How to Simplify Ratios: GCD Method with Worked Examples
Learn to reduce whole-number ratios and create equivalent ratios using the greatest common divisor.
Ratios compare quantities
A ratio expresses a comparison between two quantities in a specific order. If a group contains 12 apples and 18 oranges, the apples-to-oranges ratio is 12:18. Dividing both terms by the same positive number leaves the relationship unchanged.
Step 1: find the greatest common divisor
The greatest common divisor (GCD) is the largest whole number that divides both terms exactly. For 12 and 18, the GCD is 6.
Step 2: divide both terms
12 ÷ 6 : 18 ÷ 6 = 2:3
So 12:18 = 2:3. This is the simplest whole-number ratio because the terms 2 and 3 have no common divisor greater than 1.
Equivalent ratios
Multiply both terms of a ratio by the same positive integer to generate an equivalent ratio. For 2:3, multiplying by 2 gives 4:6; multiplying by 5 gives 10:15.
| Original ratio | Simplest form | Equivalent after ×3 |
|---|---|---|
| 12:18 | 2:3 | 6:9 |
| 20:30 | 2:3 | 6:9 |
| 15:25 | 3:5 | 9:15 |
Ratio order matters
A recipe with water to concentrate of 3:1 is different from 1:3. Always state what the first term represents and what the second represents.
Use ratios in everyday situations
Common uses include recipe scaling, classroom comparisons, model dimensions and material mixtures. When comparing quantities with different units, convert them to consistent units first.
Ratios with fractions or decimals
For a decimal ratio, first multiply both quantities by a suitable power of ten to express them as whole numbers, then simplify. For instance, 1.5:2.5 becomes 15:25, then 3:5.
Related concepts
A fraction such as 2/3 can express the quotient of a ratio 2:3, but ratios can also describe parts of an overall mixture. To solve an unknown value in a proportion, the Proportion Calculator may be more suitable.
Practice with the tool
Use the Ratio Calculator to simplify positive integer terms and generate equivalent ratios at your chosen multiplier.