# 1.5 Visualize fractions  (Page 3/12)

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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 $\frac{3}{4}.$

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.

## Fraction multiplication

If $a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then

$\frac{a}{b}·\frac{c}{d}=\frac{ac}{bd}$

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: $-\phantom{\rule{0.2em}{0ex}}\frac{11}{12}·\frac{5}{7}.$

## Solution

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

$\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{11}{12}·\frac{5}{7}\hfill \\ \text{Determine the sign of the product; multiply.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{11·5}{12·7}\hfill \\ \begin{array}{c}\text{Are there any common factors in the numerator}\hfill \\ \text{and the denominator? No}\hfill \end{array}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{55}{84}\hfill \end{array}$

Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{10}{28}·\frac{8}{15}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{4}{21}$

Multiply: $-\phantom{\rule{0.2em}{0ex}}\frac{9}{20}·\frac{5}{12}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{3}{16}$

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: $-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\left(-20x\right).$

## Solution

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

 $-\phantom{\rule{0.2em}{0ex}}\frac{12}{5}\left(-20x\right)$ 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. $48x$

Multiply: $\frac{11}{3}\left(-9a\right).$

$-33a$

Multiply: $\frac{13}{7}\left(-14b\right).$

$-36b$

## 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 $-\phantom{\rule{0.2em}{0ex}}\frac{10}{7}$ is $-\phantom{\rule{0.2em}{0ex}}\frac{7}{10},$ since $-\phantom{\rule{0.2em}{0ex}}\frac{10}{7}\left(-\phantom{\rule{0.2em}{0ex}}\frac{7}{10}\right)=1.$

## Reciprocal

The reciprocal of $\frac{a}{b}$ is $\frac{b}{a}.$

A number and its reciprocal multiply to one $\frac{a}{b}·\frac{b}{a}=1.$

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 $a,b,c\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d$ are numbers where $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0,$ then

$\frac{a}{b}÷\frac{c}{d}=\frac{a}{b}·\frac{d}{c}$

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

We need to say $b\ne 0,c\ne 0\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}d\ne 0$ to be sure we don’t divide by zero!

Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{2}{3}÷\frac{n}{5}.$

## Solution

$\begin{array}{cccccc}& & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}÷\frac{n}{5}\hfill \\ \begin{array}{c}\text{To divide, multiply the first fraction by the}\hfill \\ \text{reciprocal of the second.}\hfill \end{array}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{2}{3}·\frac{5}{n}\hfill \\ \text{Multiply.}\hfill & & & & & -\phantom{\rule{0.2em}{0ex}}\frac{10}{3n}\hfill \end{array}$

Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{3}{5}÷\frac{p}{7}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{21}{5p}$

Divide: $-\phantom{\rule{0.2em}{0ex}}\frac{5}{8}÷\frac{q}{3}.$

$-\phantom{\rule{0.2em}{0ex}}\frac{15}{8q}$

Find the quotient: $-\phantom{\rule{0.2em}{0ex}}\frac{7}{8}÷\left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right).$

## Solution

 $-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}÷\left(-\phantom{\rule{0.2em}{0ex}}\frac{14}{27}\right)$ To divide, multiply the first fraction by the reciprocal of the second. $-\phantom{\rule{0.2em}{0ex}}\frac{7}{18}\cdot -\phantom{\rule{0.2em}{0ex}}\frac{27}{14}$ 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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