### Numbers: Fractions

### Division of fractions

Division of fractions

The general rule for division of fractions is: dividing by a fraction is equal to multiplying by the reciprocal of the fraction. This means that if you want to divide by #\frac{\orange{3}}{\blue{4}}#, you need to multiply by #\frac{\blue{4}}{\orange{3}}#.

We can write:

\[\begin{array}{rcl} \dfrac{\orange{\text{numerator}_1}}{\blue{\text{denominator}_1}} : \dfrac{\orange{\text{numerator}_2}}{\blue{\text{denominator}_2}} &=& \dfrac{\orange{\text{numerator}_1}}{\blue{\text{denominator}_1}}\times\dfrac{\blue{\text{denominator}_2}}{\orange{\text{numerator}_2}} \end{array}\]

Written as a fraction of fractions, we have:

\[\begin{array}{rcl} \frac{\left(\tfrac{\orange{\text{numerator}_1}}{\blue{\text{denominator}_1}}\right)}{\left(\tfrac{\orange{\text{numerator}_2}}{\blue{\text{denominator}_2}}\right)} &=& \dfrac{\orange{\text{numerator}_1}}{\blue{\text{denominator}_1}}\times\dfrac{\blue{\text{denominator}_2}}{\orange{\text{numerator}_2}} \end{array}\]

**Examples**

\[\begin{array}{rcl}

\dfrac{\orange{3}}{\blue{5}} : \dfrac{\orange{2}}{\blue{3}} &=& \dfrac{\orange{3}}{\blue{5}}\times\dfrac{\blue{3}}{\orange{2}} \\

&=& \dfrac{9}{10} \\ \\

\frac{\tfrac{\orange{2}}{\blue{5}} }{\tfrac{\orange{3}}{\blue{7}}} &=& \dfrac{\orange{2}}{\blue{5}}\times\dfrac{\blue{7}}{\orange{3}} \\

&=& \dfrac{14}{15}

\end{array}\]

#\begin{array}{rcl}\displaystyle \dfrac{{{8}\over{5}}}{{{1}\over{5}}}&=&\displaystyle {{8}\over{5}} \times \dfrac{5}{1} \\&&\phantom{xxx}\blue{\text{dividing by a fraction equals multiplying by the reciprocal}} \\&=&\dfrac{8 \times 5}{5 \times 1} \\ &&\phantom{xxx}\blue{\text{multiplied numerator and denominator separately}}\\

&=& \dfrac{40}{5} \\ &&\phantom{xxx}\blue{\text{multiplied}}\\

&=& \displaystyle 8 \\ &&\phantom{xxx}\blue{\text{simplified}}

\end{array}#

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