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Adding and Subtracting Fractionswith Unlike Denominators
3.3
3.3 OBJECTIVES
1. Find the LCD of two fractions2. Find the LCD of a group of fractions3. Add any two fractions4. Add any group of fractions5. Subtract any two fractions
243
Step 1 Write the prime factorization for each of the denominators.Step 2 Find all the prime factors that appear in any one of the prime
factorizations.Step 3 Form the product of those prime factors, using each factor the greatest
number of times it occurs in any one factorization.
Step by Step: To Find the Least Common Denominator
In Section 3.1, you dealt with like fractions (fractions with a common denominator). What
about a sum that deals with unlike fractions, such as ?
To add unlike fractions, write them as equivalent fractions with a common denominator. Inthis case, let’s use 12 as the denominator.
Any common multiple of the denominators will work in forming equivalent fractions.
For instance, we can write as and as Our work is simplest, however, if we use the
smallest possible number for the common denominator. This is called the least commondenominator (LCD).
The LCD is the least common multiple of the denominators of the fractions. This is the
smallest number that is a multiple of all the denominators. For example, the LCD for and
is 12, not 24.1
4
1
3
6
24.
1
4
8
24
1
3
or � �31
124
or41
123
127
31
� � ?
41
1
3�
1
4
We cannot add unlike fractionsbecause they have different denominators.
We have chosen 12 because itis a multiple of 3 and 4.
is equivalent to
is equivalent to 3
12.
14
412
.13
NOTE Only like fractions canbe added.
NOTE We can now addbecause we have like fractions.
NOTE This is virtually identicalto the Step by Step on page 235for finding the LCM.
244 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
We are now ready to add unlike fractions. In this case, the fractions must be renamed asequivalent fractions that have the same denominator. We will use the following rule.
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Step 1 Find the LCD of the fractions.Step 2 Change each unlike fraction to an equivalent fraction with the LCD as a
common denominator.Step 3 Add the resulting like fractions as before.
Step by Step: To Add Unlike Fractions
Our first example shows the use of this rule.
Example 1
Adding Unlike Fractions
Add the fractions and
Step 1 We find that the LCD for fractions with denominators of 6 and 8 is 24.
Step 2 Convert the fractions so that they have the denominator 24.
Step 3 We can now add the equivalent like fractions.
1
6�
3
8�
4
24�
9
24�
13
24
9
24
3
8
4
24
1
6
3
8.
1
6
How many sixes are in 24? Thereare 4. So multiply the numeratorand denominator by 4.
How many eights are in 24? Thereare 3. So multiply the numeratorand denominator by 3.
Add the numerators and place thatsum over the common denominator.
�4
�4
�
�3
�3
�
C H E C K Y O U R S E L F 1
Add.
3
5�
1
3
Here is a similar example. Remember that the sum should always be written in simplestform.
NOTE See Section 3.2 if youwish to review how we arrivedat 24.
ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS SECTION 3.3 245
Adding Unlike Fractions that Require Simplifying
Add the fractions and
Step 1 The LCD for fractions with denominators of 10 and 15 is 30.
Step 2
Step 3
�25
30�
5
6
7
10�
2
15�
21
30�
4
30
2
15�
4
30
7
10�
21
30
2
15.
7
10
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Example 2
Do you see how the equivalentfractions are formed?
Add the resulting like fractions. Be surethe sum is in simplest form.
C H E C K Y O U R S E L F 2
Add.
1
6�
7
12
We can easily add more than two fractions by using the same procedure. Example 3illustrates this approach.
Example 3
Adding a Group of Unlike Fractions
Add
Step 1 The LCD is 90.
Step 2
Step 3
� 129
90
75
90�
20
90�
24
90�
119
90
4
15�
24
90
2
9�
20
90
5
6�
75
90
5
6�
2
9�
4
15.
Multiply the numerator anddenominator by 15.
Multiply the numerator anddenominator by 10.
Multiply the numerator anddenominator by 6.
Now add.
Remember, if the sum is animproper fraction, it should bechanged to a mixed number.
NOTE Go back and review ifyou need to.
246 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
Many of the measurements you deal with in everyday life involve fractions. Let’s lookat some typical situations.
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C H E C K Y O U R S E L F 3
Add.
2
5�
3
8�
7
20
Example 4
An Application Involving the Addition of Unlike Fractions
Jack ran mi on Monday, mi on Wednesday, and mi on Friday. How far did he run
during the week?
