Division of fractions

Algebra: Adding and subtracting fractions

Division of fractions

Division of two fractions

A division of two fractions is the same as multiplying by the inverse fraction.

Written as a fraction of fractions:

\[\begin{array}{rcl}\dfrac{\tfrac{\orange{a}}{\blue{b}}}{ \tfrac{\purple{c}}{\green{d}}} &=& \dfrac{\orange{a}}{\blue{b}} \cdot \dfrac{\green{d}}{\purple{c}} \end{array}\]

Example

\[{\begin{array}{rcl}
{\dfrac{\orange{x}}{\blue{y}} : \dfrac{\purple{5}}{\green{x^2}}}
&{=}&
{\dfrac{\orange{x}}{\blue{y}} \cdot \dfrac{{\green{x^2}}}{{\purple{5}}}} \\
&{=}&
{\dfrac{x \cdot x^2}{y \cdot 5}} \\
&{=}&
{\dfrac{x^3}{5y}}
\end{array}}\]

Brackets

Just as with multiplying fractions, when dividing fractions you should pay attention to the brackets. Both the numerator and denominator have to be placed in brackets, if they consist of multiple terms.

Example

\[\begin{array}{rcl}\dfrac{\orange{5 \cdot x}}{\blue{y+1}} : \dfrac{\purple{2 \cdot x}}{\green{y-1}} &=& \dfrac{\orange{5 \cdot x}}{\blue{y+1}} \cdot \dfrac{\green{y-1}}{\purple{2 \cdot x}} \\&=&\dfrac{5 \cdot x \cdot (y-1)}{(y+1) \cdot 2 \cdot x} \\ &=& \dfrac{5 \cdot (y-1)}{2 \cdot (y+1)} \end{array}\]

Simplify the following expression:
\[
\dfrac{x}{x\cdot y}: \dfrac{a\cdot c}{y}
\]

#{{1}\over{a\cdot c}}#

#\begin{array}{rcl}
\displaystyle \dfrac{x}{x\cdot y}: \dfrac{a\cdot c}{y}&=&\displaystyle {{1}\over{y}}\cdot {{y}\over{a\cdot c}}\\
&&\phantom{xxx}\blue{\text{calculation rule } \frac{a}{b} : \frac{c}{d}=\frac{a}{b}\cdot\frac{d}{c}}\\
&=&\dfrac{\left(1\right)\cdot \left(y\right)}{\left(y\right)\cdot \left(a\cdot c\right)}\\
&&\phantom{xxx}\blue{\text{calculation rule }\frac{a}{b} \cdot \frac{c}{d}=\frac{a\cdot c}{b\cdot d}}\\
&=&\displaystyle {{1}\over{a\cdot c}}\\
&&\phantom{xxx}\blue{\text{similar factors in numerator and denominator}}\\
&&\phantom{xxx}\blue{\text{cancelled out}}\\
\end{array}#

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