Ratio
A ratio is a comparison of two quantities showing how many times one contains the other. It can be written as 3:2, 3/2, or "3 to 2," and it underlies proportions, rates, and scale problems throughout math.
A ratio compares two quantities of the same kind, expressing their relative size. If a class has 12 girls and 8 boys, the ratio of girls to boys can be written 12:8, 12/8, or "12 to 8" — and, like a fraction, it simplifies to 3:2. Order matters: a ratio of girls to boys of 3:2 is not the same as 2:3.
Ratios become powerful through proportions — equations stating that two ratios are equal, such as 3/2 = 12/8. Cross-multiplying solves for an unknown: if a recipe uses flour and sugar in a 5:2 ratio and you have 15 cups of flour, then 5/2 = 15/x gives x = 6 cups of sugar. Ratios also describe parts of a whole: a 3:2 ratio means 3 of every 5 items are in the first group.
Closely related ideas include rates (ratios of different units, like miles per hour) and scale factors (map distances to real distances). Whenever two quantities grow or shrink together proportionally, a ratio captures the relationship.
Ratios appear across an unusually wide range of exams. The GRE tests ratio and proportion word problems directly. The PTCE relies on ratios for dose calculations, concentrations, and dilutions. Even securities exams like the Series 7 use ratio reasoning in conversion ratios for convertible bonds and preferred stock. Mastering the basic setup — write the ratio, form a proportion, cross-multiply — pays off in all of them.
Key takeaways
- A ratio compares two quantities and can be written as a:b, a/b, or "a to b."
- Ratios simplify like fractions, and the order of the terms matters.
- A proportion sets two ratios equal; cross-multiplication solves for an unknown term.
- A ratio of 3:2 means 3 out of every 5 total parts belong to the first group.
- Ratios are tested on the GRE, the PTCE (dosage and dilution math), and securities exams like the Series 7.