3.3
Adding and Subtracting Fractions
with Unlike Denominators
3.3
OBJECTIVES
1.
2.
3.
4.
5.
Find the LCD of two fractions
Find the LCD of a group of fractions
Add any two fractions
Add any group of fractions
Subtract any two fractions
In Section 3.1, you dealt with like fractions (fractions with a common denominator). What
1
1
about a sum that deals with unlike fractions, such as ?
3
4
NOTE Only like fractions can
be added.
?
We cannot add unlike fractions
because they have different
denominators.
1
4
1
3
To add unlike fractions, write them as equivalent fractions with a common denominator. In
this case, let’s use 12 as the denominator.
NOTE We can now add
because we have like fractions.
4
1
or
12
3
3
1
or
12
4
7
12
We have chosen 12 because it
is a multiple of 3 and 4.
1
4
is equivalent to .
3
12
3
1
is equivalent to .
4
12
© 2001 McGraw-Hill Companies
Any common multiple of the denominators will work in forming equivalent fractions.
1
8
1
6
For instance, we can write as
and as . Our work is simplest, however, if we use the
3 24
4 24
smallest possible number for the common denominator. This is called the least common
denominator (LCD).
The LCD is the least common multiple of the denominators of the fractions. This is the
1
smallest number that is a multiple of all the denominators. For example, the LCD for and
3
1
is 12, not 24.
4
NOTE This is virtually identical
to the Step by Step on page 235
for finding the LCM.
Step by Step: To Find the Least Common Denominator
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.
243
244
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
We are now ready to add unlike fractions. In this case, the fractions must be renamed as
equivalent fractions that have the same denominator. We will use the following rule.
Step by Step: To Add Unlike Fractions
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.
Our first example shows the use of this rule.
Example 1
Adding Unlike Fractions
Add the fractions
1
3
and .
6
8
NOTE See Section 3.2 if you
Step 1 We find that the LCD for fractions with denominators of 6 and 8 is 24.
wish to review how we arrived
at 24.
Step 2 Convert the fractions so that they have the denominator 24.
4
1
6
4
24
How many sixes are in 24? There
are 4. So multiply the numerator
and denominator by 4.
9
24
How many eights are in 24? There
are 3. So multiply the numerator
and denominator by 3.
4
3
3
8
3
Step 3 We can now add the equivalent like fractions.
Add the numerators and place that
sum over the common denominator.
CHECK YOURSELF 1
Add.
3
1
5
3
Here is a similar example. Remember that the sum should always be written in simplest
form.
© 2001 McGraw-Hill Companies
3
4
9
13
1
6
8
24
24
24
ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS
SECTION 3.3
245
Example 2
Adding Unlike Fractions that Require Simplifying
Add the fractions
7
2
and .
10
15
Step 1 The LCD for fractions with denominators of 10 and 15 is 30.
Step 2
7
21
10
30
Do you see how the equivalent
fractions are formed?
2
4
15
30
Step 3
2
21
4
7
10
15
30
30
5
25
30
6
Add the resulting like fractions. Be sure
the sum is in simplest form.
CHECK YOURSELF 2
Add.
1
7
6
12
We can easily add more than two fractions by using the same procedure. Example 3
illustrates this approach.
Example 3
Adding a Group of Unlike Fractions
Add
NOTE Go back and review if
you need to.
5
2
4
.
6
9
15
Step 1 The LCD is 90.
© 2001 McGraw-Hill Companies
Step 2
Step 3
5
75
6
90
Multiply the numerator and
denominator by 15.
2
20
9
90
Multiply the numerator and
denominator by 10.
4
24
15
90
Multiply the numerator and
denominator by 6.
75
20
24
119
90
90
90
90
1
29
90
Now add.
Remember, if the sum is an
improper fraction, it should be
changed to a mixed number.
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
CHECK YOURSELF 3
Add.
2
3
7
5
8
20
Many of the measurements you deal with in everyday life involve fractions. Let’s look
at some typical situations.
Example 4
An Application Involving the Addition of Unlike Fractions
1
2
3
mi on Monday, mi on Wednesday, and mi on Friday. How far did he run
2
3
4
during the week?
Jack ran
The three distances that Jack ran are the given information in the problem. We want to find
a total distance, so we must add for the solution.
