<s>[INST] Proof of dividing fractions $\frac{a/b}{c/d}=\frac{ad}{bc}$ For dividing two fractional expressions, how does the division sign turns into multiplication? Is there a step by step proof which proves
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$
 [/INST] Write $\frac{a}{b} \div \frac{c}{d}$ as
$$
\frac{\ \frac{a}{b}\ }{\frac{c}{d}}.
$$
Suppose you wanted to clear the denominator of this compound fraction. You could try multiplication by $\frac{d}{c}$, but you'll have to multiply the top and the bottom of the fraction to avoid changing it. So, you end up with
$$
\frac{\ \frac{a}{b}\ }{\frac{c}{d}} = \frac{\ \frac{a}{b}\ }{\frac{c}{d}} \cdot \frac{\ \frac{d}{c}\ }{\frac{d}{c}} = \frac{\ \frac{a}{b} \cdot \frac{d}{c}\ }{\frac{c}{d} \cdot \frac{d}{c}} = \frac{\ \frac{ad}{bc}\ }{1} = \frac{ad}{bc}.
$$
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