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4.2 Multiply and divide fractions

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By the end of this section, you will be able to:
  • Simplify fractions
  • Multiply fractions
  • Find reciprocals
  • Divide fractions

Before you get started, take this readiness quiz.

  1. Find the prime factorization of 48 .
    If you missed this problem, review Prime Factorization and the Least Common Multiple .
  2. Draw a model of the fraction 3 4 .
    If you missed this problem, review Visualize Fractions .
  3. Find two fractions equivalent to 5 6 .
    Answers may vary. Acceptable answers include 10 12 , 15 18 , 50 60 , etc.
    If you missed this problem, review Visualize Fractions .

Simplify fractions

In working with equivalent fractions, you saw that there are many ways to write fractions that have the same value, or represent the same part of the whole. How do you know which one to use? Often, we’ll use the fraction that is in simplified form.

A fraction is considered simplified if there are no common factors, other than 1 , in the numerator and denominator . If a fraction does have common factors in the numerator and denominator, we can reduce the fraction to its simplified form by removing the common factors.

Simplified fraction

A fraction is considered simplified if there are no common factors in the numerator and denominator.

For example,

  • 2 3 is simplified because there are no common factors of 2 and 3 .
  • 10 15 is not simplified because 5 is a common factor of 10 and 15 .

The process of simplifying a fraction is often called reducing the fraction . In the previous section, we used the Equivalent Fractions Property to find equivalent fractions. We can also use the Equivalent Fractions Property in reverse to simplify fractions. We rewrite the property to show both forms together.

Equivalent fractions property

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

a b = a · c b · c and a · c b · c = a b .

Notice that c is a common factor in the numerator and denominator . Anytime we have a common factor in the numerator and denominator, it can be removed.

Simplify a fraction.

  1. Rewrite the numerator and denominator to show the common factors. If needed, factor the numerator and denominator into prime numbers.
  2. Simplify, using the equivalent fractions property, by removing common factors.
  3. Multiply any remaining factors.

Simplify: 10 15 .

Solution

To simplify the fraction, we look for any common factors in the numerator and the denominator.

Notice that 5 is a factor of both 10 and 15. 10 15
Factor the numerator and denominator. .
Remove the common factors. .
Simplify. 2 3
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Simplify: 8 12 .

2 3

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Simplify: 12 16 .

3 4

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To simplify a negative fraction, we use the same process as in [link] . Remember to keep the negative sign.

Simplify: − 18 24 .

Solution

We notice that 18 and 24 both have factors − 18 24
Rewrite the numerator and denominator showing the common factor .
Remove common factors .
Simplify. − 3 4
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Simplify: − 21 28 .

− 3 4

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Simplify: − 16 24 .

− 2 3

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After simplifying a fraction, it is always important to check the result to make sure that the numerator and denominator do not have any more factors in common. Remember, the definition of a simplified fraction: a fraction is considered simplified if there are no common factors in the numerator and denominator .

When we simplify an improper fraction, there is no need to change it to a mixed number.

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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