# 4.3 Multiply and divide mixed numbers and complex fractions  (Page 2/4)

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Translate the phrase into an algebraic expression: “the quotient of $3x$ and $8.”$

## Solution

The keyword is quotient ; it tells us that the operation is division. Look for the words of and and to find the numbers to divide.

$\text{The quotient}\phantom{\rule{0.2em}{0ex}}\text{of}\phantom{\rule{0.2em}{0ex}}3x\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}8\text{.}$

This tells us that we need to divide $3x$ by $8.$ $\frac{3x}{8}$

Translate the phrase into an algebraic expression: the quotient of $9s$ and $14.$

$\frac{9s}{14}$

Translate the phrase into an algebraic expression: the quotient of $5y$ and $6.$

$\frac{5y}{6}$

Translate the phrase into an algebraic expression: the quotient of the difference of $m$ and $n,$ and $p.$

## Solution

We are looking for the quotient of the difference of $m$ and , and $p.$ This means we want to divide the difference of $m$ and $n$ by $p.$

$\frac{m-n}{p}$

Translate the phrase into an algebraic expression: the quotient of the difference of $a$ and $b,$ and $cd.$

$\frac{a-b}{cd}$

Translate the phrase into an algebraic expression: the quotient of the sum of $p$ and $q,$ and $r.$

$\frac{p+q}{r}$

## Simplify complex fractions

Our work with fractions so far has included proper fractions, improper fractions, and mixed numbers. Another kind of fraction is called complex fraction    , which is a fraction in which the numerator or the denominator contains a fraction.

Some examples of complex fractions are:

$\frac{\phantom{\rule{0.2em}{0ex}}\frac{6}{7}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}}\phantom{\rule{1.8em}{0ex}}\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}\phantom{\rule{1.8em}{0ex}}\frac{\phantom{\rule{0.2em}{0ex}}\frac{x}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\phantom{\rule{0.2em}{0ex}}}$

To simplify a complex fraction, remember that the fraction bar means division. So the complex fraction $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}$ can be written as $\frac{3}{4}÷\frac{5}{8}.$

Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}.$

## Solution

 $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{4}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{8}\phantom{\rule{0.2em}{0ex}}}$ Rewrite as division. $\frac{3}{4}÷\frac{5}{8}$ Multiply the first fraction by the reciprocal of the second. $\frac{3}{4}·\frac{8}{5}$ Multiply. $\frac{3·8}{4·5}$ Look for common factors. $\frac{3·4̸·2}{4̸·5}$ Remove common factors and simplify. $\frac{6}{5}$

Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{2}{3}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{5}{6}\phantom{\rule{0.2em}{0ex}}}.$

$\frac{4}{5}$

Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{3}{7}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{6}{11}\phantom{\rule{0.2em}{0ex}}}.$

$\frac{11}{14}$

## Simplify a complex fraction.

1. Rewrite the complex fraction as a division problem.
2. Follow the rules for dividing fractions.
3. Simplify if possible.

Simplify: $\frac{-\frac{6}{7}}{3}.$

## Solution

 $\frac{-\frac{6}{7}}{3}$ Rewrite as division. $-\frac{6}{7}÷3$ Multiply the first fraction by the reciprocal of the second. $-\frac{6}{7}·\frac{1}{3}$ Multiply; the product will be negative. $-\frac{6·1}{7·3}$ Look for common factors. $-\frac{3̸·2·1}{7·3̸}$ Remove common factors and simplify. $-\frac{2}{7}$

Simplify: $\frac{-\frac{8}{7}}{4}.$

$-\frac{2}{7}$

Simplify: $-\frac{\phantom{\rule{0.2em}{0ex}}3\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{9}{10}\phantom{\rule{0.2em}{0ex}}}.$

$-\frac{10}{3}$

Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{x}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{xy}{6}\phantom{\rule{0.2em}{0ex}}}.$

## Solution

 $\frac{\phantom{\rule{0.2em}{0ex}}\frac{x}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{xy}{6}\phantom{\rule{0.2em}{0ex}}}$ Rewrite as division. $\frac{x}{2}÷\frac{xy}{6}$ Multiply the first fraction by the reciprocal of the second. $\frac{x}{2}·\frac{6}{xy}$ Multiply. $\frac{x·6}{2·xy}$ Look for common factors. $\frac{\mathrm{x̸}·3·2̸}{2̸·\mathrm{x̸}·y}$ Remove common factors and simplify. $\frac{3}{y}$

Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{a}{8}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{ab}{6}\phantom{\rule{0.2em}{0ex}}}.$

$\frac{3}{4b}$

Simplify: $\frac{\phantom{\rule{0.2em}{0ex}}\frac{p}{2}\phantom{\rule{0.2em}{0ex}}}{\phantom{\rule{0.2em}{0ex}}\frac{pq}{8}\phantom{\rule{0.2em}{0ex}}}.$

$\frac{4}{q}$

Simplify: $\frac{2\frac{3}{4}}{\frac{1}{8}}.$

## Solution

 $\frac{2\frac{3}{4}}{\frac{1}{8}}$ Rewrite as division. $2\frac{3}{4}÷\frac{1}{8}$ Change the mixed number to an improper fraction. $\frac{11}{4}÷\frac{1}{8}$ Multiply the first fraction by the reciprocal of the second. $\frac{11}{4}·\frac{8}{1}$ Multiply. $\frac{11·8}{4·1}$ Look for common factors. $\frac{11·4̸·2}{4̸·1}$ Remove common factors and simplify. $22$

Simplify: $\frac{\frac{5}{7}}{1\frac{2}{5}}.$

$\frac{25}{49}.$

Simplify: $\frac{\frac{8}{5}}{3\frac{1}{5}}.$

$\frac{1}{2}$

## Simplify expressions with a fraction bar

Where does the negative sign go in a fraction? Usually, the negative sign is placed in front of the fraction, but you will sometimes see a fraction with a negative numerator or denominator. Remember that fractions represent division. The fraction $-\frac{1}{3}$ could be the result of dividing $\frac{-1}{3},$ a negative by a positive, or of dividing $\frac{1}{-3},$ a positive by a negative. When the numerator and denominator have different signs, the quotient is negative.

If both the numerator and denominator are negative, then the fraction itself is positive because we are dividing a negative by a negative.

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