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.
- Two ratios are equivalent if and only if their cross-products are equal: a:b ∼ c:d when a × d = b × c.
- The simplified form uses the GCD: divide both parts by GCD(a, b) to reach the smallest whole-number version of the ratio.
- When GCD = 1, the ratio is already in its simplest form - there is no smaller whole-number equivalent.
- The decimal value (a ÷ b) is identical for every equivalent ratio in a family - it is the invariant that defines the relationship.
- Scaling up multiplies both parts by the same integer n; scaling down divides both parts by the same integer n. Only the magnitude changes, never the proportion.
How to use this equivalent ratios 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.
Equivalent ratios explained to a beginner
Imagine a cordial drink that says "1 part syrup to 4 parts water." If you want to make a bigger jug, you could use 2 parts syrup and 8 parts water, or 3 parts syrup and 12 parts water. All three versions - 1:4, 2:8, 3:12 - taste exactly the same. That is what equivalent ratios mean: the same proportional relationship, written with different numbers.
The simplified form (1:4 here) is just the most compact way to write it. If someone hands you a 6:24 recipe, dividing both parts by their GCD (which is 6) gives 1:4. Same drink, simplest recipe. The calculator does this step automatically, then generates the full family of equivalents so you can pick whichever numbers suit your batch size.
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 for equivalent ratios
| 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 I enter a ratio and the GCD comes back as 1, people often assume something has gone wrong. It hasn't. A GCD of 1 simply means the ratio is already in its simplest form - 7:11 has no smaller whole-number equivalent.
The equivalents the calculator generates (14:22, 21:33 and so on) are still valid; there is just no more compact version. Not every ratio has an obvious common factor.
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.
In practice - pricing ratios, ingredient ratios, financial ratios - I use the cross-multiplication test to verify that a scaled version is truly equivalent and not slightly off due to rounding.
A 1:3 ratio scaled to 33:100 looks close (cross-products: 100 vs 99) but is not exactly equivalent. That one-unit gap matters in precision contexts like pharmaceutical formulations or chemical concentrations, where an approximate equivalent introduces a systematic error across every batch.
Frequently asked questions about equivalent ratios
What are equivalent ratios exactly?
Equivalent ratios are 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).
Quiz: how well do you know equivalent ratios?
1. What does it mean for two ratios to be equivalent?
2. From the worked examples table, what is the simplified form of 100:75?
3. According to the key takeaways, which value is identical across every ratio in an equivalent family?
4. For ratios 3:4 and 9:12, the cross-multiplication test gives:
5. Based on the first-person observation, what does GCD = 1 mean for a ratio like 7:11?