Cross multiplication

The rule

$$\frac{a}{b} = \frac{c}{d} \qquad\Longrightarrow\qquad ad = bc$$

Each top pairs with the opposite bottom. Multiplying straight across instead solves a different equation.

$$\frac{3}{4} = \frac{x}{12} \quad\Rightarrow\quad 3 \times 12 = 4x \quad\Rightarrow\quad x = 9$$

Why it is valid

It is not a shortcut with its own justification — it is what multiplying both sides by both denominators leaves behind.

$$\frac{a}{b} \times b \times d = \frac{c}{d} \times b \times d$$

The $b$s cancel on the left and the $d$s cancel on the right:

$$ad = cb$$

So cross multiplication is the ordinary balance rule with the cancelling already done. That is also why it needs a single fraction on each side: with two terms on one side there is nothing clean to cancel against.

Testing whether two ratios match

$$\frac{6}{9} \stackrel{?}{=} \frac{8}{12} \qquad 6 \times 12 = 72, \quad 9 \times 8 = 72$$

Equal products, so the ratios are equal. Unequal products mean they are not — and the gap can be small, so the arithmetic is worth doing rather than eyeballing.

Full worked examples are in solving proportions.

Setting one up from a word problem

Keep the same quantity on top of both sides. For ”$4$ shirts cost $52$ dollars, what do $7$ cost?”:

$$\frac{4}{52} = \frac{7}{x} \quad\Rightarrow\quad 4x = 364 \quad\Rightarrow\quad x = 91$$

Writing dollars on top of both sides gives the same answer. Mixing the two — one side shirts over dollars, the other dollars over shirts — does not.