Ratio Calculator (simplify & scale A:B)

Understanding the ratio calculator — concept and principle

A ratio A : B compares two quantities by division: it says how many times one quantity contains the other, regardless of their actual size. It is used to simplify mixing proportions, compare group sizes, check aspect ratios and scale a plan while keeping its shape.

How it is calculated

• Simplified ratio: when A and B are whole numbers, divide both by their greatest common divisor (GCD): A : B = (A ÷ GCD) : (B ÷ GCD). If either value has decimals, the calculator instead shows the unit ratio (A ÷ B) : 1.
• As 1 : x: divide both terms by A, so x = B ÷ A.
• Scaled: to make the first term equal to k, multiply both terms by k ÷ A, giving k : (B × k ÷ A).
• A as % of total: A ÷ (A + B) × 100. This is the share of A when the two parts make up a whole.

The inputs have no units, but A and B must be measured in the same unit (both grams, both people, both centimetres).

The principle

A ratio stays the same when both terms are multiplied or divided by the same non-zero number, because the quotient A ÷ B does not change. That is why 12 : 18, 6 : 9 and 2 : 3 are equivalent. Dividing by the GCD removes every common factor at once, so the result is the unique form with whole numbers that share no factor except 1. Scaling uses the same rule in the other direction: choose the multiplier that turns A into the target and apply it to B as well.

Worked example

1. Inputs: A = 12, B = 18, scale A to k = 10.
2. GCD(12, 18) = 6, so 12 ÷ 6 = 2 and 18 ÷ 6 = 3. Simplified ratio: 2 : 3.
3. As 1 : x: 18 ÷ 12 = 1.5, so 1 : 1.5.
4. Scaled: multiplier 10 ÷ 12, so B becomes 18 × 10 ÷ 12 = 15, giving 10 : 15.
5. Share of A: 12 ÷ (12 + 18) × 100 = 12 ÷ 30 × 100 = 40%.

How to read the result

1 : 1.5 says B is one and a half times A. The percentage view is often easier to communicate: 2 : 3 means A is 40% and B is 60% of the total. When comparing two ratios, convert both to the 1 : x form or to percentages so they share a common basis.

Common mistakes

• Mixing units, such as comparing 500 g with 2 kg as 500 : 2 instead of 500 : 2000.
• Confusing a part-to-part ratio (2 : 3) with a part-to-whole fraction (2/5).
• Adding the same number to both terms, which changes the ratio; only multiplication or division keeps it.

Frequently asked questions

How do I simplify a ratio with decimals, such as 1.5 : 2.5?

Multiply both terms by a power of 10 to clear the decimals (15 : 25), then divide by the GCD (5) to get 3 : 5.

Can a ratio have more than two terms?

Yes. A : B : C works the same way: divide all terms by their common GCD, for example 4 : 6 : 10 becomes 2 : 3 : 5.

How do I split an amount in a given ratio?

Add the terms to get the number of parts, divide the amount by that number, then multiply by each term. 100 split 2 : 3 is 40 and 60.

Sources: Euclid, Elements, Book V (theory of proportion) and Book VII, Proposition 2 (finding the greatest common measure) · G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Oxford University Press