Multiplication of fractions

Algebra: Adding and subtracting fractions

Multiplication of fractions

The product of two fractions

We can multiply two fractions by multiplying the numerator by the numerator and multiplying the denominator by the denominator.

\[\frac{\orange{a}}{\blue{b}} \cdot \frac{\purple{c}}{\green{d}}=\frac{\orange{a} \cdot \purple{c}}{\blue{b} \cdot \green{d}}\]

Example

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

Simplification

Just as with adding and subtracting of fractions, when multiplying, we simplify the answer if possible.

Example

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

Brackets

Just as with adding and subtracting of fractions, when multiplying 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.

Unless specified otherwise, brackets can remain in the final answer.

Example

\[\begin{aligned} \dfrac{\orange{x^2+1}}{\blue{y+1}} \cdot \dfrac{\purple{y+1}}{\green{x^2}}&=\dfrac{\orange{(x^2 +1)} \cdot \purple{(y+1)}}{\blue{(y+1)} \cdot \green{x^2}} \\ &= \dfrac{{x^2+1}}{{x^2}}\end{aligned}\]

Special case

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

Example

#\begin{array}{rcl}
\orange{x} \cdot \dfrac{\purple{2}}{\green{xy}}
&=&
\dfrac{\orange{x} \cdot \purple{2}}{ \green{xy}}\\
&=&\dfrac{2}{y}
\end{array}#

\[\begin{array}{rcl}
\orange{a} \cdot \dfrac{\purple{c}}{\green{d}}
&=& \dfrac{\orange{a}}{\blue{1}} \cdot \dfrac{\purple{c}}{\green{d}} \\
&&\qquad\blue{\text{rule \(\tfrac{a}{1}=a\)}} \\
&=& \dfrac{\orange{a} \cdot \purple{c}}{\blue{1}\cdot \green{d}} \\
&&\qquad\blue{\text{rule \(\frac{{a}}{{b}} \cdot \frac{{c}}{{d}}=\frac{{a} \cdot {c}}{{b} \cdot {d}}\)}} \\
&=& \dfrac{\orange{a} \cdot \purple{c}}{ \green{d}} \\
&&\qquad\blue{\text{rule \(1\cdot x = x\)}}
\end{array}\]

Calculate and simplify as much as possible:
\[\dfrac{x^2}{z} \cdot \dfrac{y\cdot z^2}{b\cdot x}\]

#{{x\cdot y\cdot z}\over{b}}#

#\begin{array}{rcl}
\dfrac{x^2}{z} \cdot \dfrac{y\cdot z^2}{b\cdot x} &=& \dfrac{ {x^2} \cdot {y\cdot z^2}}{{z} \cdot {b\cdot x}}\\
&& \phantom{xxx}\blue{\text{fractions multiplied by }}\\
&& \phantom{xxx}\blue{\text{multiplying numerator by numerator and denominator by denominator}}\\
&=& \displaystyle {{x\cdot y\cdot z}\over{b}}
\\ && \phantom{xxx}\blue{\text{simplified}}\\
\end{array}#

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