1
2
3
6
8
9
2
3
4
12
12
12
Jack ran 1
23
11
1 mi
12
12
Because we have no common
denominator, we must convert
to equivalent fractions before
we can add.
11
mi during the week.
12
CHECK YOURSELF 4
Susan is designing an office complex. She needs
driveways and parking, and
does she need?
1
2
acre for buildings, acre for
5
3
1
acre for walks and landscaping. How much land
6
© 2001 McGraw-Hill Companies
246
ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS
SECTION 3.3
247
Example 5
An Application Involving the Addition of Unlike Fractions
1 5
1
Sam bought three packages of spices weighing , , and pounds (lb). What was the total
4 8
2
weight?
5
8 lb
1 lb
2
1 lb
4
We need to find the total weight, so we must add.
NOTE The abbreviation for
pounds is “lb” from the Latin
libra, meaning “balance” or
“scales.”
1
5
1
2
5
4
4
8
2
8
8
8
Write each fraction with the denominator 8.
11
3
1 lb
8
8
3
The total weight was 1 lb.
8
CHECK YOURSELF 5
3 1
5
For three different recipes, Max needs , , and gallons (gal) tomato sauce. How
8 2
8
many gallons should he buy altogether?
To subtract unlike fractions, which are fractions that do not have the same denominator,
we have the following rule:
Step by Step: To Subtract Unlike Fractions
© 2001 McGraw-Hill Companies
NOTE Of course, this is the
same as our rule for adding
fractions. We just subtract
instead of add!
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.
Example 6
Subtracting Unlike Fractions
Subtract
1
5
.
8
6
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.
5
15
8
24
4
1
6
24
NOTE You can use your
calculator to check your result.
The first two steps are exactly the
same as if we were adding.
Step 3 Subtract the equivalent like fractions.
5
1
15
4
11
8
6
24
24
24
Be Careful! You cannot subtract the numerators and subtract the denominators.
CAUTION
1
5
8
6
is not
4
2
CHECK YOURSELF 6
Subtract.
7
1
10
4
The difference of two fractions should always be written in simplest form. Let’s look at
an example that applies our work in subtracting unlike fractions.
Example 7
An Application Involving the Subtraction of Unlike Fractions
7
1
yards (yd) of a handwoven linen. A pattern for a placemat calls for yd. Will
8
2
1
you have enough left for two napkins that will use yd?
3
First, find out how much fabric is left over after the placemat is made.
7
1
7
4
3
yd yd yd yd yd
8
2
8
8
8
NOTE Remember that
left over and that
is needed.
1
yd
3
3
yd is
8
Now compare the size of
9
3
yd yd
8
24
and
1
3
and .
3
8
1
8
yd yd
3
24
© 2001 McGraw-Hill Companies
You have
ADDING AND SUBTRACTING FRACTIONS WITH UNLIKE DENOMINATORS
SECTION 3.3
249
3
1
yd is more than the yd that is needed, there is enough material for the place8
3
mat and two napkins.
Because
CHECK YOURSELF 7
8
3
cubic yard (yd 3) of concrete. If you have mixed yd 3,
4
9
1
will enough concrete remain to do a project that will use yd 3?
6
A concrete walk will require
Our next application involves measurement in inches. Note that on a ruler or yardstick,
1
1
1
1
the marks divide each inch into -in., -in., and -in. sections, and on some rulers, -in.
2
4
8
16
sections. We will use denominators of 2, 4, 8, and 16 in our measurement applications.
Example 8
An Application Involving the Subtraction of Unlike Fractions
3
Alexei is cutting two slats that are each to be
in. in width from a piece of wood that is
16
3
in. across. How much will be left?
4
The two
2
3
in. pieces will total
16
3
6
3
in.
16
16
8
3
6
4
8
© 2001 McGraw-Hill Companies
6
3
3
8
8
8
The remaining strip will be
3
in. wide.
8
CHECK YOURSELF 8
Ricardo is cutting three strips from a piece of metal with a width of 1 in. Each strip
has a width of
3
in. How much metal will remain after the cuts?
16
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
CHECK YOURSELF ANSWERS
14
1
7
2
7
9
3
1
9
acre
2. 3. 1
4.
15
6
12
12
12
12
4
8
10
1
9
5. 1 gal
6.
2
20
5
7.
yd3 will remain. You do not have enough concrete for both projects.