The three distances that Jack ran are the given information in the problem. We want to finda total distance, so we must add for the solution.
Jack ran mi during the week.111
12
�23
12� 1
11
12 mi
1
2�
2
3�
3
4�
6
12�
8
12�
9
12
3
4
2
3
1
2
C H E C K Y O U R S E L F 4
Susan is designing an office complex. She needs acre for buildings, acre for
driveways and parking, and acre for walks and landscaping. How much land
does she need?
16
13
25
Because we have no commondenominator, we must convertto equivalent fractions beforewe can add.
ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS SECTION 3.3 247
An Application Involving the Addition of Unlike Fractions
Sam bought three packages of spices weighing and pounds (lb). What was the totalweight?
We need to find the total weight, so we must add.
Write each fraction with the denominator 8.
The total weight was 13
8 lb.
�11
8� 1
3
8 lb
1
4�
5
8�
1
2�
2
8�
5
8�
4
8
lb14
lb58
lb12
1
2
5
8,
1
4,
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Example 5
C H E C K Y O U R S E L F 5
For three different recipes, Max needs and gallons (gal) tomato sauce. How
many gallons should he buy altogether?
58
12
,38
,
To subtract unlike fractions, which are fractions that do not have the same denominator,we have the following rule:
Step 1 Find the LCD of the fractions.Step 2 Change each unlike fraction to an equivalent fraction with the LCD as a
common denominator.Step 3 Subtract the resulting like fractions as before.
Step by Step: To Subtract Unlike Fractions
Example 6
Subtracting Unlike Fractions
Subtract 5
8�
1
6.
NOTE The abbreviation forpounds is “lb” from the Latinlibra, meaning “balance” or“scales.”
NOTE Of course, this is thesame as our rule for addingfractions. We just subtractinstead of add!
248 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
Step 1 The LCD is 24.
Step 2 Convert the fractions so that they have the common denominator 24.
Step 3 Subtract the equivalent like fractions.
Be Careful! You cannot subtract the numerators and subtract the denominators.
5
8�
1
6 is not
4
2
5
8�
1
6�
15
24�
4
24�
11
24
1
6�
4
24
5
8�
15
24
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C H E C K Y O U R S E L F 6
Subtract.
7
10�
1
4
The difference of two fractions should always be written in simplest form. Let’s look atan example that applies our work in subtracting unlike fractions.
Example 7
An Application Involving the Subtraction of Unlike Fractions
You have yards (yd) of a handwoven linen. A pattern for a placemat calls for yd. Will
you have enough left for two napkins that will use yd?
First, find out how much fabric is left over after the placemat is made.
Now compare the size of and
3
8 yd �
9
24 yd and
1
3 yd �
8
24 yd
3
8.
1
3
7
8 yd �
1
2 yd �
7
8 yd �
4
8 yd �
3
8 yd
1
3
1
2
7
8
NOTE Remember that yd is
left over and that yd
is needed.
13
38
The first two steps are exactly thesame as if we were adding.
C A U T I O N
NOTE You can use yourcalculator to check your result.
ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS SECTION 3.3 249
Because yd is more than the yd that is needed, there is enough material for the place-
mat and two napkins.
1
3
3
8
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C H E C K Y O U R S E L F 7
A concrete walk will require cubic yard (yd3) of concrete. If you have mixed yd3,
will enough concrete remain to do a project that will use yd3?16
89
34
Our next application involves measurement in inches. Note that on a ruler or yardstick,
the marks divide each inch into -in., -in., and -in. sections, and on some rulers, -in.
sections. We will use denominators of 2, 4, 8, and 16 in our measurement applications.
1
16
1
8
1
4
1
2
Example 8
An Application Involving the Subtraction of Unlike Fractions
Alexei is cutting two slats that are each to be in. in width from a piece of wood that is
in. across. How much will be left?
The two in. pieces will total
The remaining strip will be in. wide.3
8
6
8�
3
8�
3
8
3
4�
6
8
2 �3
16�
6
16�
3
8 in.
3
16
3
4
3
16
C H E C K Y O U R S E L F 8
Ricardo is cutting three strips from a piece of metal with a width of 1 in. Each strip
has a width of in. How much metal will remain after the cuts?316
250 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
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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. yd3 will remain. You do not have enough concrete for both projects.
8. in.7
16
5
36
9
201
1
2 gal
9
10 acre1
1
8
1
6�
7
12�
2
12�
7
12�
9
12�
3
4
14
15
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Exercises
Find the least common denominator (LCD) for fractions with the given denominators.
