1.5 Operations with fractions

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. This chapter contains many examples of arithmetic techniques that are used directly or indirectly in algebra. Since the chapter is intended as a review, the problem-solving techniques are presented without being developed. Therefore, no work space is provided, nor does the chapter contain all of the pedagogical features of the text. As a review, this chapter can be assigned at the discretion of the instructor and can also be a valuable reference tool for the student.

Overview

• Multiplication of Fractions
• Division of Fractions
• Addition and Subtraction of Fractions

Multiplication of fractions

To multiply two fractions, multiply the numerators together and multiply the denominators together. Reduce to lowest terms if possible.

For example, multiply $\frac{3}{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{6}.$

$\begin{array}{lllll}\frac{3}{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{1}{6}\hfill & =\hfill & \frac{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1}{4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}6}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{3}{24}\hfill & \hfill & \text{Now\hspace{0.17em}reduce}.\hfill \\ \hfill & =\hfill & \frac{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\overline{)3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)3}}\hfill & \hfill & \text{3\hspace{0.17em}is\hspace{0.17em}the\hspace{0.17em}only\hspace{0.17em}common\hspace{0.17em}factor}.\hfill \\ \hfill & =\hfill & \frac{1}{8}\hfill & \hfill & \hfill \end{array}$
Notice that we since had to reduce, we nearly started over again with the original two fractions. If we factor first, then cancel, then multiply, we will save time and energy and still obtain the correct product.

Sample set a

Perform the following multiplications.

$\begin{array}{lllll}\frac{1}{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{8}{9}\hfill & =\hfill & \frac{1}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{1}{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \text{2\hspace{0.17em}is\hspace{0.17em}a\hspace{0.17em}common\hspace{0.17em}factor}\text{.}\hfill \\ \hfill & =\hfill & \frac{1}{1}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2}{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{2}{9}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lllll}\frac{3}{4}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{8}{9}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{12}\hfill & =\hfill & \frac{3}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\overline{)3}}{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}}{\overline{)3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \text{2\hspace{0.17em}and\hspace{0.17em}3\hspace{0.17em}are\hspace{0.17em}common\hspace{0.17em}factors}\text{.}\hfill \\ \hfill & =\hfill & \frac{1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5}{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{5}{18}\hfill & \hfill & \hfill \end{array}$

Reciprocals

Two numbers whose product is 1 are reciprocals of each other. For example, since $\frac{4}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{4}=1,\frac{4}{5}$ and $\frac{5}{4}$ are reciprocals of each other. Some other pairs of reciprocals are listed below.

$\begin{array}{lllll}\frac{2}{7},\frac{7}{2}\hfill & \hfill & \frac{3}{4},\frac{4}{3}\hfill & \hfill & \frac{6}{1},\frac{1}{6}\hfill \end{array}$

Reciprocals are used in division of fractions.

Division of fractions

To divide a first fraction by a second fraction, multiply the first fraction by the reciprocal of the second fraction. Reduce if possible.

This method is sometimes called the “invert and multiply” method.

Sample set b

Perform the following divisions.

$\begin{array}{lllll}\frac{1}{3}÷\frac{3}{4}.\hfill & \hfill & \hfill & \hfill & \text{The\hspace{0.17em}divisor\hspace{0.17em}is\hspace{0.17em}}\frac{3}{4}\text{.\hspace{0.17em}Its\hspace{0.17em}reciprocal\hspace{0.17em}is\hspace{0.17em}}\frac{4}{3}.\hfill \\ \frac{1}{3}÷\frac{3}{4}\hfill & =\hfill & \frac{1}{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{4}{3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4}{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{4}{9}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lllll}\frac{3}{8}÷\frac{5}{4}.\hfill & \hfill & \hfill & \hfill & \text{The\hspace{0.17em}divisor\hspace{0.17em}is\hspace{0.17em}}\frac{5}{4}\text{.\hspace{0.17em}Its\hspace{0.17em}reciprocal\hspace{0.17em}is\hspace{0.17em}}\frac{4}{5}.\hfill \\ \frac{3}{8}÷\frac{5}{4}\hfill & =\hfill & \frac{3}{8}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{4}{5}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{3}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{5}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{3}{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)2}}{5}\hfill & \hfill & \text{2\hspace{0.17em}is\hspace{0.17em}a\hspace{0.17em}common\hspace{0.17em}factor}\text{.}\hfill \\ \hfill & =\hfill & \frac{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}1}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}5}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{3}{10}\hfill & \hfill & \hfill \end{array}$

