Ratio Calculator
Work with ratios in two ways: solve the missing term in a proportion like a:b = c:x, or reduce any ratio such as 8:12 to its simplest form. Both modes also show the equivalent decimal value.
How this calculation works
"Solve the proportion" uses cross-multiplication. If a:b equals c:x, then a × x = b × c, so the missing term is x = (b × c) ÷ a. This is the same rule taught for scaling recipes, maps and mixtures.
"Simplify a ratio" divides both terms by their greatest common divisor (GCD) — the largest whole number that divides both evenly. Dividing a:b by their GCD gives the smallest whole-number ratio that means the same thing.
The decimal value (a ÷ b, or b ÷ a for the proportion) is shown alongside so you can compare ratios that use different scales at a glance.
Worked example
Cross-multiplication, step by step
A proportion says two ratios are equal: a:b = c:x, or equivalently a/b = c/x. Cross-multiplying clears both fractions at once: a × x = b × c. Dividing both sides by a isolates the unknown: x = (b × c) ÷ a.
This single rule covers scaling a recipe up or down, converting a map distance to real distance, mixing paint or concrete in the right proportion, and adjusting a photo or design to a new size without distorting it.
Finding the GCD to simplify a ratio
To reduce a:b to lowest terms, find the greatest common divisor (GCD) of a and b, then divide both numbers by it. The Euclidean algorithm finds the GCD quickly: repeatedly replace the larger number with the remainder of dividing it by the smaller, until the remainder is zero — the last non-zero value is the GCD.
| Ratio | GCD | Simplified |
|---|---|---|
| 4:8 | 4 | 1:2 |
| 6:9 | 3 | 2:3 |
| 10:15 | 5 | 2:3 |
| 16:24 | 8 | 2:3 |
| 21:14 | 7 | 3:2 |
Everyday uses for ratios
- Recipes: scale a dish from 4 servings to 10 by keeping every ingredient's ratio to the others constant.
- Maps and scale models: a map scale of 1:50,000 means 1 cm on the map equals 50,000 cm (500 m) in reality.
- Mixing: paint, concrete, fuel and cleaning solutions are diluted or combined in fixed ratios such as 1:4 or 3:1.
- Finance and business: price-to-earnings ratios, debt-to-income ratios and aspect ratios all compare two quantities the same way.
- Aspect ratios: a 16:9 screen is the same shape whether it is 16 cm wide or 1,600 cm wide — only the scale changes.
Ratio, fraction and percentage are the same idea
Any ratio a:b can be written as the fraction a/(a+b) of the total, or as a decimal by dividing a by b, or as a percentage by multiplying that decimal by 100. A class with a 3:2 ratio of girls to boys is 3/5 = 60% girls, and the decimal form of the ratio (girls per boy) is 3 ÷ 2 = 1.5.