# 5.5 Complex fractions

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses complex fractions. By the end of the module students should be able to distinguish between simple and complex fractions and convert a complex fraction to a simple fraction.

## Section overview

• Simple Fractions and Complex Fractions
• Converting Complex Fractions to Simple Fractions

## Simple fraction

A simple fraction is any fraction in which the numerator is any whole number and the denominator is any nonzero whole number. Some examples are the following:

$\frac{1}{2}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{4}{3}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{\text{763}}{1,\text{000}}$

## Complex fraction

A complex fraction is any fraction in which the numerator and/or the denomina­tor is a fraction; it is a fraction of fractions. Some examples of complex fractions are the following:

$\frac{\frac{3}{4}}{\frac{5}{6}}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{\frac{1}{3}}{2}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{6}{\frac{9}{\text{10}}}\phantom{\rule{6px}{0ex}},\phantom{\rule{6px}{0ex}}\frac{4+\frac{3}{8}}{7-\frac{5}{6}}$

## Converting complex fractions to simple fractions

The goal here is to convert a complex fraction to a simple fraction. We can do so by employing the methods of adding, subtracting, multiplying, and dividing fractions. Recall from [link] that a fraction bar serves as a grouping symbol separating the fractional quantity into two individual groups. We proceed in simplifying a complex fraction to a simple fraction by simplifying the numerator and the denom­inator of the complex fraction separately. We will simplify the numerator and denominator completely before removing the fraction bar by dividing. This tech­nique is illustrated in problems 3, 4, 5, and 6 of [link] .

## Sample set a

Convert each of the following complex fractions to a simple fraction.

$\frac{\frac{3}{8}}{\frac{\text{15}}{\text{16}}}$

Convert this complex fraction to a simple fraction by performing the indicated division.

$\begin{array}{cccc}\hfill \frac{\frac{3}{8}}{\frac{\text{15}}{\text{16}}}& =& \frac{3}{8}÷\frac{\text{15}}{\text{16}}\hfill & \text{The divisor is}\frac{\text{15}}{\text{16}}.\text{Invert}\frac{\text{15}}{\text{16}}\text{and multiply.}\hfill \\ & =& \frac{\stackrel{1}{\overline{)3}}}{\underset{1}{\overline{)8}}}\cdot \frac{\stackrel{2}{\overline{)16}}}{\underset{5}{\overline{)15}}}=\frac{1\cdot 2}{1\cdot 5}=\frac{2}{5}\hfill & \end{array}$

$\begin{array}{cc}\frac{\frac{4}{9}}{6}\hfill & \text{Write 6 as}\frac{6}{1}\text{and divide.}\hfill \end{array}$

$\begin{array}{ccc}\hfill \frac{\frac{4}{9}}{\frac{6}{1}}& =& \frac{4}{9}÷\frac{6}{1}\hfill \\ & =& \frac{\stackrel{2}{\overline{)4}}}{9}\cdot \frac{1}{\underset{3}{\overline{)6}}}=\frac{2\cdot 1}{9\cdot 3}=\frac{2}{\text{27}}\hfill \end{array}$

$\begin{array}{cc}\frac{5+\frac{3}{4}}{\text{46}}\hfill & \text{Simplify the numerator.}\hfill \end{array}$

$\begin{array}{cc}\frac{\frac{4\cdot 5+3}{4}}{\text{46}}=\frac{\frac{\text{20}+3}{4}}{\text{46}}=\frac{\frac{\text{23}}{4}}{\text{46}}\hfill & \text{Write 46 as}\frac{\text{46}}{1}.\hfill \end{array}$

$\begin{array}{ccc}\hfill \frac{\frac{\text{23}}{4}}{\frac{\text{46}}{1}}& =& \frac{\text{23}}{4}÷\frac{\text{46}}{1}\hfill \\ & =& \frac{\stackrel{1}{\overline{)23}}}{4}\cdot \frac{1}{\underset{2}{\overline{)46}}}=\frac{1\cdot 1}{4\cdot 2}=\frac{1}{8}\hfill \end{array}$