1. 3 and 4 2. 3 and 5
3. 4 and 8 4. 6 and 12
5. 9 and 27 6. 10 and 30
7. 8 and 12 8. 15 and 40
9. 14 and 21 10. 15 and 20
11. 48 and 80 12. 60 and 84
13. 3, 4, and 5 14. 3, 4, and 6
15. 8, 10, and 15 16. 6, 22, and 33
17. 5, 10, and 25 18. 8, 24, and 48
Add.
19. 20.
21. 22.
23. 24.
25. 26.
27. 28.
29. 30.
31. 32.
33. 34.
35. 36.1
3�
5
12�
4
5
1
9�
7
12�
5
8
2
3�
1
4�
3
8
1
5�
7
10�
4
15
5
12�
3
10
5
8�
1
12
3
10�
3
8
7
15�
2
35
7
20�
9
40
3
7�
3
14
1
6�
2
15
1
7�
3
5
4
5�
1
10
3
4�
1
8
1
3�
1
18
1
5�
3
10
3
5�
1
3
2
3�
1
4
3.3
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.
27. 28.
29. 30.
31.
32.
33.
34.
35.
36.
251
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Subtract.
37. 38.
39. 40.
41. 42.
43. 44.
Perform the following operations.
45. 46.
47. 48.
Solve the following applications.
49. Consumer buying. Paul bought pounds (lb) of peanuts and lb of cashews. Howmany pounds of nuts did he buy?
50. Countertop thickness. A countertop consists of a board inches (in.) thick and tile
in. thick. What is the overall thickness?
51. Budgets. Amy budgets of her income for housing and of her income for food.
What fraction of her income is budgeted for these two purposes? What fraction of herincome remains?
52. Daily schedule. A person spends day at work and day sleeping. What fraction of
a day do these two activities use? What fraction of the day remains?
1
3
3
8
1
6
2
5
3
8
3
4
3
8
1
2
9
10�
1
5�
1
2
15
16�
5
8�
1
4
13
24�
5
16�
3
8
33
40�
7
24�
11
30
13
15�
11
20
5
12�
3
8
9
10�
4
5
3
8�
1
4
5
6�
2
7
11
15�
3
5
7
9�
1
6
4
5�
1
3
ANSWERS
37.
38.
39.
40.
41.
42.
43.
44.
45.
46.
47.
48.
49.
50.
51.
52.
252
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53. Distance. Jose walked miles (mi) to the store, mi to a friend’s house, and then
mi home. How far did he walk?
54. Perimeter. Find the perimeter of, or the distance around, the accompanying figure.
55. Budgeting. A budget guide states that you should spend of your salary for
housing, for food, for clothing, and for transportation. What total portion of
your salary will these four expenses account for?
56. Salary. Deductions from your paycheck are made roughly as follows: for federal
tax, for state tax, for social security, and for a savings withholding plan.
What portion of your pay is deducted?
For exercises 57 and 58, find the missing dimension (?) in the given figure.
57.
58.
41
?
in.
in.3217
?in.
in.
7
43
16
1
40
1
20
1
20
1
8
1
8
1
16
3
16
1
4
21
85
43
in.
in.
in.
2
3
1
2
3
4
ANSWERS
53.
54.
55.
56.
57.
58.
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Solve the following applications.
59. Cooking. A hamburger that weighed pound (lb) before cooking weighed lb
after cooking. How much weight was lost in cooking?
60. Property. Martin owned a -acre piece of land. He sold acre. What amount of
land remains?
61. Painting. On Monday, of a house painting project remained to be done. John
painted of the house on Tuesday and of the house on Wednesday. What portion
of the project remained to be done?
62. Baking. Geraldo has cup of flour. Biscuits use cup. Will he have enough left
over for a small pie crust that requires cup?
63. Painting. You have gallons (gal) of paint. You estimate that one wall will use
gal. Can you also finish a smaller wall that will need gal?1
4
1
2
5
6
1
4
5
8
3
4
3
16
1
4
7
8
1
3
7
8
3
16
1
4
ANSWERS
59.
60.
61.
62.
63.
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64. A teenager makes a log of his activities during the course of a year.
ANSWERS
64.
65.
66.
67.
68.
255
Activity Time Fraction of the Year
Sleeping 10 hours per dayEating 2 hours per dayWeekend activities 2 days per weekSummer vacation 3 months per year
He uses these data to claim that he has no time left over for school.
