Sums of Fractions with
Different Denominators
Jen Kershaw
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Printed: November 13, 2014
AUTHOR
Jen Kershaw
www.ck12.org
C HAPTER
Chapter 1. Sums of Fractions with Different Denominators
1
Sums of Fractions with
Different Denominators
Here you’ll learn to add fractions with different denominators.
Have you ever thought of how many layers make up a wall?
Travis is having a great time working with his Uncle Larry. On his second day of working, Travis and Uncle Larry
worked on the layers of a wall. “When you first look at it you don’t realize that there are many different layers to a
wall that add to its thickness,” Uncle Larry tells Travis.
“Like what?” Travis asks.
“Well, we start with drywall that is 14 of an inch thick. Then we add the insulation. For this wall we are going to use
two different forms of insulation. One is 34 of an inch thick and the other is 12 of an inch thick. Next, we add a 12 inch
layer of wall sheathing. Finally we add the siding, that is 87 of an inch thick,” Uncle Larry explains.
“Wow, that is pretty thick.” says Travis.
But how thick is it? Travis isn’t sure. Notice that all of these fractions have different denominators. To figure
out the thickness of the wall, you will need to know how to add fractions with different denominators.
This Concept has all of the information that you will need!
Guidance
Previously we worked on how to add fractions that had the same denominator. When you add fractions with the
same denominator, you don’t have to do anything with the denominator, you can just add the numerators. Because
the wholes are divided in the same way, they are alike. Therefore, adding these fractions is very simple.
Not all fractions have common denominators. When we have fractions with unlike denominators, we can still add
them, but we will need to rename the fractions before we can add them.
How do we add fractions with different denominators?
To add fractions that have different denominators, we have to rename the fractions so that they are alike. We rename
them by changing the different denominators of the fractions to common denominators.
1 1
+ =
2 4
1
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In this example, we are trying to add halves and fourths. If something is divided into halves, it is divided into two
pieces. If something is divided into fourths, it is divided into four pieces.
Here we are trying to add fourths and halves. They are different quantities. You can see that although the bar is the
same size, the parts are different sizes. We have to rewrite these fractions so that they have common denominators.
How do we rewrite fractions to have a common denominator?
The first step in doing this is to the find the least common multiple of both of the denominators. This LCM will
become the lowest common denominator.
Let’s look at 2 and 4.
First, name the multiples of 2: 2, 4, 6, 8, 10 . . . .
Next, name the multiples of 4: 4, 8, 12, 16 . . . .
The least common multiple of 2 and 4 is 4.
Our next step is to rewrite each fraction as an equivalent fraction that has four as a denominator.
1
2
= 4 to name one-half in terms of fourths, we need to multiply the numerator and denominator by the same number.
2 × 2 = 4, so we multiply the numerator by 2 also. 1 × 2 = 2.
1 2
=
2 4
Our second fraction, 14 , is already written in terms of fourths so we don’t need to change it at all.
Next, we can add the renamed fractions.
2 1 3
+ =
4 4 4
Our answer is 34 . This answer is in simplest form, so our work is complete.
As long as you rename fractions with the lowest common denominator, you can add any number of fractions with
unlike denominators.
Try a few of these on your own. Please write your answer in simplest form.
2
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Chapter 1. Sums of Fractions with Different Denominators
Example A
1
2
+ 62 =
Solution:
5
6
Example B
2
3
+ 91 =
Solution:
7
9
Example C
4
5
+ 31 =
Solution:
17
15
2
= 1 15
Now let’s go back to our original problem.
Travis needs to add up all of the fractions to figure out how thick the wall really is. To do this, he needs to write a
numerical expression like the ones that we worked on in the last section. The expression looks like this.
1 3 1 1 7
+ + + +
4 4 2 2 8
This expression shows all of the different layers of the wall. To find a measurement for the thickness of the
wall, Travis must add all of these fractions together. To do this, he will need to rename them using the lowest
common denominator.
What is the lowest common denominator for 4, 2 and 8? Yes. It is 8.
Rename each fraction in terms of eighths.
1 2
=
4 8
3 6
=
4 8
1 4
=
2 8
Next, let’s rewrite the expression.
2 6 4 4 7
+ + + +
8 8 8 8 8
Now we can add the numerators.
2 + 6 + 4 + 4 + 7 = 23
23
7 00
=2
8
8
3
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Travis can see that the wall is almost three inches thick.
Vocabulary
Renaming fractions
renaming fractions means rewriting them with a different denominator, but not changing the value of the
fraction.
Least Common Multiple
the lowest multiple that two or more numbers have in common.
Lowest Common Denominator
the least common multiple becomes the lowest common denominator when adding or subtracting fractions
with different denominators.
Equivalent Fractions
equal fractions. Created by multiplying the numerator and the denominator of a fraction by the same number.
Guided Practice
Here is one for you to try on your own.
2
7
+ 93 =
Answer
First we need to find a common denominator. The common denominator for 7 and 9 is 63.
Next, we rename the fractions.
2
7
+ 93 =
18
63
+ 21
63 =
39
63
This is our answer.
Video Review
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/5372
James Sousa Example of Adding Fractions with Different Denominators
MEDIA
Click image to the left for use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/5373
4
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Chapter 1. Sums of Fractions with Different Denominators
James Sousa Another Example of Adding Fractions with Different Denominators
Explore More
Directions: Add the following fractions by renaming. Be sure your answer is in simplest form.
1.
2.
3.
4.
5.
6.
7.
8.
9.
3
4
6
7
2
3
2
3
1
2
3
6
+ 12 =
6
8
4
7
4
5
+ 13 =
+ 12 =
+ 13 =
+ 16 =
+ 18 =
+ 13 =
+ 12 =
+ 12 =
10.
3
7
+ 16 =
11.
5
8
+ 23 =
12.
6
1
7+3 =
9
1
12 + 6 =
8
1
10 + 2 =
13.
14.
15.
16.
6
7
3
4
+ 34 =
+ 26 =
5