# 6-4 Key Concepts Adding and Subtracting Fractions with Unlike Denominators

```Lesson
Key Concepts
6-4
Unlike Denominators
Objective
To learn how to add and subtract fractions with unlike
denominators, using the method of common denominators.
Note to the Teacher This lesson contains the most complicated
operations on fractions. Provide students with plenty of conceptual
work with sketches (such as pizzas) and manipulative materials
before introducing the algorithms.
Using Models to Add Fractions with Unlike
Denominators
the numerators the way we do when the fractions have like
denominators.
Example 1 What is
1
2
1
?
3
Solution In order to find this sum, we first need to rewrite both
fractions so they have the same denominator. This means we
must replace each fraction with an equivalent fraction having
a different denominator. In order to motivate this process to
portions of a pizza. First, draw a separate model for each of
the fractions.
The addition is modeled by taking the two parts together as
a portion of a whole pizza.
26
Lesson 6-4
It is not easy to visually determine what fraction of the
whole pizza is represented by the shaded portion of the
model. However, suppose we divided each pizza into
2 3 or 6 equal parts.
Notice that each of the smaller parts we have created is one
sixth of the pizza. Now when we merge the portions of the
pizza, we get the model below.
We can see that the shaded portion of the model consists of
five pieces, each of which is one sixth of the pizza. In other
words, the shaded portion of the pizza represents the
fraction
5
,
6
so
1
2
Example 2 What is
1
3
1
3
5
.
6
1
?
5
Solution Represent each of the fractions
pizza.
1
3
and
1
5
as portions of a
When we merge these two portions, we get this picture.
To determine what fraction is modeled by the combined
shaded regions in the figure above, divide each pizza into
3 5 or 15 equal parts.
27
Lesson 6-4
Each of the smaller parts we have created represents one
fifteenth of the pizza. The merged version of these portions
is shown below.
The shaded portion of the model above consists of eight
pieces, each of which is one fifteenth of the pizza. The
shaded portion of the model represents
1
3
1
5
8
,
15
so
8
.
15
The Algorithm and Why It Works
1
2
and
we divided the pizza portion corresponding to
1
3
1
2
(Example 1). When
into three equal
pieces, each of the three pieces represented one sixth of the pizza.
The model showed that
3
6
and
1
2
are equivalent fractions. In the
same way, when the pizza portion representing
1
3
was divided into
two equal pieces, both pieces represented one sixth of the pizza.
This showed that
2
6
and
1
3
are equivalent fractions.
31
6 2
21
6 3
To do this step without using the pizza models, we replace each
fraction by an equivalent fraction so that the two fractions have the
same denominators.
28
Lesson 6-4
1
2
1
3
equivalent fractions 
↓
 equivalent fractions
↓
3
6
2
6
We can easily add these equivalent fractions because they have like
denominators.
1
2
1
3
3
6
32
6
2
6
or
5
6
Note to the Teacher Stress to students that when they multiply both
numerator and denominator of a fraction by the same number, the
result is an equivalent fraction.
Example 3 Find
1
3
1
5
using equivalent fractions with the same
denominator.
Solution First, find equivalent fractions for both
1
3
and
1
5
that have
the same denominator. Multiply both the numerator and
denominator of
1
3
So
15
35
5
15
1
3
by 5 the denominator of
So
is equivalent to
13
53
3
15
.
5
15
and denominator of
1
5
1
5
1
5
1
.
3
Now multiply both the numerator
by 3 the denominator of
1
3
.
3
15
is equivalent to
1
.
5
Now we can add the equivalent
fractions.
1
3
1
5
5
15
3
15
53
15
or
8
15
Note to the Teacher Stress that finding equivalent fractions with
like denominators can be accomplished by multiplying both the
numerator and denominator of each fraction by the denominator of
the other fraction.
29
Lesson 6-4
Example 4 Find
2
3
3
.
7
Solution Multiply both the numerator and denominator of
the denominator of 73 .
2
3
27
37
the denominator of .
2
3
33
73
by 7
3
7
by 3
14
21
Multiply both the numerator and denominator of
3
7
2
3
9
21
Now replace each original fraction with its equivalent
2
3
3
7
14
21
14 9
21
The fraction
23
21
9
21
or
23
21
is an improper fraction, which is a fraction
whose numerator is greater than its denominator. Since 21
goes into 23 once with a remainder of 2, the sum can also be
written as a mixed number. So,
2
3
3
7
23
21
2
or 1 .
21
Here is an algorithm for adding fractions with unlike denominators.
Algorithm
Fractions
with Unlike
Denominators
1. Multiply both the numerator and denominator of each
fraction by the denominator of the other fraction. The
resulting equivalent fractions will have the same
denominator.
1
2
9
7
↓
↓
17
97
29
79
↓
↓
7
63
18
63
and retaining the like denominator.
7
63
30
18
63
25
63
Lesson 6-4
Example 5 Find
2
9
3
.
11
Solution Step 1 Multiply both the numerator and denominator of
2
9
by 11, and multiply both the numerator and
3
11
denominator of
2
9
2 11
9 11
by 9.
22
99
and
3
11
39
11 9
27
99
Step 2 Add the equivalent fractions.
22
99
So,
2
9
3
11
27
99
22 27
99
49
99
or
49
.
99
Subtract Fractions with Unlike Denominators
The algorithm for subtracting one fraction from another when they
have unlike denominators is very similar to the algorithm for adding
fractions with unlike denominators.
Algorithm for 1. Multiply both numerator and denominator of each fraction
Subtracting
by the denominator of the other fraction. The resulting
Fractions
equivalent fractions will have the same denominator.
with Unlike
2. Subtract the equivalent fractions by subtracting the
Denominators
numerators and retaining the like denominator.
Example 6 What is the value of
2
3
1
?
5
Solution Step 1 Multiply both the numerator and denominator of
by 5, and multiply both the numerator and
denominator of
1
5
by 3.
25
35
10
15
2
3
and
1
5
13
53
2
3
3
15
Step 2 Replace each original fraction with its equivalent
fraction. Subtract the equivalent fractions by
subtracting numerators and retaining the
denominator.
10
15
3
15
So,
2
3
1
5
10 3
15
or
7
15
7
.
15
31
Lesson 6-4
Example 7 Find
5
7
Solution Step 1
Step 2
So,
5
7
2
.
3
5
7
53
73
15
21
14
21
2
3
1
.
21
15
21
2
3
and
15 14
21
or
27
37
14
21
1
21
We have discussed procedures for adding and subtracting fractions,
as well as the reasoning behind them. It is very important to keep
the reasoning in mind, since it allows students to interpret practical
problems correctly, as well as to recall the procedures. One
particularly common error made by students can be avoided by
keeping the meaning of the fractions in mind. When adding fractions,
some students make the mistake of adding denominators as well as
numerators. For example,
2
3
4
5
24
35
or
6
.
8
By recognizing that each of the fractions being added is greater than
1
,
2
students will see that an answer less than 1 is unreasonable.
Note to the Teacher Divide your class into two groups. Give each
group several addition and subtraction problems involving fractions,
some of which are solved correctly and others that are solved
incorrectly. Have each group discuss which problems are solved
correctly and which are not, giving reasons for their decisions. Have
each group present their findings to the rest of the class.
End of
Lesson
32
Lesson 6-4
```