$\frac{\frac{1}{4}+\frac{3}{8}}{\frac{1}{2}+\frac{\text{13}}{\text{24}}}=\frac{\frac{2}{8}+\frac{3}{8}}{\frac{\text{12}}{\text{24}}+\frac{\text{13}}{\text{24}}}=\frac{\frac{2+3}{8}}{\frac{\text{12}+\text{13}}{\text{24}}}=\frac{\frac{5}{8}}{\frac{\text{25}}{\text{24}}}=\frac{5}{8}÷\frac{\text{25}}{\text{24}}$

$\frac{5}{8}÷\frac{\text{25}}{\text{24}}=\frac{\stackrel{1}{\overline{)5}}}{\underset{1}{\overline{)8}}}\cdot \frac{\stackrel{3}{\overline{)24}}}{\underset{5}{\overline{)25}}}=\frac{1\cdot 3}{1\cdot 5}=\frac{3}{5}$

$\begin{array}{ccc}\hfill \frac{4+\frac{5}{6}}{7-\frac{1}{3}}=\frac{\frac{4\cdot 6+5}{6}}{\frac{7\cdot 3-1}{3}}=\frac{\frac{\text{29}}{6}}{\frac{\text{20}}{3}}& =& \frac{\text{29}}{6}÷\frac{\text{20}}{3}\hfill \\ & =& \frac{\text{29}}{\underset{2}{\overline{)6}}}\cdot \frac{\stackrel{1}{\overline{)3}}}{\text{20}}=\frac{\text{29}}{\text{40}}\hfill \end{array}$

$\frac{\text{11}+\frac{3}{\text{10}}}{4\frac{4}{5}}=\frac{\frac{\text{11}\cdot \text{10}+3}{\text{10}}}{\frac{4\cdot 5+4}{5}}=\frac{\frac{\text{110}+3}{\text{10}}}{\frac{\text{20}+4}{5}}=\frac{\frac{\text{113}}{\text{10}}}{\frac{\text{24}}{5}}=\frac{\text{113}}{\text{10}}÷\frac{\text{24}}{5}$

$\frac{\text{113}}{\text{10}}÷\frac{\text{24}}{5}=\frac{\text{113}}{\underset{2}{\overline{)10}}}\cdot \frac{\stackrel{1}{\overline{)5}}}{\text{24}}=\frac{\text{113}\cdot 1}{2\cdot \text{24}}=\frac{\text{113}}{\text{48}}=2\frac{\text{17}}{\text{48}}$

## Practice set a

Convert each of the following complex fractions to a simple fraction.

$\frac{\frac{4}{9}}{\frac{8}{\text{15}}}$

$\frac{5}{6}$

$\frac{\frac{7}{\text{10}}}{\text{28}}$

$\frac{1}{\text{40}}$

$\frac{5+\frac{2}{5}}{3+\frac{3}{5}}$

$\frac{3}{2}$

$\frac{\frac{1}{8}+\frac{7}{8}}{6-\frac{3}{\text{10}}}$

$\frac{\text{10}}{\text{57}}$

$\frac{\frac{1}{6}+\frac{5}{8}}{\frac{5}{9}-\frac{1}{4}}$

$2\frac{\text{13}}{\text{22}}$

$\frac{\text{16}-\text{10}\frac{2}{3}}{\text{11}\frac{5}{6}-7\frac{7}{6}}$

$1\frac{5}{\text{11}}$

## Exercises

Simplify each fraction.