$\begin{array}{lllll}\frac{5}{6}÷\frac{5}{12}.\hfill & \hfill & \hfill & \hfill & \text{The\hspace{0.17em}divisor\hspace{0.17em}is\hspace{0.17em}}\frac{5}{12}\text{.\hspace{0.17em}Its\hspace{0.17em}reciprocal\hspace{0.17em}is\hspace{0.17em}}\frac{12}{5}.\hfill \\ \frac{5}{6}÷\frac{5}{12}\hfill & =\hfill & \frac{5}{6}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{12}{5}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{5}{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}{5}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\overline{)5}}{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)3}}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{\overline{)2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\overline{)3}}{\overline{)5}}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{1}\hfill & \hfill & \hfill \\ \hfill & =\hfill & 2\hfill & \hfill & \hfill \end{array}$

Fractions with like denominators

To add (or subtract) two or more fractions that have the same denominators, add (or subtract) the numerators and place the resulting sum over the common denominator. Reduce if possible.

CAUTION

Add or subtract only the numerators. Do not add or subtract the denominators!

Sample set c

Find the following sums.

$\begin{array}{l}\begin{array}{lll}\frac{3}{7}+\frac{2}{7}.\hfill & \hfill & \text{The\hspace{0.17em}denominators\hspace{0.17em}are\hspace{0.17em}the\hspace{0.17em}same}.\text{\hspace{0.17em}Add\hspace{0.17em}the\hspace{0.17em}numerators\hspace{0.17em}and\hspace{0.17em}place\hspace{0.17em}the\hspace{0.17em}sum\hspace{0.17em}over\hspace{0.17em}7}.\hfill \end{array}\\ \frac{3}{7}+\frac{2}{7}=\frac{3+2}{7}=\frac{5}{7}\end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{7}{9}-\frac{4}{9}.\hfill & \hfill & \text{The\hspace{0.17em}denominators\hspace{0.17em}are\hspace{0.17em}the\hspace{0.17em}same}.\text{\hspace{0.17em}Subtract\hspace{0.17em}4\hspace{0.17em}from\hspace{0.17em}7\hspace{0.17em}and\hspace{0.17em}place\hspace{0.17em}the\hspace{0.17em}difference\hspace{0.17em}over\hspace{0.17em}9}.\hfill \end{array}\\ \frac{7}{9}-\frac{4}{9}=\frac{7-4}{9}=\frac{3}{9}=\frac{1}{3}\end{array}$

Fractions can only be added or subtracted conveniently if they have like denominators.

Fractions with unlike denominators

To add or subtract fractions having unlike denominators, convert each fraction to an equivalent fraction having as the denominator the least common multiple of the original denominators.

The least common multiple of the original denominators is commonly referred to as the least common denominator (LCD). See Section ( [link] ) for the technique of finding the least common multiple of several numbers.

Sample set d

Find each sum or difference.

$\begin{array}{l}\begin{array}{lll}\frac{1}{6}+\frac{3}{4}.\hfill & \hfill & \text{The\hspace{0.17em}denominators\hspace{0.17em}are\hspace{0.17em}not\hspace{0.17em}alike}.\text{\hspace{0.17em}Find\hspace{0.17em}the\hspace{0.17em}LCD\hspace{0.17em}of\hspace{0.17em}6\hspace{0.17em}and\hspace{0.17em}4}.\hfill \\ \left\{\begin{array}{l}6=2\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\\ 4={2}^{2}\end{array}\hfill & \hfill & \text{The\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}}{2}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3=4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3=12.\hfill \end{array}\\ \text{Convert\hspace{0.17em}each\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}original\hspace{0.17em}fractions\hspace{0.17em}to\hspace{0.17em}equivalent\hspace{0.17em}fractions\hspace{0.17em}having\hspace{0.17em}the\hspace{0.17em}common\hspace{0.17em}denominator\hspace{0.17em}12}\text{.}\\ \begin{array}{lll}\frac{1}{6}=\frac{1\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}{6\text{\hspace{0.17em}}·\text{\hspace{0.17em}}2}=\frac{2}{12}\hfill & \hfill & \frac{3}{4}=\frac{3\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}{4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}=\frac{9}{12}\hfill \end{array}\\ \text{Now\hspace{0.17em}we\hspace{0.17em}can\hspace{0.17em}proceed\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}addition}.\\ \begin{array}{lll}\frac{1}{6}+\frac{3}{4}\hfill & =\hfill & \frac{2}{12}+\frac{9}{12}\hfill \\ \hfill & =\hfill & \frac{2+9}{12}\hfill \\ \hfill & =\hfill & \frac{11}{12}\hfill \end{array}\end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{5}{9}-\frac{5}{12}.\hfill & \hfill & \text{The\hspace{0.17em}denominators\hspace{0.17em}are\hspace{0.17em}not\hspace{0.17em}alike}.\text{\hspace{0.17em}Find\hspace{0.17em}the\hspace{0.17em}LCD\hspace{0.17em}of\hspace{0.17em}9\hspace{0.17em}and\hspace{0.17em}12}.\hfill \\ \left\{\begin{array}{l}9={3}^{2}\\ 12={2}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3\end{array}\hfill & \hfill & \text{The\hspace{0.17em}LCD\hspace{0.17em}is\hspace{0.17em}}{2}^{2}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}{3}^{2}=4\text{\hspace{0.17em}}·\text{\hspace{0.17em}}9=36.\hfill \end{array}\\ \text{Convert\hspace{0.17em}each\hspace{0.17em}of\hspace{0.17em}the\hspace{0.17em}original\hspace{0.17em}fractions\hspace{0.17em}to\hspace{0.17em}equivalent\hspace{0.17em}fractions\hspace{0.17em}having\hspace{0.17em}the\hspace{0.17em}common\hspace{0.17em}denominator\hspace{0.17em}36}\text{.}\\ \begin{array}{lll}\frac{5}{9}=\frac{5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4}{9\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4}=\frac{20}{36}\hfill & \hfill & \frac{5}{12}=\frac{5\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}{12\text{\hspace{0.17em}}·\text{\hspace{0.17em}}3}=\frac{15}{36}\hfill \end{array}\\ \text{Now\hspace{0.17em}we\hspace{0.17em}can\hspace{0.17em}proceed\hspace{0.17em}with\hspace{0.17em}the\hspace{0.17em}subtraction}.\\ \begin{array}{lll}\frac{5}{9}-\frac{5}{12}\hfill & =\hfill & \frac{20}{36}-\frac{15}{36}\hfill \\ \hfill & =\hfill & \frac{20-15}{36}\hfill \\ \hfill & =\hfill & \frac{5}{36}\hfill \end{array}\end{array}$

