1.5 Visualize fractions  (Page 3/12)

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

Now we’ll take $\frac{1}{2}$ of $\frac{3}{4}.$

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

To multiply fractions, we 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.

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 $\frac{a}{1}.$ So, for example, $3=\frac{3}{1}.$

Multiply: $-\frac{12}{5}\left(\mathrm{-20}x\right).$

Solution

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

	$-\frac{12}{5}(\mathrm{-20}x)$
Write $20x$ as a fraction.	$\frac{12}{5}\left(\frac{20x}{1}\right)$
Multiply.
Rewrite 20 to show the common factor 5 and divide it out.
Simplify.	$48x$

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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 $\frac{2}{3}$ is $\frac{3}{2}.$

Notice that $\frac{2}{3}·\frac{3}{2}=1.$ 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 $-\frac{10}{7}$ is $-\frac{7}{10},$ since $-\frac{10}{7}\left(-\frac{7}{10}\right)=1.$

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

We need to say $b\ne 0,c\ne 0\text{and}d\ne 0$ to be sure we don’t divide by zero!

Find the quotient: $-\frac{7}{8}÷\left(-\frac{14}{27}\right).$

Solution

	$-\frac{7}{18}÷\left(\mathrm{-}\frac{14}{27}\right)$
To divide, multiply the first fraction by the reciprocal of the second.	$-\frac{7}{18}\cdot \mathrm{-}\frac{27}{14}$
Determine the sign of the product, and then multiply..	$\frac{7\cdot 27}{18\cdot 14}$
Rewrite showing common factors.
Remove common factors.	$\frac{3}{2\cdot 2}$
Simplify.	$\frac{3}{4}$

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