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By the end of this section, you will be able to:
  • Model fraction addition
  • Add fractions with a common denominator
  • Model fraction subtraction
  • Subtract fractions with a common denominator

Before you get started, take this readiness quiz.

  1. Simplify: 2 x + 9 + 3 x 4 .
    If you missed this problem, review Evaluate, Simplify and Translate Expressions .
  2. Draw a model of the fraction 3 4 .
    If you missed this problem, review Visualize Fractions .
  3. Simplify: 3 + 2 6 .
    If you missed this problem, review Multiply and Divide Mixed Numbers and Complex Fractions .

Model fraction addition

How many quarters are pictured? One quarter plus 2 quarters equals 3 quarters.

Three U.S. quarters are shown. One is shown on the left, and two are shown on the right.

Remember, quarters are really fractions of a dollar. Quarters are another way to say fourths. So the picture of the coins shows that

1 4 2 4 3 4 one quarter + two quarters = three quarters

Let’s use fraction circles to model the same example, 1 4 + 2 4 .

Start with one 1 4 piece. . .
Add two more 1 4 pieces. . .
The result is 3 4 . . .

So again, we see that

1 4 + 2 4 = 3 4
Doing the Manipulative Mathematics activity "Model Fraction Addition" will help you develop a better understanding of adding fractions

Use a model to find the sum 3 8 + 2 8 .

Solution

Start with three 1 8 pieces. . .
Add two 1 8 pieces. . .
How many 1 8 pieces are there? . .

There are five 1 8 pieces, or five-eighths. The model shows that 3 8 + 2 8 = 5 8 .

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Use a model to find each sum. Show a diagram to illustrate your model.

1 8 + 4 8

5 8

A circle divided into 8 sections, 5 of which are shaded.

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Use a model to find each sum. Show a diagram to illustrate your model.

1 6 + 4 6

5 6

A circle divided into 6 sections, 5 of which are shaded.

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Add fractions with a common denominator

[link] shows that to add the same-size pieces—meaning that the fractions have the same denominator —we just add the number of pieces.

Fraction addition

If a , b , and c are numbers where c 0 , then

a c + b c = a + b c

To add fractions with a common denominators, add the numerators and place the sum over the common denominator.

Find the sum: 3 5 + 1 5 .

Solution

3 5 + 1 5
Add the numerators and place the sum over the common denominator. 3 + 1 5
Simplify. 4 5
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Find each sum: 3 6 + 2 6 .

5 6

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Find each sum: 3 10 + 7 10 .

1

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Find the sum: x 3 + 2 3 .

Solution

x 3 + 2 3
Add the numerators and place the sum over the common denominator. x + 2 3

Note that we cannot simplify this fraction any more. Since x and 2 are not like terms, we cannot combine them.

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Find the sum: x 4 + 3 4 .

x + 3 4

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Find the sum: y 8 + 5 8 .

y + 5 8

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Find the sum: 9 d + 3 d .

Solution

We will begin by rewriting the first fraction with the negative sign in the numerator.

a b = a b

9 d + 3 d
Rewrite the first fraction with the negative in the numerator. −9 d + 9 d
Add the numerators and place the sum over the common denominator. −9 + 3 d
Simplify the numerator. −6 d
Rewrite with negative sign in front of the fraction. 6 d
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Find the sum: 7 d + 8 d .

1 d

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Find the sum: 6 m + 9 m .

3 m

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Find the sum: 2 n 11 + 5 n 11 .

Solution

2 n 11 + 5 n 11
Add the numerators and place the sum over the common denominator. 2 n + 5 n 11
Combine like terms. 7 n 11
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Find the sum: 3 p 8 + 6 p 8 .

9 p 8

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Find the sum: 2 q 5 + 7 q 5 .

9 q 5

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Find the sum: 3 12 + ( 5 12 ) .

Solution

3 12 + ( 5 12 )
Add the numerators and place the sum over the common denominator. −3 + ( −5 ) 12
Add. −8 12
Simplify the fraction. 2 3
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Find each sum: 4 15 + ( 6 15 ) .

2 3

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Find each sum: 5 21 + ( 9 21 ) .

2 3

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Model fraction subtraction

Subtracting two fractions with common denominators is much like adding fractions. Think of a pizza that was cut into 12 slices. Suppose five pieces are eaten for dinner. This means that, after dinner, there are seven pieces (or 7 12 of the pizza) left in the box. If Leonardo eats 2 of these remaining pieces (or 2 12 of the pizza), how much is left? There would be 5 pieces left (or 5 12 of the pizza).

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Source:  OpenStax, Prealgebra. OpenStax CNX. Jul 15, 2016 Download for free at http://legacy.cnx.org/content/col11756/1.9
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