Understand the idea

Equivalent ratios: how to make and check them

See why 2:3 and 8:12 describe the same relationship.

Equivalent ratios describe the same relative relationship using different numbers. For example, 2:3 and 8:12 are equivalent because multiplying both entries of 2:3 by 4 gives 8:12. The absolute quantities differ; their relative sizes do not.

Multiply both parts by the same factor

Starting with 2:3, multiply by 2 to get 4:6, by 3 to get 6:9, and by 4 to get 8:12. The common scale factor must be nonzero when the ratio is written as a fraction with a nonzero denominator.

Simplify to a common form

Divide both parts by their greatest common factor. For 18:24, the greatest common factor is 6, so the simplified ratio is 3:4. A ratio of 15:20 also simplifies to 3:4. That establishes that 18:24 and 15:20 are equivalent.

Check without simplifying

For a/b and c/d, compare ad with bc. To test 18/24 and 15/20, calculate 18 × 20 = 360 and 24 × 15 = 360. The matching products confirm equality. Fill all four fields in the proportion calculator to use check mode.

Part-to-part is not part-to-whole

If a mixture has 2 cups of concentrate and 3 cups of water, concentrate to water is 2:3. Concentrate to the total mixture is 2:5, because the total is 5 cups. These answer different questions. Always label what each position in the ratio represents.

Why adding the same number fails

Adding 2 to both parts of 2:3 produces 4:5. But 2/3 and 4/5 are not equal: the cross products are 10 and 12. Equivalent ratios depend on a common multiplicative factor, not a common additive change.

Apply the idea to recipe scaling or use practice mode to check your understanding with small numbers before moving on to fractions.