Ratios and Proportions — Mathematics Knowledge System

Overview

A ratio compares two quantities by division, showing their relative sizes — for example, a recipe might call for flour and sugar in a ratio of 3 to 1. A proportion is a statement that two ratios are equal, and it's the tool used to scale quantities up or down consistently: if you know 3 apples cost $2, a proportion tells you how much 12 apples cost. Together, ratios and proportions are the backbone of scaling, unit conversion, map reading, and mixing.

Intuition

A ratio is just a comparison written as a fraction-like relationship: 'for every 3 of this, there are 1 of that.' A proportion says two such comparisons describe the same underlying rate, just at different scales — like a photograph enlarged to a poster: the ratio of width to height stays the same even though the actual measurements change. Solving a proportion means finding the missing number that keeps that rate consistent.

Interactive Graph

Formal Definition

Definition

A ratio of a to b (with b ≠ 0) is written a:b or the fraction a/b. A proportion is an equation stating that two ratios are equal; it can be solved for an unknown term using cross-multiplication.

\frac{a}{b} = \frac{c}{d}

Two equal ratios

Proportion

ad = bc

Equivalent form obtained by multiplying both sides by bd

Cross-multiplication

Properties

Cross-multiplication

\frac{a}{b} = \frac{c}{d} \iff ad = bc

Condition: b, d ≠ 0

Scaling

\frac{a}{b} = \frac{ka}{kb}

Condition: k ≠ 0 — multiplying both terms of a ratio by the same nonzero number preserves it

Applications

Scale models and blueprints use fixed ratios (e.g. 1:100) between drawn measurements and real-world dimensions.

Worked Examples

Divide both terms by their greatest common divisor, 4.
$8:12 = (8\div4):(12\div4) = 2:3$

Answer: 2:3

Practice Problems

Difficulty 1/10

Simplify the ratio 15:25.

Difficulty 2/10

Solve the proportion 4/9 = x/36 for x.

Common Mistakes

Common Mistake

Cross-multiplying incorrectly, e.g. multiplying numerator by numerator instead of numerator by the opposite denominator.

In a/b = c/d, cross-multiplication gives ad = bc — each numerator is multiplied by the OTHER fraction's denominator.

Common Mistake

Setting up a proportion with mismatched units, e.g. putting miles over hours on one side and hours over miles on the other.

Both ratios in a proportion must compare quantities in the same order (e.g. always miles/hours on both sides), or the equation describes an unrelated relationship.

Summary

A ratio a:b compares two quantities; a proportion states that two ratios are equal.
Proportions are solved with cross-multiplication: a/b = c/d implies ad = bc.
Ratios can be scaled by multiplying or dividing both terms by the same nonzero number.
Keep the order and units of quantities consistent on both sides of a proportion.

References

WebsiteWikipedia — Ratio