Simplify a ratio, or solve a proportion A : B = C : ? (leave the last box empty to just simplify A : B).
Ratios are everywhere — scaling a recipe, mixing paint colours, resizing an image, or solving a math problem. This free ratio calculator does two things: it simplifies a ratio to its lowest terms, and it solves for the missing value in a proportion. Enter your numbers and get the answer instantly, with the working shown so you understand exactly how it was derived.
How to Use the Ratio Calculator
- To simplify a ratio, enter both values (A and B) and press Simplify. The tool divides both by their greatest common divisor to give the reduced form.
- To solve a proportion (A:B = C:X), enter any three values and leave the missing one blank. Press Solve to find the fourth value.
How It Works: The Formula
Simplifying a ratio means dividing both terms by their greatest common divisor (GCD). The GCD is the largest integer that divides both A and B without a remainder. For example, the ratio 18:24 — GCD(18, 24) = 6. Dividing both by 6 gives 3:4. That is the simplest form.
Solving a proportion uses cross-multiplication. If A:B = C:X, then A × X = B × C, so X = (B × C) ÷ A. For example, if 3:4 = 9:X, then X = (4 × 9) ÷ 3 = 12. So 3:4 = 9:12.
Worked Examples
Simplify 36:48. GCD(36, 48): 36 = 2² × 3², 48 = 2&sup4; × 3. GCD = 2² × 3 = 12. 36 ÷ 12 = 3; 48 ÷ 12 = 4. Simplified ratio: 3:4.
Solve 5:8 = 15:X. X = (8 × 15) ÷ 5 = 120 ÷ 5 = 24. So 5:8 = 15:24 (and 15:24 simplifies back to 5:8, confirming the answer).
Recipe scaling: A recipe calls for flour and sugar in a 3:1 ratio. You want to use 750 g of flour. How much sugar? 3:1 = 750:X. X = (1 × 750) ÷ 3 = 250 g.
Why Ratios Are Useful
A ratio expresses the relative size of two quantities. Because it captures a relationship rather than an absolute amount, it scales cleanly — the ratio stays the same whether you are working with small or large numbers. In cooking, ratios let you scale recipes up or down without recalculating each ingredient from scratch. In design, maintaining the aspect ratio of an image prevents distortion when resizing. In finance, ratios like price-to-earnings or debt-to-income let you compare companies or households of wildly different sizes on a level playing field.
Ratios in Everyday Life
You encounter ratios constantly without always recognizing them. Map scales (1:50,000 means one unit on the map equals 50,000 units in reality). Odds in betting (3:1 against). Mixing concrete (cement:sand:gravel in specific proportions). Aspect ratios on screens (16:9 for widescreen). Paint tinting (a certain number of drops of pigment per litre of base). Interest rate comparisons. Even the golden ratio (approximately 1:1.618) appears in art, architecture, and nature. Understanding how to simplify and solve ratios is a foundational quantitative skill.
Finding the GCD: The Euclidean Algorithm
The most efficient way to find the GCD of two numbers is the Euclidean algorithm. Divide the larger by the smaller and take the remainder. Then divide the previous smaller number by that remainder. Repeat until the remainder is zero; the last non-zero remainder is the GCD. Example: GCD(48, 36). 48 ÷ 36 = 1 remainder 12. 36 ÷ 12 = 3 remainder 0. GCD = 12. This approach is exactly what the calculator uses internally, and it works efficiently even for very large numbers.
Frequently Asked Questions
How do I simplify a ratio?
Divide both terms by their greatest common divisor (GCD). The result is the ratio in its simplest or lowest terms, where the two numbers share no common factor other than 1.
What is a proportion?
A proportion is a statement that two ratios are equal: A:B = C:D. If you know three of the four values, you can find the fourth by cross-multiplication: D = (B × C) ÷ A.
Is this ratio calculator free?
Yes — it is completely free, requires no signup, and runs entirely in your browser with no data sent to any server.
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