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Ratio Calculator
Simplify a ratio of two or three numbers, find the missing term in A : B = C : x, or share a total in a given ratio. Decimals are accepted and the steps are shown.
What a ratio tells you
A ratio compares quantities by size, not by absolute amount. The ratio 12 : 18 says that for every 12 of the first thing there are 18 of the second, exactly as 2 : 3 does. Multiplying or dividing every term by the same non-zero number gives an equivalent ratio. Simplifying means dividing by the greatest common factor until no whole number greater than 1 divides every term.
Ratios with decimals are handled by scaling first. The ratio 0.5 : 1.5 becomes 5 : 15 after multiplying by 10, and then 1 : 3 after dividing by the greatest common factor of 5.
Solving for a missing term
If A : B = C : x, the two fractions A ÷ B and C ÷ x are equal, so cross-multiplying gives x = B × C ÷ A. For 3 : 4 = 9 : x, the scale factor from 3 to 9 is 3, so x is 4 × 3 = 12. This is the same step used in scaling a recipe, enlarging a map distance or converting a model measurement to a real one.
A cannot be zero in this form, because no scale factor turns zero into a non-zero amount.
Splitting a total
To share a total in a ratio, add the ratio numbers to get the number of equal parts, divide the total by that count to get the value of one part, and multiply by each ratio number. Sharing 100 in the ratio 2 : 3 gives five parts of 20, so the shares are 40 and 60. The shares always add back to the total, which is a quick check on your own working.
Common mistakes
- Adding the same number to both terms. Going from 2 : 3 to 4 : 5 does not keep the ratio; only multiplication and division do.
- Mixing units. Convert 2 hours : 30 minutes to the same unit first, giving 120 : 30 or 4 : 1.
- Reading a part-to-part ratio as a fraction of the whole. In 2 : 3, the first part is 2 out of 5 parts, or 40%, not two thirds.
- Ignoring the order. 2 : 3 and 3 : 2 describe opposite relationships.
For percentages and fractions of the whole, see the percentage calculator and the fraction calculator.
Worked example
Three partners invest in the ratio 2 : 3 : 5 and agree to share a profit of 1,500 in the same proportion.
Inputs
| What do you want to do? | Split a total in a ratio |
|---|---|
| A | 2 |
| B | 3 |
| Third part (optional) | 5 |
| Total to split | 1500 |
Result
1,500 shared in the ratio 2 : 3 : 5.
| Total ratio parts | 10 |
|---|---|
| Value of one part | 150 |
Shares
| Part | Ratio number | Share of total | Amount |
|---|---|---|---|
| A | 2 | 20% | 300 |
| B | 3 | 30% | 450 |
| C | 5 | 50% | 750 |
Accuracy note: Results use standard double-precision arithmetic, so very large or very small values may be rounded. Check important results by another method.
Frequently asked questions
How do I simplify a ratio?
Divide every term by their greatest common factor. For 12 : 18 the greatest common factor is 6, so the simplest form is 2 : 3.
Can I simplify a ratio with decimals?
Yes. Multiply every term by a power of ten until all are whole numbers, then divide by the greatest common factor. This calculator does that automatically for up to six decimal places.
How do I find a missing number in a proportion?
Cross-multiply. In A : B = C : x, the missing value is B times C divided by A.
How do I divide an amount in the ratio 2 : 3?
Add the parts to get 5, divide the amount by 5, then multiply by 2 and by 3. Sharing 100 gives 40 and 60.
Can a ratio contain a zero?
A ratio such as 0 : 5 is allowed and simplifies to 0 : 1, but at least one term must be non-zero. A ratio of all zeros has no meaning.
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References and sources
- Standard school mathematics treatment of ratio, proportion and equivalent ratios.
- Euclid, Elements Book V and Book VII, for the classical theory of ratio and the greatest common divisor.