3.1
Adding and Subtracting Fractions
with Like Denominators
3.1
OBJECTIVES
1. Add two like fractions
2. Add a group of like fractions
3. Subtract two like fractions
Recall from our work in Chapter 1 that adding can be thought of as combining groups of
the same kind of objects. This is also true when you think about adding fractions.
Fractions can be added only if they name a number of the same parts of a whole. This
means that we can add fractions only when they are like fractions, that is, when they have
the same (a common) denominator.
As long as we are dealing with like fractions, addition is an easy matter. Just use the
following rule.
NOTE For instance, we can
add two nickels and three
nickels to get five nickels. We
cannot directly add two nickels
and three dimes!
Step by Step: To Add Like Fractions
Step 1 Add the numerators.
Step 2 Place the sum over the common denominator.
Step 3 Simplify the resulting fraction when necessary.
Our first example illustrates the use of this rule.
Example 1
Adding Like Fractions
Add.
1
3
5
5
Step 1 Add the numerators.
134
Step 2 Write that sum over the common denominator, 5. We are done at this point
4
because the answer, , is in the simplest possible form.
5
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Step 1
Step 2
1
3
13
4
5
5
5
5
Let’s illustrate with a diagram.
NOTE Combining 1 of the 5
parts with 3 of the 5 parts gives
a total of 4 of the 5 equal parts.
1
5
3
5
4
5
225
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
CHECK YOURSELF 1
Add.
2
5
9
9
CAUTION
Be Careful! In adding fractions, do not follow the rule for multiplying fractions. To
1
3
multiply , you would multiply both the numerators and the denominators:
5
5
1
3
3
5
5
25
1
3
3
of 5
5
25
Also, because to add fractions, you must have a common denominator, you do not add
the denominators. Thus
1
3
4
5
5
5
1
3
4
5
5
10
Step 3 of the addition rule for like fractions tells us to simplify the sum. The sum of fractions should always be written in lowest terms. Consider Example 2.
Example 2
Adding Like Fractions That Require Simplifying
Add and simplify.
Step 3
5
8
2
3
12
12
12
3
8
is not in lowest terms.
12
Divide the numerator and denominator
by 4 to simplify the result.
The sum
CHECK YOURSELF 2
Add.
4
6
15
15
If the sum of two fractions is an improper fraction, we will usually write that sum as a
mixed number.
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226
ADDING AND SUBTRACTING FRACTIONS WITH LIKE DENOMINATORS
SECTION 3.1
227
Example 3
Adding Like Fractions That Result in Mixed Numbers
NOTE Add as before. Then
Add
convert the sum to a mixed
number.
5
8
4
13
1
9
9
9
9
Write the sum
13
as a mixed number.
9
CHECK YOURSELF 3
Add.
7
10
12
12
We can also easily extend our addition rule to find the sum of more than two fractions as
long as they all have the same denominator. This is shown in Example 4.
Example 4
Adding a Group of Like Fractions
Add.
2
3
6
11
7
7
7
7
1
Add the numerators: 2 3 6 11.
4
7
CHECK YOURSELF 4
Add.
1
3
5
8
8
8
Many applications can be solved by adding fractions.
Example 5
An Application Involving the Adding of Like Fractions
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9
7
Noel walked
miles (mi) to Jensen’s house and then walked
mi to school. How far did
10
10
Noel walk?
To find the total distance Noel walked, add the two distances.
9
7
16
6
3
1 1
10
10
10
10
5
3
Noel walked 1 mi.
5
228
CHAPTER 3
ADDING AND SUBTRACTING FRACTIONS
CHECK YOURSELF 5
Emir bought
11
7
pounds (lb) of candy at one store and
lb at another store. How
16
16
much candy did Emir buy?
NOTE Like fractions have the
If a problem involves like fractions, then subtraction, like addition, is not difficult.
same denominator.
Step by Step: To Subtract Like Fractions
Step 1 Subtract the numerators.
Step 2 Place the difference over the common denominator.
Step 3 Simplify the resulting fraction when necessary.
NOTE Note the similarity to
our rule for adding like
fractions.
Example 6
Subtracting Like Fractions
Subtract.
Step 1
(a)
Subtract the numerators: 4 2 2.
Write the difference over the common
denominator, 5. Step 3 is not necessary
because the difference is in simplest
form.
Step 2
4
2
42
2
5
5
5
5
Illustrating with a diagram:
NOTE Subtracting 2 of the
5 parts from 4 of the 5 parts
leaves 2 of the 5 parts.
4
5
(b)
in lowest terms.
2
5
2
5
5
3
2
1
53
8
8
8
8
4
CHECK YOURSELF 6
Subtract.
5
11
12
12
CHECK YOURSELF ANSWERS
1.
2
5
25
7
9
9
9
9
2.
2
3
3. 1
5
12
4. 1
1
8
1
5. 1 lb
8
6.
