# Multiplication and Division of Fraction

## Basic Concept

#### Multiplying Fractions

$$\large \dfrac{\text{a}}{\text{b}}\times \dfrac{\text{c}}{\text{d}} = \dfrac{\text{a} \times \text{c}}{\text{b} \times \text{d}}$$

Example:

Evaluate  $$5\dfrac{2}{3} \times 2\dfrac{2}{17}$$

\begin{equation*}
\begin{split}
5\dfrac{2}{3} \times 2\dfrac{2}{17}& = 5\frac{2}{3} \times 2\frac{2}{17}\:\:\:\:\:\color {blue}\text{change into improper fraction}\\\\
5\dfrac{2}{3} \times 2\dfrac{2}{17}& = \left(\frac{15}{3} + \frac{2}{3}\right) \times \left(\frac{34}{17} + \frac{2}{17}\right)\\\\
5\dfrac{2}{3} \times 2\dfrac{2}{17}& = \frac{17}{3} \times \frac{36}{17}\\\\
5\dfrac{2}{3} \times 2\dfrac{2}{17}& =  \frac{17 \times 36}{3 \times 17}\:\:\:\:\color{blue}\text{multiply the numerators and multiply the denominators}\\\\
5\dfrac{2}{3} \times 2\dfrac{2}{17}& = \frac{\cancel {17} \times \cancelto {12}{36}}{\cancelto {1}{3} \times \cancel {17}}\:\:\:\:\:\color {blue} \text{simplify}\\\\
5\dfrac{2}{3} \times 2\dfrac{2}{17}& = 12
\end{split}
\end{equation*}

#### Dividing Fractions

$$\large \dfrac{\text{a}}{\text{b}}\div \dfrac{\text{c}}{\text{d}} = \dfrac{\text{a}}{\text{b}}\times\dfrac{\text{d}}{\text{c}}$$

Example:

Evaluate  $$3\dfrac{3}{4} \div \dfrac{5}{8}$$

\begin{equation*}
\begin{split}
3\dfrac{3}{4} \div \dfrac{5}{8}& =  \left(\frac{12}{4} + \frac{3}{4}\right) \div \frac{5}{8}\:\:\:\:\:\color {blue}\text{change into improper fraction}\\\\
3\dfrac{3}{4} \div \dfrac{5}{8}& =  \frac{15}{4} \times \color{red} \frac{8}{5}\:\:\:\:\:\color{blue} \text{turn the second fraction upside down}\\\\
3\dfrac{3}{4} \div \dfrac{5}{8}& =  \frac{\cancelto {3}{15} \times \cancelto {2}{8}}{\cancelto {1}{4} \times \cancelto {1}{5}}\\\\
3\dfrac{3}{4} \div \dfrac{5}{8}& =  6
\end{split}
\end{equation*}