1.4 Equivalent fractions

$\frac{15}{25},\frac{12}{20},\frac{3}{5}$

$\frac{1}{3},\frac{2}{6},\frac{3}{9},\frac{4}{12}$

$\frac{7}{6},\frac{14}{12},\frac{21}{18},\frac{28}{24},\frac{35}{30}$

$$\begin{array}{lllll}\frac{6}{18}\hfill & =\hfill & \frac{2·3}{2·3·3}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\cancel{2}·\cancel{3}}{\cancel{2}·\cancel{3}·3}\hfill & \hfill & \text{2\hspace{0.17em}and\hspace{0.17em}3\hspace{0.17em}are\hspace{0.17em}common\hspace{0.17em}factors}.\hfill \\ \hfill & =\hfill & \frac{1}{3}\hfill & \hfill & \hfill \end{array}$$

$$\begin{array}{lllll}\frac{16}{20}\hfill & =\hfill & \frac{2·2·2·2}{2·2·5}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\cancel{2}·\cancel{2}·2·2}{\cancel{2}·\cancel{2}·5}\hfill & \hfill & \text{2\hspace{0.17em}is\hspace{0.17em}the\hspace{0.17em}only\hspace{0.17em}common\hspace{0.17em}factor}\text{.}\hfill \\ \hfill & =\hfill & \frac{4}{5}\hfill & \hfill & \hfill \end{array}$$

$$\begin{array}{lllll}\frac{56}{70}\hfill & =\hfill & \frac{2·4·7}{2·5·7}\hfill & \hfill & \hfill \\ \hfill & =\hfill & \frac{\cancel{2}·4·\cancel{7}}{\cancel{2}·5·\cancel{7}}\hfill & \hfill & \text{2\hspace{0.17em}and\hspace{0.17em}7\hspace{0.17em}are\hspace{0.17em}common\hspace{0.17em}factors}.\hfill \\ \hfill & =\hfill & \frac{4}{5}\hfill & \hfill & \hfill \end{array}$$

$$\begin{array}{l}\begin{array}{lllll}\frac{8}{15}\hfill & =\hfill & \frac{2·2·2}{3·5}\hfill & \hfill & \text{There\hspace{0.17em}are\hspace{0.17em}no\hspace{0.17em}common\hspace{0.17em}factors}.\hfill \end{array}\\ \text{Thus},\frac{8}{15}\text{\hspace{0.17em}is\hspace{0.17em}reduced\hspace{0.17em}to\hspace{0.17em}lowest\hspace{0.17em}terms}.\end{array}$$

$\begin{array}{l}\begin{array}{lllll}\frac{3}{7}\hfill & =\hfill & \frac{?}{35}.\hfill & \hfill & \text{Divide\hspace{0.17em}the\hspace{0.17em}original\hspace{0.17em}denominator},\text{\hspace{0.17em}7},\text{\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}new\hspace{0.17em}denominator},\hfill \end{array}35.35÷7=5.\\ \text{Multiply\hspace{0.17em}the\hspace{0.17em}original\hspace{0.17em}numerator\hspace{0.17em}by\hspace{0.17em}5}.\\ \begin{array}{lllll}\frac{3}{7}\hfill & =\hfill & \frac{3·5}{7·5}\hfill & =\hfill & \frac{15}{35}\hfill \end{array}\end{array}$

$\begin{array}{l}\begin{array}{lllll}\frac{5}{6}\hfill & =\hfill & \frac{45}{?}.\hfill & \hfill & \text{Divide\hspace{0.17em}the\hspace{0.17em}original\hspace{0.17em}numerator},\text{\hspace{0.17em}5},\text{\hspace{0.17em}into\hspace{0.17em}the\hspace{0.17em}new\hspace{0.17em}numerator},\hfill \end{array}45.45÷5=9.\\ \text{Multiply\hspace{0.17em}the\hspace{0.17em}original\hspace{0.17em}denominator\hspace{0.17em}by\hspace{0.17em}9}.\\ \begin{array}{lllll}\frac{5}{6}\hfill & =\hfill & \frac{5·9}{6·9}\hfill & =\hfill & \frac{45}{54}\hfill \end{array}\end{array}$