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Add: 13 42 + 17 35 .

37 210

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Add: 19 24 + 17 32 .

25 96

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In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

Add: 3 5 + x 8 .

Solution

The fractions have different denominators.

3 5 + x 8
Find the LCD.
.
Rewrite as equivalent fractions with the LCD. .
Simplify the numerators and denominators. 24 40 + 5 x 8
Add. 24 + 5 x 40

We cannot add 24 and 5 x since they are not like terms, so we cannot simplify the expression any further.

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Add: y 6 + 7 9 .

3 y + 14 18

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Add: x 6 + 7 15 .

5 x + 14 30

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Identify and use fraction operations

By now in this chapter, you have practiced multiplying, dividing, adding, and subtracting fractions. The following table summarizes these four fraction operations. Remember: You need a common denominator to add or subtract fractions, but not to multiply or divide fractions

Summary of fraction operations

Fraction multiplication: Multiply the numerators and multiply the denominators.

a b · c d = a c b d

Fraction division: Multiply the first fraction by the reciprocal of the second.

a b ÷ c d = a b · d c

Fraction addition: Add the numerators and place the sum over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

a c + b c = a + b c

Fraction subtraction: Subtract the numerators and place the difference over the common denominator. If the fractions have different denominators, first convert them to equivalent forms with the LCD.

a c b c = a b c

Simplify:

  1. 1 4 + 1 6
  2. 1 4 ÷ 1 6

Solution

First we ask ourselves, “What is the operation?”

The operation is addition.

Do the fractions have a common denominator? No.

1 4 + 1 6
Find the LCD.
.
Rewrite each fraction as an equivalent fraction with the LCD. .
Simplify the numerators and denominators. 3 12 + 2 12
Add the numerators and place the sum over the common denominator. 1 12
Check to see if the answer can be simplified. It cannot.

The operation is division. We do not need a common denominator.

1 4 ÷ 1 6
To divide fractions, multiply the first fraction by the reciprocal of the second. 1 4 · 6 1
Multiply. 6 4
Simplify. 3 2
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Simplify each expression:

  1. 3 4 1 6
  2. 3 4 · 1 6

  1. 11 12
  2. 1 8

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Simplify each expression:

  1. 5 6 ÷ ( 1 4 )
  2. 5 6 ( 1 4 )

  1. 10 3
  2. 13 12

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Simplify:

  1. 5 x 6 3 10
  2. 5 x 6 · 3 10

Solution

The operation is subtraction. The fractions do not have a common denominator.

5 x 6 3 10
Rewrite each fraction as an equivalent fraction with the LCD, 30. 5 x · 5 6 · 5 3 · 3 10 · 3
25 x 30 9 30
Subtract the numerators and place the difference over the common denominator. 25 x 9 30

The operation is multiplication; no need for a common denominator.

5 x 6 · 3 10
To multiply fractions, multiply the numerators and multiply the denominators. 5 x · 3 6 · 10
Rewrite, showing common factors. 5 · x · 3 2 · 3 · 2 · 5
Remove common factors to simplify. x 4
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Simplify:

  1. 3 a 4 8 9
  2. 3 a 4 · 8 9

  1. 27 a 32 36
  2. 2 a 3

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Simplify:

  1. 4 k 5 + 5 6
  2. 4 k 5 ÷ 5 6

  1. 24 k + 25 30
  2. 24 k 25

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Use the order of operations to simplify complex fractions

In Multiply and Divide Mixed Numbers and Complex Fractions , we saw that a complex fraction is a fraction in which the numerator or denominator contains a fraction. We simplified complex fractions by rewriting them as division problems. For example,

Questions & Answers

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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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