Cross-multiplication test for equal fractions

Direction 1 ($\Rightarrow$). Suppose $\dfrac{a}{b}=\dfrac{c}{d}$. Since $b,d\neq0$, we may multiply both sides by the nonzero number $bd$: $\frac{a}{b}\cdot bd = \frac{c}{d}\cdot bd$. On the left, $\frac{a}{b}\cdot bd = a\cdot d$ because the factor $b$ cancels; on the right, $\frac{c}{d}\cdot bd = c\cdot b$ because the factor $d$ cancels. So $ad=bc$.

Direction 2 ($\Leftarrow$). Suppose $ad=bc$. Divide both sides by the nonzero number $bd$: $\dfrac{ad}{bd}=\dfrac{bc}{bd}$. Cancel the common factor $d$ from the left fraction and the common factor $b$ from the right fraction: $\dfrac{a}{b}=\dfrac{c}{d}$.

Both directions hold, so the two statements are equivalent: $\dfrac{a}{b}=\dfrac{c}{d}$ exactly when $ad=bc$.