### Summary

• The reciprocal of a fraction is the fraction we obtain by swapping the numerator and the denominator. The product of a number and its reciprocal is 1.
• Division and multiplication are inverse operations.
• Dividing a fraction by a fraction is the same as multiplying the first fraction by the reciprocal of the second.

#### Example 1

1. $$5÷\dfrac{7}{12}$$
2. $$\dfrac{3}{5}÷\dfrac{4}{7}$$

#### Solution

1. \begin{align}5\div\dfrac{7}{12}&=5\times\dfrac{12}{7}\\\\ &=\dfrac{60}{7}\\\\ &=8\dfrac{4}{7}\end{align}
2. \begin{align}\dfrac{3}{5}\div\dfrac{4}{7}&=\dfrac{3}{5}\times\dfrac{7}{4}\\\\ &=\dfrac{21}{20}\\\\ &=1\dfrac{1}{20}\end{align}

#### Example 2

1. $$\dfrac{3}{4}\times\dfrac{11}{3}\div\dfrac{11}{2}$$
2. $$\dfrac{7}{8}\div\dfrac{1}{2}\times\dfrac{2}{7}$$

#### Solution

1. \begin{align}\dfrac{3}{4}\times\dfrac{11}{3}\div\dfrac{11}{2}&=\dfrac{3^1\hspace{-6mm}\color{darkred}{\setminus}}{4^2\hspace{-6mm}\color{darkred}{\setminus}}\times\dfrac{11^1\hspace{-7mm}\color{darkred}{\setminus}}{3^1\hspace{-6mm}\color{darkred}{\setminus}}\times\dfrac{2^1\hspace{-6mm}\color{darkred}{\setminus}}{11^1\hspace{-7mm}\color{darkred}{\setminus}}\\\\ &=\dfrac{1}{2}\end{align}
2. \begin{align}\dfrac{7}{8}\div\dfrac{1}{2}\times\dfrac{2}{7}&=\dfrac{7^1\hspace{-6mm}\color{darkred}{\setminus}}{8^{4^2\hspace{-3mm}\color{darkred}{\setminus}}\hspace{-6mm}\color{darkred}{\setminus}}\times\dfrac{2^1\hspace{-6mm}\color{darkred}{\setminus}}{1}\times\dfrac{2^1\hspace{-6mm}\color{darkred}{\setminus}}{7^1\hspace{-6mm}\color{darkred}{\setminus}}\\\\ &=\dfrac{1}{2}\end{align}