Ratio Calculator

Simplifying a ratio

A ratio compares quantities: 12 : 18 says there are 12 of one thing for every 18 of another. Dividing every term by the same number does not change the comparison, so a ratio is simplified by dividing by the greatest common divisor of all its terms. The greatest common divisor of 12 and 18 is 6, and 12 : 18 = 2 : 3.

To add, subtract, multiply or divide fractions and mixed numbers, use the fraction calculator.

When the terms are decimals or fractions, first multiply them all by the same number to make them whole. 0.3 : 0.45 × 100 is 30 : 45, which simplifies to 2 : 3. For 1/2 : 1/3, multiplying by 6 gives 3 : 2.

Solving a proportion

A proportion states that two ratios are equal, a : b = c : d. In any proportion the product of the outer terms equals the product of the inner terms, a × d = b × c. That is enough to find any one missing term.

1. For 3 : 4 = 9 : x, multiply the outer terms: 3 × x.
2. Multiply the inner terms: 4 × 9 = 36.
3. So 3 × x = 36, and x = 36 ÷ 3 = 12.

The same method resizes a picture: a 16 : 9 screen that is 1,080 pixels tall is 16 × 1,080 ÷ 9 = 1,920 pixels wide.

Dividing an amount in a ratio

To share 12,000 in the ratio 2 : 3 : 1, add the parts (2 + 3 + 1 = 6), find one part (12,000 ÷ 6 = 2,000) and multiply: 4,000, 6,000 and 2,000. When the amount does not divide evenly the calculator gives the exact fractions rather than rounding them away, so you can decide where a remainder goes.

A ratio is not a fraction of the total

In a class where boys and girls are in the ratio 2 : 3, the boys are not two thirds of the class. There are 2 + 3 = 5 parts, so the boys are 2/5 of it — 40% — and the girls 3/5, or 60%. The fraction of the total always has the sum of the parts underneath.

Mixing works the same way. A concrete mix of cement, sand and gravel at 1 : 2 : 3 has 6 parts; for 60 kg of dry mix each part is 10 kg, so 10 kg of cement, 20 kg of sand and 30 kg of gravel. Enter the total and the ratio under “Divide in a ratio” and the calculator does the same split.

More worked examples

• Simplify 30 : 45 : 60. The greatest common divisor of all three terms is 15, so it reduces to 2 : 3 : 4.
• Solve 2 : 5 = x : 30. The inner terms multiply to 5 × x, the outer terms to 2 × 30 = 60, so x = 60 ÷ 5 = 12.
• Divide 7,500 in the ratio 3 : 2. One part is 7,500 ÷ 5 = 1,500, giving 4,500 : 3,000.
• Simplify a ratio that does not divide evenly, 8 : 12 : 15. 8 and 12 share a factor of 4, but 15 shares no factor greater than 1 with either, so the greatest common divisor of all three is 1 and the ratio is already in lowest terms: 8 : 12 : 15.

Edge cases and blocked input

• Zero is not accepted in any term. A ratio compares two positive quantities, and a term of 0 cannot be scaled or used as a divisor, so the calculator shows a field error instead of a result.
• Negative numbers are not accepted for the same reason — ratios in this calculator describe quantities, not signed values.
• Leaving more than one box empty when solving a proportion does not give a unique answer, so exactly one of the four terms must be blank.
• Leaving all four proportion boxes full still runs, but there is nothing to solve for — leave one empty to see the missing term.
• An amount that does not divide evenly, such as 100 split 1 : 3 (25 and 75, which happens to be exact) or 100 split 1 : 6 (which does not divide evenly), is not rounded away. The calculator gives the exact fractional split, so nothing is silently lost to rounding.

Common mistakes

• Reading a ratio as a fraction of the total — see "A ratio is not a fraction of the total" above. 2 : 3 means 2/5 and 3/5, not 2/3.
• Cross-multiplying the wrong pair of terms in a proportion. The rule is outer × outer = inner × inner for a : b = c : d written in a line; sketch which terms are which before multiplying.
• Forgetting to convert decimals to whole numbers first when simplifying by hand. The calculator does this automatically, but doing it by hand means multiplying every term by the same power of ten before finding the greatest common divisor.
• Rounding a split too early. If an amount does not divide evenly, round only the final shares you report, not the "amount per part" figure used to get there.

Limitations

This calculator handles ratios and proportions with two or three terms, using whole numbers, decimals or simple fractions such as 1/3. It does not solve equations with more than one unknown, and it does not accept zero or negative terms. All the arithmetic runs in your browser; nothing you type is sent to a server.

Related calculators

For work with numerators and denominators directly — adding, subtracting, multiplying or dividing — use the fraction calculator. To simplify a fraction down to a whole-number ratio using the greatest common divisor by hand, the GCD and LCM calculator shows the working. For splitting a bill or amount by percentage rather than by ratio, see the percentage calculator.

Frequently asked questions