Ratios, Rates, and Proportions

Concise answer

Keep corresponding quantities in the same numerator and denominator positions, preserve units, and solve a proportion only after confirming the relationship is multiplicative.

Definitions

Ratio: A comparison between quantities measured in the same unit.
Rate: A comparison between quantities measured in different units.
Proportion: An equation stating that two ratios are equal.

Intuition

A proportion preserves one multiplicative relationship while both quantities scale.

Unit labels reveal reversed ratios before the algebra begins.

Concept walkthrough

A ratio compares two quantities that share the same unit — 3 cups of flour to 2 cups of sugar is the ratio $3$ to $2$ , written $3:2$ or $\tfrac{3}{2}$ . A rate compares quantities with different units, such as $120$ miles in $2$ hours; dividing gives a unit rate of $60$ miles per hour. On the GRE, deciding whether a comparison is a ratio or a rate is usually the first move.

A proportion sets two ratios equal, like $\tfrac{3}{2} = \tfrac{x}{10}$ . Because the two fractions are equal, cross-multiplying ( $3 \times 10 = 2 \times x$ ) turns the proportion into a linear equation you can solve. The GRE rewards doing this cleanly: keep the same quantity in the numerator of both fractions, and the units on each side will line up.

These ideas sit inside the arithmetic content ETS describes for the Quantitative Reasoning measure, which stays at or below a second course in algebra. That means the difficulty on the GRE comes from careful setup and reading, not from advanced techniques.

After this page, you should be able to

Write a comparison as a ratio or a rate, and decide which one a problem is describing.
Reduce a ratio and find a unit rate so quantities are easy to compare.
Solve a proportion for an unknown using cross-multiplication.
Keep units consistent across a proportion so the setup cannot silently invert.

Formulas and assumptions

Proportion solved by cross-multiplication

\frac{a}{b} = \frac{c}{d} \;\Rightarrow\; a\,d = b\,c

Variables

a, c: the numerators (the same type of quantity in both ratios)
b, d: the denominators (the same type of quantity in both ratios)

Assumptions

b and d are not zero.
The numerators measure the same quantity as each other, and so do the denominators.

Unit rate

r = \dfrac{y}{x}

Variables

y: the first quantity (for example, distance)
x: the second quantity in different units (for example, time)
r: the amount of y per one unit of x

Assumptions

x is not zero.
y and x are measured in different units, so r is a rate, not a plain ratio.

Worked example

Scaling a recipe with a proportion

A recipe uses flour and sugar in the ratio $3:2$ . If you use $12$ cups of flour and keep the same ratio, how many cups of sugar do you need?

1Write the known ratio as a fraction with flour on top: $\tfrac{3}{2}$ .
2Write the scaled ratio the same way, with the unknown sugar as $x$ : $\tfrac{12}{x}$ .
3Set the two equal as a proportion: $\tfrac{3}{2} = \tfrac{12}{x}$ .
4Cross-multiply: $3 \times x = 2 \times 12$ , so $3x = 24$ .
5Divide by $3$ : $x = 8$ .

You need $8$ cups of sugar.

Common traps

Flipping one fraction so the same quantity is not on top of both — for example, writing $\tfrac{3}{2} = \tfrac{x}{12}$ when $12$ is flour, not sugar.
Treating a rate like a ratio and dropping the units, then comparing numbers that measure different things.
Cross-multiplying before the two ratios are actually set equal, or when the relationship is additive rather than proportional.

GRE General overview Sign in for current practice

Question depth and domain coverage vary by exam. Practice answers are checked after submission.

Sources

GRE General Test Quantitative Reasoning Overview — ETS. Accessed 2026-07-11. Use as a cited source for exam facts; do not imply affiliation or reproduce protected test material.
Prealgebra 2e, Section 5.6: Ratios and Rate — OpenStax. Accessed 2026-07-11. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.
Prealgebra 2e, Section 6.5: Solve Proportions and Their Applications — OpenStax. Accessed 2026-07-11. OpenStax textbook content is CC BY 4.0; attribute and avoid verbatim reuse beyond short cited references.

Review and maintenance

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Last source check: 2026-07-06
Next scheduled review: 2026-10-11

Recheck the ETS Quantitative Reasoning content page and the OpenStax Prealgebra references before each major GRE preparation cycle.

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