Equivalent Ratio Calculator

What Are Equivalent Ratios?

Equivalent ratios are two or more ratios that express the exact same mathematical relationship or proportion between quantities, even though their individual numbers may differ. Just like equivalent fractions, you create equivalent ratios by multiplying or dividing both parts of the ratio by the same non-zero constant factor.

For instance, the ratio $3 : 5$ is equivalent to $6 : 10$, $9 : 15$, and $12 : 20$. In each case, dividing the first term by the second yields the exact same decimal value:

$$\frac{3}{5} = \frac{6}{10} = \frac{9}{15} = 0.6$$

How to Find and Generate Equivalent Ratios

There are two primary methods to find equivalent ratios:

• Multiplication (Scaling Up): Multiply each term of the given ratio $A : B$ by any positive integer or scalar $k \neq 0$: $$A : B \implies (A \times k) : (B \times k)$$ For example, multiplying $4 : 7$ by $3$ gives $(4 \times 3) : (7 \times 3) = 12 : 21$.
• Division (Simplifying / Scaling Down): Divide each term by their greatest common divisor (GCD) to reach the simplest whole-number form: $$A : B \implies (A \div \gcd(A, B)) : (B \div \gcd(A, B))$$ For example, simplifying $18 : 24$ by dividing both terms by $\gcd(18, 24) = 6$ yields $3 : 4$.

How to Check If Two Ratios Are Equivalent

To determine whether two ratios $A : B$ and $C : D$ are equivalent, you can use the cross-multiplication method:

$$A \times D = B \times C$$

If the product of the extremes ($A \times D$) equals the product of the means ($B \times C$), the two ratios form a valid proportion and are mathematically equivalent. For example, testing $4 : 6$ and $10 : 15$:

$$4 \times 15 = 60 \quad \text{and} \quad 6 \times 10 = 60 \implies 60 = 60 \quad (\text{Equivalent})$$

Solving for an Unknown Term in a Proportion

When you have an incomplete proportion like $A : B = C : x$, you can solve for the missing term $x$ using cross-multiplication:

$$x = \frac{B \times C}{A}$$

This formula is useful for scaling recipes, calculating gear ratios, resizing images, and solving financial currency conversions. For related calculations, check our Ratio Calculator, Equivalent Fractions Calculator, and Ratio Simplifier.