1.5 Operations with fractions

For example, multiply $\frac{3}{4}·\frac{1}{6}.$

$$\begin{array}{lllll}\frac{3}{4}·\frac{1}{6}\hfill & =\hfill & \frac{3·1}{4·6}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{3}{24}\hfill & \hfill & \text{Now\hspace{0.17em}reduce}.\hfill \\ \hfill & =\hfill & \frac{3·1}{2·2·2·3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\cancel{3}·1}{2·2·2·\cancel{3}}\hfill & \hfill & \text{3\hspace{0.17em}is\hspace{0.17em}the\hspace{0.17em}only\hspace{0.17em}common\hspace{0.17em}factor}.\hfill \\ \hfill & =\hfill & \frac{1}{8}\hfill & \hfill & \hfill \end{array}$$
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.

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

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

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

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

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

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

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

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

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