Ratio Calculator

Simplify a ratio down to its lowest terms, or solve for the missing number in a proportion — either way, see the full working, not just the final answer.

What is Ratio Calculator?

A ratio compares two quantities, showing how many times one contains the other. Ratios can always be simplified to their lowest terms by dividing both numbers by their greatest common divisor (GCD) — just like simplifying a fraction.

A proportion is a statement that two ratios are equal, written as a:b = c:x. Proportions are commonly used to scale recipes, convert measurements, adjust map distances, or solve for an unknown quantity when you know it's proportional to a known one. This calculator both simplifies ratios and solves proportions for a missing value using cross-multiplication.

Formula

Simplify: a:b = (a÷GCD(a,b)) : (b÷GCD(a,b))

Decimal: a/b

Proportion: a/b = c/x, solved as x = (b × c) / a

How to Calculate

Choose Simplify Ratio mode to reduce a:b to its lowest terms.
Or choose Solve Proportion mode to find a missing value in a:b = c:x.
Enter the known values.
Review the full step-by-step calculation.

Example

Simplifying 12:18 — the GCD of 12 and 18 is 6, so the simplified ratio is 2:3. Solving the proportion 4:6 = 10:x — cross-multiplying gives 4x = 60, so x = 15.

Key Benefits

Simplifies any ratio to its lowest terms using the greatest common divisor
Solves proportions for a missing value using cross-multiplication
Shows the ratio as a decimal and as a percentage split
Full step-by-step solution for every calculation, not just the final answer

Common Mistakes to Avoid

Simplifying only one side of a ratio instead of dividing both numbers by the same GCD
Setting up a proportion with mismatched units on each side (e.g., mixing minutes with hours)
Cross-multiplying incorrectly by multiplying the wrong pair of terms
Confusing a ratio (a comparison, a:b) with a fraction of a whole (a part-to-whole relationship, like a/(a+b))

Pro Tips

To scale a ratio up, multiply both numbers by the same value; to simplify it, divide both by their GCD
If a ratio can't be simplified further, its two numbers are already coprime (their GCD is 1)
When solving a proportion, always double-check that the units are consistent on both sides before cross-multiplying
A ratio of 1:1 means the two quantities are exactly equal

Key Terms Explained

Ratio: A comparison of two quantities showing their relative sizes, written as a:b
Proportion: A statement that two ratios are equal, written as a:b = c:d
Greatest Common Divisor (GCD): The largest number that divides both terms of a ratio without a remainder
Cross-Multiplication: A technique for solving proportions by multiplying diagonally across the equals sign

Common Use Cases

Simplifying recipe ingredient ratios when scaling up or down
Solving map-scale and blueprint-scale proportions
Business and finance ratios, like price-to-earnings or debt-to-income
Math homework involving ratios, rates, and proportional reasoning

Frequently Asked Questions

How do I simplify a ratio?

Find the greatest common divisor (GCD) of both numbers, then divide each number by that GCD. For example, to simplify 12:18, the GCD is 6, so dividing both by 6 gives the simplified ratio 2:3.

How do I solve a proportion for an unknown value?

Use cross-multiplication. For the proportion a/b = c/x, multiply a by x and b by c to get a × x = b × c, then divide both sides by a to solve for x: x = (b × c) / a.

What is the difference between a ratio and a fraction?

A ratio compares two separate quantities (a:b), while a fraction typically represents a part of a whole (a/(a+b)). However, a ratio a:b can always be converted into the fraction a/b to express it as a single number.

Can a ratio have more than two numbers?

Yes — ratios can compare three or more quantities at once, such as 2:3:5 for mixing paint or ingredients. This calculator focuses on two-term ratios (a:b), which are the most common case, but the same simplification principle (dividing by the GCD of all terms) applies to ratios with more parts.

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