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Doing the Manipulative Mathematics activity “Model Fraction Multiplication” will help you develop a better understanding of multiplying fractions.

We’ll use a model to show you how to multiply two fractions and to help you remember the procedure. Let’s start with 3 4 .

A rectangle made up of four squares in a row. The first three squares are shaded.

Now we’ll take 1 2 of 3 4 .

A rectangle made up of four squares in a row. The first three squares are shaded. The bottom halves of the first three squares are shaded darker with diagonal lines.

Notice that now, the whole is divided into 8 equal parts. So 1 2 · 3 4 = 3 8 .

To multiply fractions, we multiply the numerators and multiply the denominators.

Fraction multiplication

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

a b · c d = a c b d

To multiply fractions, multiply the numerators and multiply the denominators.

When multiplying fractions , the properties of positive and negative numbers still apply, of course. It is a good idea to determine the sign of the product as the first step. In [link] , we will multiply negative and a positive, so the product will be negative.

Multiply: 11 12 · 5 7 .

Solution

The first step is to find the sign of the product. Since the signs are the different, the product is negative.

11 12 · 5 7 Determine the sign of the product; multiply. 11 · 5 12 · 7 Are there any common factors in the numerator and the denominator? No 55 84

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Multiply: 10 28 · 8 15 .

4 21

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Multiply: 9 20 · 5 12 .

3 16

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When multiplying a fraction by an integer, it may be helpful to write the integer as a fraction. Any integer, a , can be written as a 1 . So, for example, 3 = 3 1 .

Multiply: 12 5 ( −20 x ) .

Solution

Determine the sign of the product. The signs are the same, so the product is positive.

12 5 ( −20 x )
Write 20 x as a fraction. 12 5 ( 20 x 1 )
Multiply.
Rewrite 20 to show the common factor 5 and divide it out. .
Simplify. 48 x
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Multiply: 11 3 ( −9 a ) .

−33 a

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Multiply: 13 7 ( −14 b ) .

−36 b

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Divide fractions

Now that we know how to multiply fractions, we are almost ready to divide. Before we can do that, that we need some vocabulary.

The reciprocal    of a fraction is found by inverting the fraction, placing the numerator in the denominator and the denominator in the numerator. The reciprocal of 2 3 is 3 2 .

Notice that 2 3 · 3 2 = 1 . A number and its reciprocal multiply to 1.

To get a product of positive 1 when multiplying two numbers, the numbers must have the same sign. So reciprocals must have the same sign.

The reciprocal of 10 7 is 7 10 , since 10 7 ( 7 10 ) = 1 .

Reciprocal

The reciprocal of a b is b a .

A number and its reciprocal multiply to one a b · b a = 1 .

Doing the Manipulative Mathematics activity “Model Fraction Division” will help you develop a better understanding of dividing fractions.

To divide fractions, we multiply the first fraction by the reciprocal of the second.

Fraction division

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

a b ÷ c d = a b · d c

To divide fractions, we multiply the first fraction by the reciprocal of the second.

We need to say b 0 , c 0 and d 0 to be sure we don’t divide by zero!

Divide: 2 3 ÷ n 5 .

Solution

2 3 ÷ n 5 To divide, multiply the first fraction by the reciprocal of the second. 2 3 · 5 n Multiply. 10 3 n

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Divide: 3 5 ÷ p 7 .

21 5 p

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Divide: 5 8 ÷ q 3 .

15 8 q

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Find the quotient: 7 8 ÷ ( 14 27 ) .

Solution

7 18 ÷ ( 14 27 )
To divide, multiply the first fraction by the reciprocal of the second. 7 18 27 14
Determine the sign of the product, and then multiply.. 7 27 18 14
Rewrite showing common factors. .
Remove common factors. 3 2 2
Simplify. 3 4
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Practice Key Terms 7

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Source:  OpenStax, Elementary algebra. OpenStax CNX. Jan 18, 2017 Download for free at http://cnx.org/content/col12116/1.2
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