1
2
© 2001 McGraw-Hill Companies
NOTE Always write the result
Name
3.1
Exercises
Section
Date
Add. Write all answers in lowest terms.
1.
3
1
5
5
2.
4
1
7
7
3.
4
6
11
11
4.
5
4
16
16
ANSWERS
1.
2.
5.
2
3
10
10
6.
5
1
12
12
7.
3
4
7
7
8.
3
7
20
20
3.
4.
5.
9.
11.
9
11
30
30
10.
13
23
48
48
12.
4
5
9
9
17
31
60
60
6.
7.
8.
13.
3
6
7
7
14.
3
4
5
5
15.
7
9
10
10
16.
5
7
8
8
9.
10.
11.
17.
11
10
12
12
18.
13
11
18
18
19.
1
1
3
8
8
8
20.
3
3
1
10
10
10
1
4
5
21.
9
9
9
7
11
1
22.
12
12
12
© 2001 McGraw-Hill Companies
Subtract. Write all answers in lowest terms.
23.
3
1
5
5
7
4
25.
9
9
27.
13
3
20
20
24.
5
2
7
7
7
3
26.
10
10
28.
19
17
30
30
12.
13.
14.
15.
16.
17.
18.
19.
20.
21.
22.
23.
24.
25.
26.
27.
28.
229
ANSWERS
29.
29.
19
5
24
24
30.
25
13
36
36
31.
11
7
12
12
32.
9
6
10
10
33.
8
3
9
9
34.
1
5
8
8
30.
31.
32.
33.
34.
Evaluate each of the following. Write all answers in lowest terms.
35.
35.
4
3
7
12
12
12
36.
3
5
8
9
9
9
37.
6
3
11
13
13
13
38.
9
3
7
11
11
11
39.
18
13
3
23
23
23
40.
17
11
5
18
18
18
36.
37.
38.
39.
40.
Solve the following applications. Write each answer in lowest terms.
41.
41. Money. You collect 3 dimes, 2 dimes, and then 4 dimes. How much money do you
have as a fraction of a dollar?
42.
42. Money. You collect 7 nickels, 4 nickels, and then 5 nickels. How much money do
43.
you have as a fraction of a dollar?
44.
43. Work. You work 7 hours (h) one day, 5 h the second day, and 6 h the third day. How
long did you work, as a fraction of a 24-h day?
44. Time. One task took 7 minutes (min), a second task took 12 min, and a third task
took 21 min. How long did the three tasks take, as a fraction of an hour?
45. Perimeter. What is the perimeter of a rectangle if each of the lengths is
7
2
inches (in.) and each of the widths is
in.?
10
10
230
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45.
ANSWERS
46. Perimeter. Find the perimeter of a rectangular picture if each width is
5
each length is yd.
9
7
yd and
9
46.
47.
48.
49.
4
7
of an hour (h) in the batting cages on Friday, and of an
9
9
hour on Saturday. He wants to spend 2 h total on the weekend. How much time
should he spend on Sunday to accomplish this goal?
47. Athletics. Patrick spent
17
of a mile of road. If she
30
11
has already inspected
of a mile, how much more does she need to inspect?
30
© 2001 McGraw-Hill Companies
48. Quality Control. Maria, a road inspector, must inspect
49. Perimeter. Find the perimeter of the following figure:
3
8
in.
2
8
5
8
in.
in.
4
8
7
8
in.
3
8
in.
in.
231
ANSWERS
Find the perimeters of the following triangles.
50.
3
8
50.
51.
cm
3
4
51.
52.
3
8
3
8
cm
5
4
in.
7
4
cm
in.
in.
53.
54.
52.
53.
15
16
55.
19
16
in.
18
16
in.
7
8
9
8
in.
9
8
in.
in.
in.
Find the perimeters of the following polygons.
54.
7
8
7
8
in.
7
8
in.
55.
in.
7
8
in.
7
8
7
8
7
8
in.
in.
15
8
in.
11
8
in.
15
8
in.
7
8
in.
5
8
in.
in.
Answers
19.
33.
45.
53.
232
2
3
5
1
4
7
3
1
7.
10
10
10
2
7
7
7
2
7
9
16
6
3
3
1 1
13. 1
15.
17. 1
7
10
10
10
10
5
4
5
1
2
1
1
7
1
21. 1
23.
25.
27.
29.
31.
8
9
5
3
2
12
3
5
1
1
2
9
3
of a dollar
day
35.
37. 1
39.
41.
43.
9
2
13
23
10
4
9
4
7
15
3
in. or 1 in.
in. 3 in.
h
49. 3 in.
51.
47.
5
5
9
4
4
25
1
60
1
in. 3 in.
in. 7 in.
55.
8
8
8
2
10
11
3
11.
4
3.
5.
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4
5
2
9.
3
1.