A ratio calculator solves the proportion A:B = C:D when you know any three of the four values. Ratios compare two quantities — 2 cups of flour to 3 cups of water, 8 wins to 12 losses — and two ratios are equivalent when they represent the same relative amounts, even if the raw numbers differ.
Enter three values, leave the one you want to find blank, and the calculator solves it with cross-multiplication. It also simplifies A:B to its lowest whole-number terms and shows the decimal value of the ratio, which is handy for scaling recipes, mixing solutions, resizing images, and converting map distances.
How Cross-Multiplication Solves a Ratio
The statement A:B = C:D means the fraction A/B equals the fraction C/D. Multiplying both sides by B × D gives the cross-product rule:
A × D = B × C
Whichever value is missing, isolate it algebraically:
- Missing A: A = (B × C) ÷ D
- Missing B: B = (A × D) ÷ C
- Missing C: C = (A × D) ÷ B
- Missing D: D = (B × C) ÷ A
For example, with 2 : 3 = 8 : D, the rule gives 2 × D = 3 × 8 = 24, so D = 24 ÷ 2 = 12. The check is built in: 2/3 = 0.667 and 8/12 = 0.667, so the ratios match.
Simplifying and Scaling Ratios
A ratio is simplified by dividing both terms by their greatest common divisor (GCD), exactly like reducing a fraction:
- 8 : 12 → GCD is 4 → 2 : 3
- 15 : 25 → GCD is 5 → 3 : 5
- 0.5 : 1.5 → multiply both by 10 to get 5 : 15, then divide by 5 → 1 : 3
Scaling works in reverse: multiply both terms by the same number to get an equivalent ratio. A pancake recipe using 2 : 3 flour-to-milk scales to 6 : 9 for a triple batch. Ratios also appear as unit rates — 2 : 3 is equivalent to 0.667 : 1 — which is what the decimal equivalent in the results represents.
Worked Example: Scaling a Paint Mix
A paint recipe calls for 2 parts blue to 3 parts yellow. You have 8 liters of blue — how much yellow keeps the same shade?
Set up the ratio equation: 2 : 3 = 8 : D, where D is the yellow needed.
Step 1: Cross-multiply: 2 × D = 3 × 8, so 2D = 24. Step 2: Divide both sides by 2: D = 12.
You need 12 liters of yellow. Verify with decimals: 2 ÷ 3 = 0.6667 and 8 ÷ 12 = 0.6667 — identical, so 8 : 12 is an equivalent ratio to 2 : 3. Note that 8 : 12 also simplifies back to 2 : 3 when both terms are divided by their GCD of 4.
Frequently Asked Questions
How do you solve a ratio with one missing value?
Write the ratios as a proportion A:B = C:D, then use cross-multiplication: A × D = B × C. Solve the resulting equation for the missing letter. For 2:3 = 8:D, cross-multiplying gives 2D = 24, so D = 12.
How do you simplify a ratio?
Divide both terms by their greatest common divisor. For 8:12, the GCD is 4, so the simplified ratio is 2:3. If the terms are decimals, first multiply both by a power of 10 to make them whole numbers, then reduce.
What are equivalent ratios?
Equivalent ratios express the same relationship with different numbers, like 2:3, 4:6, and 8:12. You get one from another by multiplying or dividing both terms by the same nonzero number. Their cross-products are always equal, and they reduce to the same simplest form.
What is the difference between a ratio and a fraction?
A fraction represents a part of a whole, while a ratio compares two separate quantities. The ratio 2:3 corresponds to the fraction 2/3 only when comparing the first quantity to the second. If 2 of 5 students are boys, the boy-to-girl ratio is 2:3 but the fraction of boys is 2/5.
Can a ratio include decimals or zero?
Ratios can include decimals — 0.5:1.5 is valid and simplifies to 1:3. A zero is allowed in some positions, but a term used as a divisor during solving cannot be zero, and B cannot be zero when computing the decimal equivalent A ÷ B, since division by zero is undefined.