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} |
#{{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}#
#\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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