Multiplying Fractions

${\displaystyle \frac{\text{a}}{\text{b}}}\times {\displaystyle \frac{\text{c}}{\text{d}}}={\displaystyle \frac{\text{a}\times \text{c}}{\text{b}\times \text{d}}}$

Example:

Evaluate  $5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}$

$$\begin{array}{rl}5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}& =5\frac{2}{3}\times 2\frac{2}{17}\text{change into improper fraction}\\ \\ 5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}& =(\frac{15}{3}+\frac{2}{3})\times (\frac{34}{17}+\frac{2}{17})\\ \\ 5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}& =\frac{17}{3}\times \frac{36}{17}\\ \\ 5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}& =\frac{17\times 36}{3\times 17}\text{multiply the numerators and multiply the denominators}\\ \\ 5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}& =\frac{\cancel{17}\times {\cancel{36}}^{12}}{{\cancel{3}}^1\times \cancel{17}}\text{simplify}\\ \\ 5{\displaystyle \frac{2}{3}}\times 2{\displaystyle \frac{2}{17}}& =12\end{array}$$

Dividing Fractions

${\displaystyle \frac{\text{a}}{\text{b}}}÷{\displaystyle \frac{\text{c}}{\text{d}}}={\displaystyle \frac{\text{a}}{\text{b}}}\times {\displaystyle \frac{\text{d}}{\text{c}}}$

Example:

Evaluate  $3{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{5}{8}}$

$$\begin{array}{rl}3{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{5}{8}}& =(\frac{12}{4}+\frac{3}{4})÷\frac{5}{8}\text{change into improper fraction}\\ \\ 3{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{5}{8}}& =\frac{15}{4}\times \frac{8}{5}\text{turn the second fraction upside down}\\ \\ 3{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{5}{8}}& =\frac{{\cancel{15}}^3\times {\cancel{8}}^2}{{\cancel{4}}^1\times {\cancel{5}}^1}\\ \\ 3{\displaystyle \frac{3}{4}}÷{\displaystyle \frac{5}{8}}& =6\end{array}$$