# multiplication and division of fractions

## Primary tabs

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The familiar methods
\begin{align}
\frac{a}{b}\cdot\frac{c}{d} = \frac{ac}{bd}
\end{align}
and
\begin{align}
\end{align}
for multiplying and dividing fractions are justified simply by using only the

For (1), we show that its left hand side is the quotient of $ac$ and $bd$:
$$bd\cdot\left(\frac{a}{b}\cdot\frac{c}{d}\right) = \left(b\cdot\frac{a}{b}\right)\left(d\cdot\frac{c}{d}\right) = ac.$$

For (2), we show that its right hand side is the quotient of
$\frac{a}{b}$ and $\frac{c}{d}$:
$$\frac{c}{d}\cdot\frac{ad}{bc} = \frac{c}{d}\cdot\frac{a}{b}\cdot\frac{d}{c} = \frac{cd}{dc}\cdot\frac{a}{b} = \frac{a}{b}$$

Thus the formulas (1) and (2) are true not only for
\PMlinkname{integers}{integer} $a, b, c, d$ but for all