Ratio Calculator & Simplifier

What Is a Ratio?

A ratio is a comparison of two quantities that shows how much of one thing there is relative to another. When a paint recipe calls for two parts blue and three parts white, it is using a ratio of 2:3. When a map uses a scale of 1:50,000, it means one centimetre on the map represents 50,000 centimetres on the ground. Ratios appear constantly in cooking, construction, photography, finance, science, and everyday problem-solving.

A ratio and a fraction represent the same mathematical relationship. The ratio 3:4 is equivalent to the fraction 3/4 and the decimal 0.75. This calculator converts between all three for every simplification result, so you can use whichever form suits your context.

How to Simplify a Ratio to Lowest Terms

To simplify a ratio, find the Greatest Common Divisor (GCD) of both numbers and divide each term by it. The GCD is the largest whole number that divides into both values without leaving a remainder. For a ratio of 12:8, the factors of 12 are 1, 2, 3, 4, 6, 12 and the factors of 8 are 1, 2, 4, 8. The largest common factor is 4. Dividing both terms by 4 gives 3:2, which is fully simplified.

The fastest way to find the GCD is the Euclidean algorithm. Divide the larger number by the smaller, take the remainder, then repeat until the remainder is zero. The last non-zero remainder is the GCD. This works even for ratios with decimal values — multiply both sides by 10 or 100 first to get whole numbers, then apply the same process.

How to Solve a Ratio Proportion (A:B = C:?)

Proportion problems ask you to find a missing value that keeps two ratios equal. If a recipe for 4 servings uses 2 cups of flour and you want to scale it to 10 servings, the proportion is 2:4 = ?:10. Cross-multiply to get 2 × 10 = 4 × ?, so ? = 20/4 = 5 cups. The same logic applies to map distances, gear ratios, chemical dilutions, and any scaling problem.

Cross-multiplication works because both ratios are equal fractions. Writing A:B = C:D as A/B = C/D and cross-multiplying gives A × D = B × C. Rearranging to isolate D gives D = (B × C) / A. This is the formula the proportion mode uses.

Equivalent Ratios Explained

Equivalent ratios are different ratios that express the same relationship. The ratios 1:3, 2:6, 4:12, and 10:30 are all equivalent because each one simplifies to 1:3. You can generate equivalent ratios by multiplying both terms by the same whole number. If you divide both terms by their GCD you move in the opposite direction, eventually reaching the simplest form.

Equivalent ratios are useful whenever you need to scale something up or down while preserving its proportions. A concrete mix specified at 1:2:3 (cement:sand:gravel) stays at those proportions whether you are mixing 6 kg or 600 kg of material. The numbers change but the relationship between them stays constant.

Real-World Ratio Examples

Common Ratio Formulas at a Glance

Ratio Calculator FAQ

How do you simplify a ratio?

Find the Greatest Common Divisor (GCD) of both numbers using the Euclidean algorithm, then divide each term by the GCD. For 15:10, the GCD is 5, so 15 ÷ 5 = 3 and 10 ÷ 5 = 2, giving the simplified ratio 3:2. A ratio is fully simplified when its two terms share no common factors other than 1.

What is the GCD and why does it matter for ratios?

The Greatest Common Divisor is the largest whole number that divides exactly into two values. For ratios, the GCD tells you the largest reduction factor — dividing by it in one step gets you directly to the lowest-terms ratio without needing multiple reductions. The Euclidean algorithm finds the GCD efficiently for any two positive integers.

How do you solve a proportion when one value is missing?

Write the two ratios as fractions, then cross-multiply. If A:B = C:D and D is unknown, write A/B = C/D, which gives A × D = B × C. Dividing both sides by A gives D = (B × C) / A. Substitute your known values and compute. For example, 3:4 = 9:? gives D = (4 × 9) / 3 = 12.

What is the difference between a ratio and a fraction?

A ratio and a fraction both compare two values, but they are used differently. The ratio 3:4 means "3 of A for every 4 of B" and can describe quantities that are separate (e.g. boys to girls in a class). The fraction 3/4 represents 3 parts out of a whole of 4. Mathematically they are equivalent, but ratios are often used when the two quantities remain distinct rather than being parts of one whole.

Can a ratio have decimals?

Yes. A ratio like 1.5:2.5 is valid, though it is usually cleaner to express it as a whole-number ratio. Multiply both terms by 10 to get 15:25, then find the GCD (which is 5) to simplify to 3:5. This calculator handles decimal inputs automatically by scaling to integers before applying the GCD algorithm.

How do you scale a ratio up or down?

To scale up, multiply both terms by the same whole number. The ratio 2:3 scaled by 4 becomes 8:12. To scale down, divide both terms by a common factor (ideally the GCD). For ratios that represent real-world measurements like cooking or mixing, scaling preserves the proportions while changing the actual quantities. The equivalent ratios table below each simplification result shows common scale multiples.