Ratio Calculator
Enter two ratios A : B and C : D. The calculator simplifies each to lowest terms, shows the decimal and percentage values, tests whether the ratios are equivalent by cross-multiplication, and reports the scale factor between them.
Simplified first ratio (A : B)
2 : 3 → decimal 0.666667
- Simplified second ratio (C : D)
- 2 : 3
- First ratio as a decimal (A ÷ B)
- 0.67
- Second ratio as a decimal (C ÷ D)
- 0.67
- First ratio as a percentage
- 66.67%
- A × D (cross-product)
- 72
- B × C (cross-product)
- 72
- Equivalent ratios?
- Yes — A : B equals C : D
- Scale factor from A to C
- ×0.333333
Each ratio is simplified by dividing both terms by their greatest common divisor (Euclid). Two ratios A:B and C:D are equivalent when A·D = B·C (cross-multiplication). The scale factor C ÷ A tells you what to multiply the first ratio by to reach the second.
How to use this calculator
Type any four numbers into A, B, C and D. Treat the first pair as one ratio (A : B) and the second pair as another (C : D). The calculator immediately shows both ratios in lowest terms, their decimal and percentage values, and whether the two ratios are equivalent. Set C and D to small numbers if you just want to simplify A : B — ignore the second-ratio outputs.
How the calculation works
A ratio A : B is simplified by dividing both terms by their greatest common divisor (GCD), which is found using the Euclidean algorithm. The decimal value is simply A ÷ B. Two ratios A : B and C : D are equivalent when A·D = B·C — this is the cross-multiplication property of proportions and avoids floating-point error from comparing two divisions. The scale factor between the two ratios is C ÷ A: multiply A and B by this factor and you get C and D when the ratios are equivalent.
Worked example
Take the default values: A=12, B=18, C=4, D=6. GCD(12,18) = 6, so 12 : 18 simplifies to 2 : 3. GCD(4,6) = 2, so 4 : 6 simplifies to 2 : 3 — same simplified form. Cross-multiplying: 12 · 6 = 72 and 18 · 4 = 72, so the ratios are equivalent. Scale factor from 12 to 4 is 4 ÷ 12 ≈ 0.333, i.e. the second ratio is one-third the first. As a decimal, 12 ÷ 18 ≈ 0.667, or 66.67%.
Frequently asked questions
What does it mean to simplify a ratio?
Simplifying a ratio means dividing both terms by their greatest common divisor so that the two terms share no common factor other than 1. For example, 12 : 18 simplifies to 2 : 3 because both 12 and 18 are divisible by 6. The simplified ratio expresses the same relationship in the smallest whole numbers possible.
How do I know if two ratios are equal?
Cross-multiply: ratios A : B and C : D are equivalent if and only if A · D = B · C. This avoids the floating-point pitfalls of comparing A/B against C/D directly. The calculator does the cross-multiplication for you and reports a yes/no answer along with the two products.
What is the scale factor?
The scale factor is the number you multiply the first ratio by to reach the second. The calculator computes it as C ÷ A. For equivalent ratios, multiplying both A and B by the scale factor gives C and D exactly; for non-equivalent ratios the scale factor only matches one term and the other will not match.
Can I use decimals, not just whole numbers?
Yes. The simplification step uses a real-number GCD, so non-integer inputs like 1.5 : 4.5 still get reduced (to 1 : 3 in that example). For exact arithmetic with very small fractions, it is best to multiply through by a power of ten first so the inputs are whole numbers.
Why does the calculator show "Invalid input"?
A ratio with a zero second term — A : 0 or C : 0 — is undefined because it would require dividing by zero. Enter a non-zero value for both B and D. The calculator also rejects non-finite numbers like NaN or infinity.
How is the percentage figure calculated?
The percentage is the decimal value of A : B multiplied by 100. So 1 : 4 is 25% and 3 : 4 is 75%. Note that this is the percentage A is of B, not a "part of a whole" figure — for percentage shares of a total, divide each term by the sum of all terms.