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
$$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \cdot \frac{d}{c}=\frac{ad}{bc}?$$

Suppose $\frac{a}{b}$ and $\frac{c}{d}$ are fractions.  That is, $a$, $b$, $c$, $d$ are whole numbers and $b\ne0$, $d\ne0$.  In addition we require $c\ne0$.
Let $\frac{a}{b}\div\frac{c}{d}=A$.  Then by definition of division of fractions , $A$ is a unique fraction so that $A\times\frac{c}{d}=\frac{a}{b}$.  
However, $(\frac{a}{b}\times\frac{d}{c})\times\frac{c}{d}=\frac{a}{b}\times(\frac{d}{c}\times\frac{c}{d})=\frac{a}{b}\times(\frac{dc}{cd})=\frac{a}{b}\times(\frac{dc}{dc})=\frac{a}{b}$.
Then by uniqueness (of $A$), $A=\frac{a}{b}\times\frac{d}{c}$.
