Quick answer
Multiply or divide both parts of a ratio by the same number to get an equivalent ratio. For 6:9 — GCD = 3, simplified = 2:3. Equivalents: 4:6, 6:9, 8:12, 10:15… All express the same proportional relationship.
How to use this calculator
Enter any two positive values. The calculator simplifies the ratio to its lowest terms using the GCD, then generates a table of 12 equivalent ratios (the simplified form scaled by 1 through 12). Use this to find a ratio that uses convenient whole numbers for your context.
What are equivalent ratios?
Equivalent ratios represent the same proportional relationship between two quantities. Just as 1/2 and 3/6 are equivalent fractions, 1:2 and 3:6 are equivalent ratios — both state that the first quantity is half the second.
Any ratio can be scaled up or down by multiplying or dividing both parts by the same non-zero number. The simplified (lowest-terms) form is the ratio with the smallest possible whole-number values.
Formula and generation
To generate equivalent ratios, multiply both parts by any non-zero integer n:
$$a{:}b = (a \times n){:}(b \times n) \quad \text{for any } n \neq 0$$
To simplify a ratio to lowest terms, divide both parts by their GCD:
$$\frac{a}{\gcd(a,b)} : \frac{b}{\gcd(a,b)}$$
Example — generating equivalents of 3:4:
$$3{:}4 = 6{:}8 = 9{:}12 = 12{:}16 = 15{:}20$$
Verifying that two ratios are equivalent
Use cross-multiplication: two ratios a:b and c:d are equivalent if and only if the cross-products are equal:
$$a{:}b \sim c{:}d \iff a \times d = b \times c$$
For 3:4 and 9:12: 3 × 12 = 36 and 4 × 9 = 36. Equal — so they are equivalent.
For 3:4 and 9:13: 3 × 13 = 39 and 4 × 9 = 36. Not equal — so they are not equivalent.
Worked examples
| Input ratio | GCD | Simplified | Some equivalents |
|---|---|---|---|
| 6:9 | 3 | 2:3 | 4:6, 6:9, 8:12, 10:15 |
| 12:8 | 4 | 3:2 | 6:4, 9:6, 12:8, 15:10 |
| 15:25 | 5 | 3:5 | 6:10, 9:15, 12:20, 15:25 |
| 100:75 | 25 | 4:3 | 8:6, 12:9, 16:12, 20:15 |
| 7:7 | 7 | 1:1 | 2:2, 3:3, 4:4, 5:5 |
| 1:2 | 1 | 1:2 | 2:4, 3:6, 4:8, 5:10 |
When equivalent ratios matter
Recipe scaling — a recipe serving 4 in ratio 2:3:5 can be scaled to serve 12 by tripling all parts to 6:9:15. The ratio remains equivalent; only the total quantity changes.
Unit conversion and rates — a car travelling 120 km in 2 hours has a speed ratio of 120:2 = 60:1 (60 km per hour). Finding the equivalent with denominator 1 gives the unit rate.
Map and scale drawing — a scale of 1:50,000 is equivalent to 2:100,000 or 0.5:25,000. All represent the same relationship; the choice depends on the unit or context.
Mixing and formulation — a paint colour mix of 4 parts red : 6 parts blue simplifies to 2:3. Knowing the simplified form makes it easy to scale up or down for any batch size.
Frequently asked questions
What are equivalent ratios?
Ratios that express the same proportional relationship, generated by multiplying or dividing both parts by the same number. 2:3, 4:6, and 6:9 are all equivalent.
How do you find equivalent ratios?
Simplify first using the GCD. Then multiply both parts of the simplified ratio by 2, 3, 4… to generate equivalents.
How do you check if two ratios are equivalent?
Cross-multiply: a:b ∼ c:d if a × d = b × c.
What is the simplest form of a ratio?
The form where both parts share no common factors other than 1. Divide both parts by GCD(a, b).