Simplify ratios, convert to fractions and percentages, divide quantities, and solve proportions with step-by-step working.
Solve for x in: a : b = c : x
A ratio compares two or more quantities by division. It shows how much of one thing there is compared to another. For example, a ratio of 2:3 means "2 parts of A for every 3 parts of B".
Simplifying a ratio means reducing it to its smallest whole-number form. This is done by dividing all terms by their greatest common divisor (GCD). For example, 8:12 simplifies to 2:3 because both 8 and 12 are divisible by 4.
Converting a ratio to a fraction expresses each part as a fraction of the whole. In ratio 2:3, the parts are 2 and 3, totalling 5. So the first part is 2/5 and the second is 3/5 of the whole.
Dividing a quantity in a ratio splits an amount according to given proportions. If $500 is divided in ratio 2:3, you calculate: total parts = 2+3 = 5, then each part = $500÷5 = $100, so first share = 2×$100 = $200 and second share = 3×$100 = $300.
A proportion is an equation that states two ratios are equal (e.g. 2:3 = 4:6). Solving a proportion means finding the unknown value using cross-multiplication: if a:b = c:x, then a×x = b×c, so x = (b×c)÷a.
| Original Ratio | Simplified | GCD |
|---|---|---|
| 2:4 | 1:2 | 2 |
| 3:6 | 1:2 | 3 |
| 4:8 | 1:2 | 4 |
| 6:9 | 2:3 | 3 |
| 10:15 | 2:3 | 5 |
| 12:18 | 2:3 | 6 |
| 15:25 | 3:5 | 5 |
| 20:30 | 2:3 | 10 |
| 8:12 | 2:3 | 4 |
| 5:10 | 1:2 | 5 |
A ratio compares two or more quantities (e.g. 2:3 means "2 parts to 3 parts"). A fraction represents a part of a whole (e.g. 2/5 means "2 out of 5 total parts"). A ratio can be converted to fractions: the ratio 2:3 becomes 2/5 and 3/5 when expressed as parts of the whole.
Find the greatest common divisor (GCD) of all three numbers, then divide each number by the GCD. For example, to simplify 12:18:24, find GCD(12, 18, 24) = 6, then divide: 12÷6 = 2, 18÷6 = 3, 24÷6 = 4, giving the simplified ratio 2:3:4.
Add the ratio parts: 2+3 = 5. Divide the total by the sum: $500÷5 = $100 per part. Multiply each ratio number by this amount: first share = 2×$100 = $200, second share = 3×$100 = $300. Check: $200 + $300 = $500 ✓.
A ratio of 1:1 means the two quantities are equal in size. For example, a 1:1 ratio of salt to sugar in a recipe means use the same amount of each. It represents perfect balance between the two parts.
Use cross-multiplication. If a:b = c:x, then a×x = b×c. Solve for x: x = (b×c)÷a. For example, if 2:3 = 4:x, then 2×x = 3×4, so 2x = 12, and x = 6.
Simplifying makes ratios easier to understand and work with. A simplified ratio like 1:2 is clearer than 8:16 at a glance. It also helps identify equivalent ratios: 2:3 = 4:6 = 6:9 all simplify to the same thing.