How ratios and proportions work
A ratio compares two quantities, written A:B. Two ratios that describe the same relationship form a proportion, A:B = C:D. Simplifying a ratio means dividing both sides by their greatest common divisor; solving a proportion means finding the one missing value that keeps the two ratios equal.
Simplifying and cross-multiplying
simplify: A:B = (A÷g) : (B÷g), g = gcd(A,B)
solve: A:B = C:D ⇒ A × D = B × C
missing D = (B × C) ÷ A
The cross-multiplication rule — the product of the outer terms equals the product of the inner terms — is what lets you solve for any single unknown.
Worked example
Simplify 18:24, then solve 3:4 = 9:?:
gcd(18, 24) = 6, so 18:24 = 3:4.
3:4 = 9:? → ? = (4 × 9) ÷ 3 = 12.
Tip: scaling a recipe or a map uses exactly this. To double a 2:3 mix you keep the ratio — 4:6 — which simplifies right back to 2:3.