Proportions

A proportion is an equation that sets two ratios equal to each other: a/b = c/d. It says that two comparisons have the same relative size — 3 parts to 4 parts is the same relationship as 6 parts to 8 parts. Any statement like "2 cups of flour for every 3 cups of milk, so 8 cups of flour for 12 cups of milk" is a proportion in disguise.

The key solving technique is cross-multiplication: if a/b = c/d, then ad = bc. Suppose a car travels 120 miles on 4 gallons of gas, and you want to know how far it goes on 7 gallons. Set up 120/4 = x/7, cross-multiply to get 4x = 840, and solve: x = 210 miles. Setting up the proportion correctly — keeping matching quantities in matching positions — is where most errors happen.

Proportions appear throughout math and daily life: converting units, reading map scales, resizing recipes, computing percentages, and working with similar triangles, where corresponding side lengths form equal ratios. In statistics, the word takes on a related meaning — a sample proportion is the fraction of a sample with some characteristic — and AP Statistics builds confidence intervals and hypothesis tests around comparing proportions.

The ACT and CLT test classic ratio-and-proportion problems in their algebra and pre-algebra sections, while the AP Statistics exam works with sample proportions and their sampling distributions. For all of them, the core skills are the same: set up the ratio equation carefully and cross-multiply.