 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
whole number and the numerator and denominator. For example, to enter , press
3 7 12a b/ca b/c
3
7
12
271
Example 1
Adding Mixed Numbers
Add.
The keystroke sequence is
3 7 12 2 11 16
The result is
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 .
For the problem in Example 1, the keystroke sequence is
3 7 12 2 11 16
The display will read 301
48.
Enter�Frac���
3
7
12� 2
11
16,
Enter
6
13
48.
�a b/ca b/c�a b/ca b/c
3
7
12� 2
11
16
C H E C K Y O U R S E L F 1
Find the sum.
4
3
7�4
5
6
C H E C K Y O U R S E L F A N S W E R S
1. 9
11
42
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Name
Section Date
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
272
Calculator ExercisesAdd or subtract the following.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
Answers
1. 3. 5. 7. 9. 11.
13. 15. 224
5
6
3
13
482
1
637
31
5416
11
128
1
95
11
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7
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27
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