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

The fraction  $x = \frac{a}{b}$ is the solution to $b \cdot x = a$. Similarly $y = \frac{c}{d}$ solves $d \cdot y = c$. Now, multiply the left-hand-side with $b \cdot x$ and the right hand-size with $a$ which preserves the equality since $a = b \cdot x$. We get
$$
 \begin{eqnarray}
    (b \cdot x) \cdot ( d \cdot y) &=& c \cdot a \\
    (b \cdot d) \cdot (x \cdot y) & = & (c \cdot a)
 \end{eqnarray}
$$
Thus $ ( x \cdot y) $ is the fraction $\frac{a \cdot c}{b \cdot d}$.