(a) Complete the above chart.
(b) What is the flaw in the teenager’s reasoning?
65. Manny, Moe, and Jack each have equal shares in a automotive store. Moe decides to
retire and sell his shares. He sells of his shares to Manny and the remainder to Jack.
What is Jack’s share of the store now?
66. Measure the length of an unstretched rubber band to the nearest eighth of an inch.Then stretch the band as far as you can (without breaking it), and again measure thelength to the nearest eighth of an inch.
(a) What is the difference between the lengths of the stretched and unstretchedrubber band?
(b) Repeat this process for other rubber bands of different thicknesses. What is therelationship between the thickness of the rubber band and the distance it can bestretched?
67. A door is ft wide. Two hooks are to be attached to the door so that they are in.
apart and the same distance from each edge. How far from the edge of the door
should each hook be located?
68. Complete the following:
Based on these results, predict the answer to the following:
Now, do the addition, and check your prediction.
1
2�
1
4�
1
8�
1
16�
1
32� ________
1
2�
1
4�
1
8�
1
16� ________.
1
2�
1
4�
1
8� ________.
1
2�
1
4� ________.
11
24
1
4
1
4
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Answers1. 12 3. 8 5. 277. 8 � 2 � 2 � 2; 12 � 2 � 2 � 3; The LCD is 2 � 2 � 2 � 3 � 24 9. 4211. 240 13. 60 15. 120 17. 50
19. 21. 23. 25. 27. 29. 31.
33. 35.
37. 39. 41. 43.
45. 47. 49.
51. 53. 55. 57. 59. 61.
63. Yes— gal remains 65. 67. 2 ft or 21
16 ft
3
4 in.
7
12
1
3
7
16
1
16 lb
5
16 in.
5
81
11
12 mi
17
30,
13
30
7
8 lb1
5
16
33
40�
7
24�
11
30�
99
120�
35
120�
44
120�
108
120�
9
10
1
24
1
8
2
15
4
5�
1
3�
12
15�
5
15�
7
15
123
72
1
5�
7
10�
4
15�
6
30�
21
30�
8
30�
35
30� 1
5
30� 1
1
6
17
24
11
21
9
14
26
35
7
8
1
2
11
12
256
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Using Your Calculator to Addand Subtract Fractions
Adding or subtracting fractions on the calculator is very much like the multiplicationand division you did in the previous chapter. The only thing that changes is theoperation.
Scientific Calculator
Here’s where the fraction calculator is a great tool for checking your work. No muss, nofuss, no searching for a common denominator. Just enter the fractions and get the rightanswer!
257
Example 1
Adding Fractions
Find the sum or difference.
(a)
The keystroke sequence is
3 14 7 12
The result is .
(b)
5 8 7 18
The result is .17
72
�a b/c�a b/c
5
8�
7
18
67
84
�a b/c�a b/c
3
14�
7
12
Graphing Calculator
Use your graphing calculator to find the sum.
7
24�
17
42
258 CHAPTER 3 ADDING AND SUBTRACTING FRACTIONS
The keystroke sequence is
7 24 17 42
The result is .39
56
Enter1: � Frac���
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C H E C K Y O U R S E L F A N S W E R
1. (a) ; (b)16
33
59
72
C H E C K Y O U R S E L F 1
Find the sum or difference.
(a) (b)9
11�
1
3
5
24�
11
18
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Calculator Exercises
Find the following sums or differences using your calculator.
1. 2.
3. 4.
5. 6.
7. 8.
9. 10.
11. 12.
13. 14.
15. 16.
17. 18.
19. 20.8
9�
3
4
5
8�
3
5
11
13�
2
3
4
9�
2
5
31
43�
18
53
15
17�
9
11
5
8�
4
9
11
18�
5
12
7
15�
2
5
8
9�
6
7
7
15�
17
24
1
10�
7
12
7
16�
9
24
8
15�
7
12
2
7�
3
8
5
12�
1
6
7
11�
5
6
3
4�
7
9
11
12�
5
6
2
3�
1
2
Name
Section Date
ANSWERS
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
16.
17.
18.
19. 20.
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Answers
1. 3. 5. 7. 9.
11. 13. 15. 17. 19.1
40
2
45
12
187
37
36 or 1
1
36
110
63 or 1
47
60
41
60
67
60 or 1
7
60
7
12
55
36 or 1
19
36
7
6 or 1
1
6
260260
