Ratio Calculator

How to Simplify a Ratio

Simplifying a ratio means reducing both numbers to the smallest whole numbers that still describe the same proportion — the same idea as reducing a fraction to lowest terms. The way to do it is to divide both numbers by their greatest common divisor (GCD), the largest number that divides evenly into both.

$$A : B \;\longrightarrow\; \frac{A}{\gcd(A,B)} : \frac{B}{\gcd(A,B)}$$

For example, a ratio of 20 to 30: the GCD of 20 and 30 is 10, so dividing both sides by 10 gives 2 to 3 — the simplest form of that same ratio. A recipe that calls for 20 cups of flour to 30 cups of water has exactly the same proportions as one calling for 2 cups of flour to 3 cups of water.

Solving a Proportion — The Rule of Three

A proportion states that two ratios are equal: A is to B as C is to D. If three of the four values are known, the fourth can be found with cross-multiplication — historically called the "rule of three."

$$A : B = C : D \;\;\Rightarrow\;\; D = \frac{B \times C}{A}$$

For example, if 2 is to 5 as 8 is to D: D equals (5 times 8) divided by 2, which is 20. So 2:5 = 8:20 — both ratios reduce to the same simplified proportion. This is exactly how recipe scaling, map-reading, unit conversion, and currency exchange problems are typically solved by hand.

Ratios With Decimals

A ratio doesn't have to start out as whole numbers — 2.5 to 7.5 is a perfectly valid ratio. To simplify it, both numbers are first scaled up (multiplied by the same power of 10) until they're whole numbers, then reduced normally using the GCD. 2.5 to 7.5 scales up to 25 to 75, which reduces to 1 to 3, the same clean whole-number ratio the original decimals represented all along.

Common Ratio Mistakes

Treating a ratio like a fraction and adding a constant to both sides is a common error — a ratio's relationship only survives multiplication or division by the same factor, not addition. Setting up a proportion with the terms in the wrong order (writing B:A = C:D when the problem describes A:B = C:D) is another frequent slip, since swapping which value goes on top silently inverts the result. Forgetting that a ratio compares quantities of the same kind — mixing units, like flour in cups against water in liters, without converting first — also produces a ratio that looks correct but doesn't mean anything useful.

Ratio Terms You Should Know

Proportion — a statement that two ratios are equal, such as A:B = C:D.

Greatest Common Divisor (GCD) — the largest whole number that divides evenly into two or more numbers; also called the greatest common factor (GCF).

Cross-Multiplication — the technique of multiplying across a proportion's equals sign to solve for an unknown value; also called the rule of three.

Rate — a ratio comparing two quantities with different units, like miles per hour, as opposed to a plain ratio comparing quantities of the same kind.