Ratio Calculator
Simplify a ratio to lowest terms, solve a missing value in a proportion (A : B = C : X), scale a ratio to a target, or split a total across multi-part ratios like 1 : 2 : 3 — with the Euclidean algorithm shown step by step.
Written by Suhaib Hassan. The method is Euclid’s greatest-common-divisor algorithm and cross-multiplication, both re-derived by hand here. Our methodology.
Decimals are scaled to whole numbers, divided by their greatest common divisor, and scaled back — so 1.5 : 2.5 becomes 3 : 5 with no rounding.
Ratio questions come in different shapes. One page covers simplifying, missing values, scaling and splitting, and shows the working for each.
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Step-by-step working
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Ratio types compared
Four different jobs that all use the word “ratio”.
| Type | Form | Units | Key property | Example |
|---|---|---|---|---|
| Part-to-part | A : B | Same units | Unchanged when both parts are scaled | 16 wide : 9 high |
| Part-to-whole | A ÷ (A + B) | Fraction or percent, at most 100% | Parts add to 1 | 16 of 25 units = 64% |
| Rate | Δy ÷ Δx | Compound units | Constant rate means a straight line | 60 miles per hour |
| Inverse proportion | y = k ÷ x | Reciprocal pairing | The product x·y stays constant | Speed vs time for a fixed distance |
Worked examples
Illustrative figures, each checked by hand. All can be reproduced above.
Video aspect ratio
A 16 : 9 video player is 1,280 px wide. How tall must it be?
X = 9 × 1280 ÷ 16 = 720 px
Concrete mix
A 1 : 2 : 3 mix (cement : sand : gravel, by volume) is needed for 18 m³.
Each part = 18 ÷ 6 = 3 m³
cement 3 · sand 6 · gravel 9 m³
Splitting a bill
A $1,200 bill is shared 5 : 3 : 2.
$600 · $360 · $240
What is a ratio?
A ratio compares two or more quantities of the same kind, written A : B or A ÷ B. Two ratios are equivalent if they have the same value: 16 : 9 and 1920 : 1080 are the same ratio. A proportion states that two ratios are equal, A : B = C : D.
How to calculate with ratios
- Simplify: divide every part by their greatest common divisor (GCD).
- Missing value: cross-multiply. From A : B = C : D, A × D = B × C, so D = B × C ÷ A.
- Scale: multiply every part by the same factor.
- Split a total: add the parts, divide the total by that sum to get the value of one part, then multiply by each part.
By Suhaib Hassan · Reviewed 28 September 2026 · How CalculatePilot verifies formulas →
Sources
- Euclid, Elements, Book VII, Propositions 1 and 2 — the Euclidean algorithm for the greatest common divisor.
- ISO 80000-2:2019, Quantities and units — Part 2: Mathematics.
- NIST Special Publication 811, Guide for the Use of the International System of Units (SI) — quantities of dimension one.
Frequently asked questions
What is the difference between a ratio and a proportion?
A ratio compares quantities, such as 16 : 9. A proportion is an equation that says two ratios are equal, such as 16 : 9 = 1920 : 1080. Proportions let you find a missing value by cross-multiplying.
How does Euclid’s algorithm simplify a ratio?
It finds the greatest common divisor by repeatedly replacing the larger number with the remainder of dividing it by the smaller, until the remainder is zero. The last non-zero remainder is the GCD, and dividing both parts by it gives the lowest terms. It never needs to factorise the numbers.
Can ratios include decimals or negative numbers?
Decimals work: the calculator scales them to whole numbers first (1.5 : 2.5 becomes 15 : 25, then 3 : 5). Ratios of physical quantities are normally positive, so this calculator asks for values above zero.
What does a ratio of 1 : 2 mean as a fraction or percent?
As a part-to-part ratio, 1 : 2 means the second part is twice the first. As shares of the whole, the parts are 1 ÷ 3 = 33.3% and 2 ÷ 3 = 66.7%. Always check whether a ratio is part-to-part or part-to-whole.
How do I scale a recipe using a ratio?
Use “Scale ratio” to set the quantity you want for one ingredient (or the total), and the calculator returns the factor and scaled amounts. Multiply every ingredient by that same factor.
What if one part of the ratio is zero?
A ratio with a zero part, such as 5 : 0, is degenerate — it cannot be scaled back or used to find a missing value, and dividing by zero is undefined. This calculator asks for parts greater than zero.
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