Exercises

For the following problems, perform each indicated operation.

$\frac{1}{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{4}{3}$

$\frac{4}{9}$

$\frac{1}{3}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{2}{3}$

$\frac{2}{5}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{5}{6}$

$\frac{1}{3}$

$\frac{5}{6}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{14}{15}$

$\frac{9}{16}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{20}{27}$

$\frac{5}{12}$

$\frac{35}{36}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{48}{55}$

$\frac{21}{25}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{15}{14}$

$\frac{9}{10}$

$\frac{76}{99}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{66}{38}$

$\frac{3}{7}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{14}{18}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{6}{2}$

1

$\frac{14}{15}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{21}{28}\text{\hspace{0.17em}}·\text{\hspace{0.17em}}\frac{45}{7}$

$\frac{5}{9}÷\frac{5}{6}$

$\frac{2}{3}$

$\frac{9}{16}÷\frac{15}{8}$

$\frac{4}{9}÷\frac{6}{15}$

$\frac{10}{9}$

$\frac{25}{49}÷\frac{4}{9}$

$\frac{15}{4}÷\frac{27}{8}$

$\frac{10}{9}$

$\frac{24}{75}÷\frac{8}{15}$

$\frac{57}{8}÷\frac{7}{8}$

$\frac{57}{7}$

$\frac{7}{10}÷\frac{10}{7}$

$\frac{3}{8}+\frac{2}{8}$

$\frac{5}{8}$

$\frac{3}{11}+\frac{4}{11}$

$\frac{5}{12}+\frac{7}{12}$

1

$\frac{11}{16}-\frac{2}{16}$

$\frac{15}{23}-\frac{2}{23}$

$\frac{13}{23}$

$\frac{3}{11}+\frac{1}{11}+\frac{5}{11}$

$\frac{16}{20}+\frac{1}{20}+\frac{2}{20}$

$\frac{19}{20}$

$\frac{3}{8}+\frac{2}{8}-\frac{1}{8}$

$\frac{11}{16}+\frac{9}{16}-\frac{5}{16}$

$\frac{15}{16}$

$\frac{1}{2}+\frac{1}{6}$

$\frac{1}{8}+\frac{1}{2}$

$\frac{5}{8}$

$\frac{3}{4}+\frac{1}{3}$

$\frac{5}{8}+\frac{2}{3}$

$\frac{31}{24}$

$\frac{6}{7}-\frac{1}{4}$

$\frac{8}{15}-\frac{3}{10}$

$\frac{5}{6}$

$\frac{1}{15}+\frac{5}{12}$

$\frac{25}{36}-\frac{7}{10}$

$\frac{-1}{180}$

$\frac{9}{28}-\frac{4}{45}$

$\frac{8}{15}-\frac{3}{10}$

$\frac{7}{30}$

$\frac{1}{16}+\frac{3}{4}-\frac{3}{8}$

$\frac{8}{3}-\frac{1}{4}+\frac{7}{36}$

$\frac{47}{18}$

$\frac{3}{4}-\frac{3}{22}+\frac{5}{24}$

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