$\frac{\frac{3}{5}}{\frac{9}{\text{15}}}$

1

$\frac{\frac{1}{3}}{\frac{1}{9}}$

$\frac{\frac{1}{4}}{\frac{5}{\text{12}}}$

$\frac{3}{5}$

$\frac{\frac{8}{9}}{\frac{4}{\text{15}}}$

$\frac{6+\frac{1}{4}}{\text{11}+\frac{1}{4}}$

$\frac{5}{9}$

$\frac{2+\frac{1}{2}}{7+\frac{1}{2}}$

$\frac{5+\frac{1}{3}}{2+\frac{2}{\text{15}}}$

$\frac{5}{2}$

$\frac{9+\frac{1}{2}}{1+\frac{8}{\text{11}}}$

$\frac{4+\frac{\text{10}}{\text{13}}}{\frac{\text{12}}{\text{39}}}$

$\frac{\text{31}}{2}$

$\frac{\frac{1}{3}+\frac{2}{7}}{\frac{\text{26}}{\text{21}}}$

$\frac{\frac{5}{6}-\frac{1}{4}}{\frac{1}{\text{12}}}$

7

$\frac{\frac{3}{\text{10}}+\frac{4}{\text{12}}}{\frac{\text{19}}{\text{90}}}$

$\frac{\frac{9}{\text{16}}+\frac{7}{3}}{\frac{\text{139}}{\text{48}}}$

1

$\frac{\frac{1}{\text{288}}}{\frac{8}{9}-\frac{3}{\text{16}}}$

$\frac{\frac{27}{\text{429}}}{\frac{5}{11}-\frac{1}{\text{13}}}$

$\frac{1}{6}$

$\frac{\frac{1}{3}+\frac{2}{5}}{\frac{3}{5}+\frac{\text{17}}{\text{45}}}$

$\frac{\frac{9}{\text{70}}+\frac{5}{\text{42}}}{\frac{\text{13}}{\text{30}}-\frac{1}{\text{21}}}$

$\frac{\text{52}}{\text{81}}$

$\frac{\frac{1}{\text{16}}+\frac{1}{\text{14}}}{\frac{2}{3}-\frac{\text{13}}{\text{60}}}$

$\frac{\frac{3}{\text{20}}+\frac{\text{11}}{\text{12}}}{\frac{\text{19}}{7}-1\frac{\text{11}}{\text{35}}}$

$\frac{\text{16}}{\text{21}}$

$\frac{2\frac{2}{3}-1\frac{1}{2}}{\frac{1}{4}+1\frac{1}{\text{16}}}$

$\frac{3\frac{1}{5}+3\frac{1}{3}}{\frac{6}{5}-\frac{\text{15}}{\text{63}}}$

$\frac{\text{686}}{\text{101}}$

$\frac{\frac{1\frac{1}{2}+\text{15}}{5\frac{1}{4}-3\frac{5}{\text{12}}}}{\frac{8\frac{1}{3}-4\frac{1}{2}}{\text{11}\frac{2}{3}-5\frac{\text{11}}{\text{12}}}}$

$\frac{\frac{5\frac{3}{4}+3\frac{1}{5}}{2\frac{1}{5}+\text{15}\frac{7}{\text{10}}}}{\frac{9\frac{1}{2}-4\frac{1}{6}}{\frac{1}{8}+2\frac{1}{\text{120}}}}$

$\frac{1}{3}$

## Exercises for review

( [link] ) Find the prime factorization of 882.

( [link] ) Convert $\frac{\text{62}}{7}$ to a mixed number.

$8\frac{6}{7}$

( [link] ) Reduce $\frac{\text{114}}{\text{342}}$ to lowest terms.

( [link] ) Find the value of $6\frac{3}{8}-4\frac{5}{6}$ .

$1\frac{\text{13}}{\text{24}}$ or $\frac{\text{37}}{\text{24}}$

( [link] ) Arrange from smallest to largest: $\frac{1}{2}$ , $\frac{3}{5}$ , $\frac{4}{7}$ .

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Source:  OpenStax, Fundamentals of mathematics. OpenStax CNX. Aug 18, 2010 Download for free at http://cnx.org/content/col10615/1.4
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