Can someone very intuitively explain why the cross-product works, with respect to proportions?

Question

Note: unrelated to matrices.

I have to teach this concept, and I want to know what is the best way to show that the cross product is always equal if the two fractions are equal.

Thank you so much.

Answer 1

Claim: Let $b$ and $d$ be nonzero. Then: $$ \frac{a}{b} = \frac{c}{d} \iff ad = bc $$

Proof: Observe that: \begin{align*} \frac{a}{b} = \frac{c}{d} &\iff bd\left(\frac{a}{b}\right) = bd\left(\frac{c}{d}\right) &\text{multiply both sides by $bd$} \\ &\iff ad\left(\frac{b}{b}\right) = bc\left(\frac{d}{d}\right) &\text{by rearranging} \\ &\iff ad = bc &\text{by cancellation} \end{align*}

Answer 2

You are given $\frac ab=\frac cd$ If you multiply both sides by $bd$, which cannot be $0$ if the fractions make sense, you have $ad=bc$