Solving Proportions

Cross multiplication

In $\dfrac{a}{b} = \dfrac{c}{d}$, the cross products are $a \cdot d$ and $b \cdot c$, and they're equal: $ad = bc$. For $\dfrac{x}{6} = \dfrac{12}{9}$, that gives $9x = 6 \cdot 12$, so $9x = 72$ and $x = 8$.

It isn't a magic trick — it's just multiplying both sides by both denominators at once. Multiplying $\dfrac{x}{6} = \dfrac{12}{9}$ by $6$ and by $9$ clears both fractions, and what survives is exactly the cross products.

When the variable is in the denominator

Cross multiplication doesn't care where the variable sits. In $\dfrac{4}{x} = \dfrac{6}{15}$, the cross products are $4 \cdot 15$ and $6x$, so $6x = 60$ and $x = 10$.

This is the big advantage of the method: top or bottom, left or right, the variable always ends up in a simple one-step equation after one cross multiply.

Setting up proportions from a situation

The setup rule: keep matching quantities in matching positions. If the left fraction is cups of flour over batches, the right fraction must also be cups of flour over batches — tops match, bottoms match.

A proportion set up with the units crossed, like flour over batches equal to batches over flour, will cross multiply into a wrong answer that looks perfectly reasonable. Label your setup with units before you compute.

Worked examples

Try one yourself

Common questions