Ratios & Proportions

Equal ratios and the cross-multiplication that solves them.

The idea

A ratio compares two quantities by division. We write it $a : b$, or $\dfrac{a}{b}$ with $b \neq 0$, and it records relative size: how much of the first there is for each unit of the second. A ratio rescaled so the second quantity is $1$ is a unit rate, such as miles per gallon or dollars per hour.

A proportion states that two ratios are equal: $\frac{a}{b} = \frac{c}{d}, \qquad b, d \neq 0.$ To solve one, multiply both sides by $b$ and by $d$. The $b$ cancels on the left and the $d$ on the right, leaving $ad = bc.$ The products $ad$ and $bc$ are the cross products, and passing from the proportion to $ad = bc$ is called cross-multiplying. It turns the proportion into a linear equation whatever position the unknown occupies, including a denominator.

Ways to work on it

• Walkthrough. Define ratio and proportion; solve by cross-multiplying.
• Practice. Solve a random proportion for the unknown.
• Hardest. A word problem that sets up and solves a proportion.

Not sure where to start? Take the ten-question placement test.