36
7
8.
in.
16
1.
© 2001 McGraw-Hill Companies
250
Name
3.3
Exercises
Section
Date
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
1.
2.
6. 10 and 30
3.
4.
7. 8 and 12
8. 15 and 40
5.
6.
9. 14 and 21
10. 15 and 20
7.
8.
11. 48 and 80
12. 60 and 84
9.
10.
11.
12.
13. 3, 4, and 5
14. 3, 4, and 6
13.
14.
15. 8, 10, and 15
16. 6, 22, and 33
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
29.
30.
17. 5, 10, and 25
18. 8, 24, and 48
Add.
2
1
19.
3
4
3
1
20.
5
3
21.
1
3
5
10
22.
1
1
3
18
23.
3
1
4
8
24.
4
1
5
10
25.
1
3
7
5
26.
2
1
6
15
27.
3
3
7
14
28.
7
9
20
40
7
2
29.
15
35
© 2001 McGraw-Hill Companies
ANSWERS
3
3
30.
10
8
31.
5
1
8
12
32.
5
3
12
10
33.
1
7
4
5
10
15
34.
2
1
3
3
4
8
31.
32.
33.
34.
35.
36.
1
7
5
35.
9
12
8
1
5
4
36.
3
12
5
251
ANSWERS
Subtract.
37.
37.
4
1
5
3
38.
7
1
9
6
39.
11
3
15
5
40.
5
2
6
7
41.
3
1
8
4
42.
9
4
10
5
43.
5
3
12
8
44.
11
13
15
20
38.
39.
40.
41.
42.
43.
Perform the following operations.
44.
45.
33
7
11
40
24
30
46.
13
5
3
24
16
8
47.
15
5
1
16
8
4
48.
9
1
1
10
5
2
45.
46.
47.
Solve the following applications.
48.
49. Consumer buying. Paul bought
many pounds of nuts did he buy?
1
3
pounds (lb) of peanuts and lb of cashews. How
2
8
49.
50. Countertop thickness. A countertop consists of a board
50.
3
in. thick. What is the overall thickness?
8
3
inches (in.) thick and tile
4
51.
2
1
of her income for housing and of her income for food.
5
6
What fraction of her income is budgeted for these two purposes? What fraction of her
income remains?
52.
1
3
day at work and day sleeping. What fraction of
8
3
a day do these two activities use? What fraction of the day remains?
52. Daily schedule. A person spends
252
© 2001 McGraw-Hill Companies
51. Budgets. Amy budgets
ANSWERS
3
1
miles (mi) to the store, mi to a friend’s house, and then
4
2
2
mi home. How far did he walk?
3
53. Distance. Jose walked
54. Perimeter. Find the perimeter of, or the distance around, the accompanying figure.
53.
54.
55.
56.
1
2
5
8
in.
in.
57.
3
4
in.
58.
1
of your salary for
4
3
1
1
housing,
for food,
for clothing, and for transportation. What total portion of
16
16
8
your salary will these four expenses account for?
55. Budgeting. A budget guide states that you should spend
1
for federal
8
1
1
1
tax,
for state tax,
for social security, and
for a savings withholding plan.
20
20
40
What portion of your pay is deducted?
56. Salary. Deductions from your paycheck are made roughly as follows:
For exercises 57 and 58, find the missing dimension (?) in the given figure.
© 2001 McGraw-Hill Companies
57.
7
16 in.
?
3
4
in.
58.
?
17
in.
32
1
4
in.
253
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?
7
of a house painting project remained to be done. John
8
1
3
painted of the house on Tuesday and
of the house on Wednesday. What portion
4
16
of the project remained to be done?
61. Painting. On Monday,
3
5
cup of flour. Biscuits use cup. Will he have enough left
4
8
1
over for a small pie crust that requires cup?
4
5
gallons (gal) of paint. You estimate that one wall will use
6
1
1
gal. Can you also finish a smaller wall that will need gal?
2
4
63. Painting. You have
© 2001 McGraw-Hill Companies
62. Baking. Geraldo has
ANSWERS
64. A teenager makes a log of his activities during the course of a year.
64.
Activity
Time
Sleeping
Eating
Weekend activities
Summer vacation
10 hours per day
2 hours per day
2 days per week
3 months per year
Fraction of the Year
65.
66.
67.
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?
68.
