Ratio Calculator
Simplify a ratio to lowest terms, or solve a proportion for its unknown term.
Divided both terms by their GCD, 6.
To simplify a ratio, divide both terms by their greatest common divisor: 12 : 18 simplifies to 2 : 3, dividing both by the GCD 6. To solve a proportion a : b = c : x, cross-multiply: x = b × c ÷ a. So 2 : 3 = 8 : 12, because x = 3 × 8 ÷ 2 = 12.
What a ratio is
A ratio compares two quantities — written a : b, read “a to b.” Like a fraction, a ratio has equivalent forms: 12 : 18, 6 : 9, and 2 : 3 all describe the same relationship. The simplest form divides both terms by their greatest common divisor (GCD) so they share no common factor. A proportion is a statement that two ratios are equal, which lets you solve for a missing term.
to simplify, divide both terms by GCD(a, b); to solve a proportion, cross-multiply
Worked example
Solve 2 : 3 = 8 : x:
- 1 To simplify, find the GCD. For 12 : 18, GCD(12, 18) = 6.
- 2 Divide both terms by it. 12 ÷ 6 = 2 and 18 ÷ 6 = 3, so 12 : 18 = 2 : 3.
- 3 To solve a proportion, cross-multiply. From a : b = c : x you get a × x = b × c, so x = b × c ÷ a.
- 4 Plug in the numbers. For 2 : 3 = 8 : x, x = 3 × 8 ÷ 2 = 24 ÷ 2 = 12.
Common ratios simplified
Each ratio reduced by its greatest common divisor.
| Ratio | GCD | Simplified |
|---|---|---|
| 12 : 18 | 6 | 2 : 3 |
| 10 : 15 | 5 | 2 : 3 |
| 8 : 12 | 4 | 2 : 3 |
| 9 : 6 | 3 | 3 : 2 |
| 100 : 40 | 20 | 5 : 2 |
| 14 : 21 | 7 | 2 : 3 |
Equivalent ratios and scaling
Equivalent ratios are made by multiplying or dividing both terms by the same number — 2 : 3 scales up to 4 : 6, 8 : 12, or 200 : 300. Proportions are how you scale things in the real world: doubling a recipe that calls for 2 : 3 flour-to-sugar keeps the same taste because the ratio is unchanged, and a map scale of 1 : 50000 tells you every unit on paper equals 50000 in reality.
Part-to-part vs part-to-whole. A ratio of 2 : 3 boys to girls is part-to-part; the part-to-whole version is 2 : 5 (boys to total) and 3 : 5 (girls to total), since 2 + 3 = 5. Keep track of which you mean — the same situation gives different numbers depending on the comparison.