Ratio

Three ways to write one

Form	Example
colon	$3 : 2$
words	$3$ to $2$
fraction	$\tfrac{3}{2}$

All three carry the same comparison. The colon form is most common in a ratio question; the fraction form is the one that lets you calculate with it.

Part-to-part and part-to-whole

For a class of $3$ girls and $2$ boys:

Comparison	Type	Ratio
girls to boys	part-to-part	$3 : 2$
girls to students	part-to-whole	$3 : 5$

A part-to-whole ratio needs the total, so the parts have to be added first. It behaves exactly like a fraction — girls are $\tfrac{3}{5}$ of the class.

Mixing these up is the most common error in ratio work, and what a ratio is sets out how to tell which one a question is asking for.

A ratio holds at any size

$$2 : 3 \qquad 4 : 6 \qquad 10 : 15$$

These are equivalent ratios. Multiplying or dividing both parts by the same number keeps the comparison true, which is why a recipe can be doubled without changing how it tastes. Adding to both parts does not work, because it helps the smaller number more.

Related ideas

A rate is a ratio comparing different units, such as miles to hours. A unit rate is a rate scaled so the second quantity is $1$. A percent is a ratio whose second part is fixed at $100$.