65. Manny, Moe, and Jack each have equal shares in a automotive store. Moe decides to
1
of his shares to Manny and the remainder to Jack.
4
What is Jack’s share of the store now?
retire and sell his shares. He sells
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 the
length to the nearest eighth of an inch.
(a) What is the difference between the lengths of the stretched and unstretched
rubber band?
(b) Repeat this process for other rubber bands of different thicknesses. What is the
relationship between the thickness of the rubber band and the distance it can be
stretched?
1
1
4
2
apart and the same distance from each edge. How far from the edge of the door
should each hook be located?
67. A door is 4 ft wide. Two hooks are to be attached to the door so that they are 1 in.
© 2001 McGraw-Hill Companies
68. Complete the following:
1
1
________.
2
4
1
1
1
________.
2
4
8
1
1
1
1
________.
2
4
8
16
Based on these results, predict the answer to the following:
1
1
1
1
1
________
2
4
8
16
32
Now, do the addition, and check your prediction.
255
Answers
1. 12
3. 8
5. 27
7. 8 2 2 2; 12 2 2 3; The LCD is 2 2 2 3 24
11. 240
13. 60
15. 120
17. 50
19.
33.
37.
45.
51.
11
1
7
26
9
11
17
21.
23.
25.
27.
29.
31.
12
2
8
35
14
21
24
1
7
4
6
21
8
35
5
1
23
1 1
35. 1
5
10
15
30
30
30
30
30
6
72
4
1
12
5
7
2
1
1
39.
41.
43.
5
3
15
15
15
15
8
24
33
7
11
99
35
44
108
9
5
7
47. 1
49. lb
40
24
30
120
120
120
120
10
16
8
17 13
11
5
5
1
7
,
mi
in.
lb
53. 1
55.
57.
59.
61.
30 30
12
8
16
16
16
1
7
3
1
Yes— gal remains
65.
67. 2 ft in. or 2
ft
3
12
4
16
© 2001 McGraw-Hill Companies
63.
9. 42
256
Using Your Calculator to Add
and Subtract Fractions
Adding or subtracting fractions on the calculator is very much like the multiplication
and division you did in the previous chapter. The only thing that changes is the
operation.
Scientific Calculator
Here’s where the fraction calculator is a great tool for checking your work. No muss, no
fuss, no searching for a common denominator. Just enter the fractions and get the right
answer!
Example 1
Adding Fractions
Find the sum or difference.
(a)
3
7
14
12
The keystroke sequence is
3 a b/c 14 7 a b/c 12 The result is
(b)
67
.
84
5
7
8
18
5 a b/c 8 7 a b/c 18 © 2001 McGraw-Hill Companies
The result is
17
.
72
Graphing Calculator
Use your graphing calculator to find the sum.
17
7
24
42
257
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
The keystroke sequence is
7 24 17 42 1: Frac
The result is
Enter
39
.
56
CHECK YOURSELF 1
Find the sum or difference.
(a)
5
11
24
18
(b)
9
1
11
3
CHECK YOURSELF ANSWER
1. (a)
16
59
; (b)
72
33
© 2001 McGraw-Hill Companies
258
Name
Calculator Exercises
Section
Date
Find the following sums or differences using your calculator.
1.
2
1
3
2
2.
11
5
12
6
ANSWERS
1.
3.
3
7
4
9
4.
7
5
11
6
2.
3.
5.
5
1
12
6
6.
3
2
7
8
4.
5.
7.
8
7
15
12
8.
7
9
16
24
6.
7.
8.
1
7
9.
10
12
7
17
10.
15
24
9.
10.
11.
8
6
9
7
12.
7
2
15
5
11.
12.
13.
11
5
18
12
14.
5
4
8
9
13.
14.
© 2001 McGraw-Hill Companies
15.
15
9
17
11
16.
31
18
43
53
15.
16.
17.
4
2
9
5
18.
11
2
13
3
17.
18.
19.
5
3
19.
8
5
20.
8
3
20.
9
4
259
Answers
7
1
55
19
7
67
7
3.
5.
7.
or 1
or 1
or 1
6
6
36
36
12
60
60
110
47
37
1
12
2
11.
or 1
13.
or 1
15.
17.
63
60
36
36
187
45
9.
41
60
19.
1
40
© 2001 McGraw-Hill Companies
1.
260