# 4.3 Multiply and divide mixed numbers and complex fractions

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By the end of this section, you will be able to:
• Multiply and divide mixed numbers
• Translate phrases to expressions with fractions
• Simplify complex fractions
• Simplify expressions written with a fraction bar

Before you get started, take this readiness quiz.

1. Divide and reduce, if possible: $\left(4+5\right)÷\left(10-7\right).$
If you missed this problem, review Add Integers .
2. Multiply and write the answer in simplified form: $\frac{1}{8}·\frac{2}{3}.$
If you missed this problem, review Multiply and Divide Fractions .
3. Convert $2·\frac{3}{5}$ into an improper fraction.
If you missed this problem, review Visualize Fractions .

## Multiply and divide mixed numbers

In the previous section, you learned how to multiply and divide fractions. All of the examples there used either proper or improper fractions. What happens when you are asked to multiply or divide mixed numbers? Remember that we can convert a mixed number    to an improper fraction . And you learned how to do that in Visualize Fractions .

Multiply: $3\frac{1}{3}·\frac{5}{8}$

## Solution

 $3\frac{1}{3}·\frac{5}{8}$ Convert $3\frac{1}{3}$ to an improper fraction. $\frac{10}{3}·\frac{5}{8}$ Multiply. $\frac{10·5}{3·8}$ Look for common factors. $\frac{2̸·5·5}{3·2̸·4}$ Remove common factors. $\frac{5·5}{3·4}$ Simplify. $\frac{25}{12}$

Notice that we left the answer as an improper fraction, $\frac{25}{12},$ and did not convert it to a mixed number. In algebra, it is preferable to write answers as improper fractions instead of mixed numbers. This avoids any possible confusion between $2\frac{1}{12}$ and $2·\frac{1}{12}.$

Multiply, and write your answer in simplified form: $5\frac{2}{3}·\frac{6}{17}.$

2

Multiply, and write your answer in simplified form: $\frac{3}{7}·5\frac{1}{4}.$

$\frac{9}{4}$

## Multiply or divide mixed numbers.

1. Convert the mixed numbers to improper fractions.
2. Follow the rules for fraction multiplication or division.
3. Simplify if possible.

Multiply, and write your answer in simplified form: $2\frac{4}{5}\phantom{\rule{0.2em}{0ex}}\left(-1\frac{7}{8}\right).$

## Solution

 $2\frac{4}{5}\phantom{\rule{0.2em}{0ex}}\left(-1\frac{7}{8}\right)$ Convert mixed numbers to improper fractions. $\frac{14}{5}\phantom{\rule{0.2em}{0ex}}\left(-\frac{15}{8}\right)$ Multiply. $-\phantom{\rule{0.2em}{0ex}}\frac{14·15}{5·8}$ Look for common factors. $-\phantom{\rule{0.2em}{0ex}}\frac{2̸·7·5̸·3}{5̸·2̸·4}$ Remove common factors. $-\phantom{\rule{0.2em}{0ex}}\frac{\phantom{\rule{0.2em}{0ex}}7·3}{4}$ Simplify. $-\phantom{\rule{0.2em}{0ex}}\frac{\phantom{\rule{0.2em}{0ex}}21}{4}$

Multiply, and write your answer in simplified form. $5\frac{5}{7}\phantom{\rule{0.2em}{0ex}}\left(-2\frac{5}{8}\right).$

−15

Multiply, and write your answer in simplified form. $-3\frac{2}{5}·4\frac{1}{6}.$

$-\frac{85}{6}$

Divide, and write your answer in simplified form: $3\frac{4}{7}\phantom{\rule{0.2em}{0ex}}÷\phantom{\rule{0.2em}{0ex}}5.$

## Solution

 $3\frac{4}{7}\phantom{\rule{0.2em}{0ex}}÷\phantom{\rule{0.2em}{0ex}}5$ Convert mixed numbers to improper fractions. $\frac{25}{7}\phantom{\rule{0.2em}{0ex}}÷\phantom{\rule{0.2em}{0ex}}\frac{5}{1}$ Multiply the first fraction by the reciprocal of the second. $\frac{25}{7}·\frac{1}{5}$ Multiply. $\frac{25·1}{7·5}$ Look for common factors. $\frac{5̸·5·1}{7·5̸}$ Remove common factors. $\frac{5·1}{7}$ Simplify. $\frac{5}{7}$

Divide, and write your answer in simplified form: $4\frac{3}{8}÷7.$

$\frac{5}{8}$

Divide, and write your answer in simplified form: $2\frac{5}{8}÷3.$

$\frac{7}{8}$

Divide: $2\frac{1}{2}÷1\frac{1}{4}.$

## Solution

 $2\frac{1}{2}÷1\frac{1}{4}$ Convert mixed numbers to improper fractions. $\frac{5}{2}÷\frac{5}{4}$ Multiply the first fraction by the reciprocal of the second. $\frac{5}{2}·\frac{4}{5}$ Multiply. $\frac{5·4}{2·5}$ Look for common factors. $\frac{5̸·2̸·2}{2̸·1·5̸}$ Remove common factors. $\frac{2}{1}$ Simplify. $2$

Divide, and write your answer in simplified form: $2\frac{2}{3}÷1\frac{1}{3}.$

2

Divide, and write your answer in simplified form: $3\frac{3}{4}÷1\frac{1}{2}.$

$\frac{5}{2}$

## Translate phrases to expressions with fractions

The words quotient and ratio are often used to describe fractions. In Subtract Whole Numbers , we defined quotient as the result of division. The quotient of $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ is the result you get from dividing $a\phantom{\rule{0.2em}{0ex}}\text{by}\phantom{\rule{0.2em}{0ex}}b\phantom{\rule{0.2em}{0ex}},$ or $\frac{a}{b}.$ Let’s practice translating some phrases into algebraic expressions using these terms.

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