Solving Proportions and Cross Multiplication

A proportion is two equal ratios

$$\frac{3}{4} = \frac{9}{12}$$

That statement is a proportion. It says the two ratios describe the same comparison, which they do: $\tfrac{9}{12}$ scales down to $\tfrac{3}{4}$.

Most proportion questions hide one of the four numbers:

$$\frac{3}{4} = \frac{x}{12}$$

Cross multiplication

Multiply each top by the other bottom, and set the two results equal:

$$\frac{3}{4} \;\; \begin{matrix} \diagup \\[-6pt] \diagdown \end{matrix} \;\; \frac{x}{12} \qquad\Longrightarrow\qquad 3 \times 12 = 4 \times x$$ $$36 = 4x \quad\Rightarrow\quad x = 9$$

The fractions are gone after one step, and what remains is a one-step equation.

Why cross multiplication works

It is not a trick. It is what multiplying both sides by both denominators leaves behind.

Start with the proportion:

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

Multiply both sides by $b$ and by $d$ — legal, because doing the same thing to both sides keeps an equation balanced:

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

On the left the $b$s cancel. On the right the $d$s cancel:

$$a \times d = c \times b$$

Those are the cross products. Cross multiplication is the balance rule with the cancelling already done, which is why it is safe and why it only works when each side is a single fraction.

Checking whether two ratios match

Cross multiplying also settles whether a proportion is true.

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

The products match, so the ratios are equal.

$$\frac{4}{5} \stackrel{?}{=} \frac{7}{9}$$ $$4 \times 9 = 36 \qquad 5 \times 7 = 35$$

$36 \neq 35$, so these are not in proportion — and the gap of $1$ shows how close a false proportion can look.

Setting one up from a word problem

The labels have to line up. Put the same quantity on top of both sides.

If $4$ shirts cost $52$ dollars, what do $7$ shirts cost?

$$\frac{\text{shirts}}{\text{dollars}}: \quad \frac{4}{52} = \frac{7}{x}$$ $$4x = 52 \times 7 = 364 \quad\Rightarrow\quad x = 91$$

Writing it the other way up works equally well:

$$\frac{\text{dollars}}{\text{shirts}}: \quad \frac{52}{4} = \frac{x}{7}$$ $$4x = 364 \quad\Rightarrow\quad x = 91$$

Same answer. What breaks a proportion is mixing the two — shirts over dollars on one side and dollars over shirts on the other.

Worked examples

Common mistakes

Practice problems