What ratios and proportions are
A ratio compares two quantities by division and is written A:B (read “A to B”). If a class has 12 girls and 8 boys, the girls-to-boys ratio is 12:8. A proportion is a statement that two ratios are equal: A:B = C:D. It says both comparisons have the same value even when the numbers differ. The proportion 2:3 = 4:6 is true because both ratios equal 0.6667.
This calculator does two related jobs. The first solves a proportion when one term is missing: leave a box blank and the value comes out by cross multiplication. The second simplifies a ratio to lowest terms by dividing through the greatest common divisor (GCD), and also reports its decimal value. Everything runs in your browser, with nothing sent anywhere.
How to use the calculator
On the Solve proportion tab you’ll see four boxes laid out as A : B = C : D. Type three numbers and leave exactly one box empty; the unknown appears instantly alongside the completed proportion. Leave two boxes empty or type letters and you’ll get a gentle notice instead of a wrong answer.
On the Simplify ratio tab you type two positive integers, say 18 and 24. You get the reduced ratio (3:4), the GCD used (6), and the decimal value (0.75). The Copy button sends the result to your clipboard, and Use example loads a ready-made case.
The formula
A proportion A:B = C:D is the same as saying the cross products are equal:
A × D = B × C
From that identity you can isolate whichever term is missing:
| Missing term | Formula | Example |
|---|---|---|
| A | A = (B × C) ÷ D | ?:5 = 9:15 → 3 |
| B | B = (A × D) ÷ C | 8:? = 12:9 → 6 |
| C | C = (A × D) ÷ B | 7:2 = ?:6 → 21 |
| D | D = (B × C) ÷ A | 2:3 = 4:? → 6 |
To simplify A:B, compute the GCD with the Euclidean algorithm (divide, keep the remainder, repeat until it reaches zero) and divide each term by it: A÷GCD : B÷GCD. The decimal value is simply A ÷ B.
Worked example
Let’s solve 2:3 = 4:x. The fourth term (D) is missing, so we apply the formula:
x = (B × C) ÷ A = (3 × 4) ÷ 2 = 12 ÷ 2 = 6
Check it with the cross product: 2 × 6 = 12 and 3 × 4 = 12. They match, so x = 6 and the full proportion is 2:3 = 4:6.
Now let’s simplify the ratio 18:24. The GCD of 18 and 24 comes from Euclid: 24 = 18 × 1 + 6, then 18 = 6 × 3 + 0, so GCD = 6. Divide each term: 18 ÷ 6 = 3 and 24 ÷ 6 = 4. The reduced ratio is 3:4 and its decimal value is 18 ÷ 24 = 0.75.
Frequently asked questions
What’s the difference between a ratio and a proportion?
A ratio is a single comparison, like 18:24. A proportion is an equality between two ratios, like 18:24 = 3:4. In other words, the proportion asserts that two ratios are equivalent because they share the same decimal value.
How is this different from the rule of three?
They’re close cousins: both rely on cross multiplication. The rule of three focuses on quantities with units (“if 3 kg cost 150, how much do 5 kg cost?”), while here the focus is the pure ratio in A:B form and its reduction with the GCD. If you want the setup with real-world magnitudes, the rule of three calculator fits better.
Can I use decimals when solving a proportion?
Yes. Cross multiplication works with any real number, so 1.5:2 = 3:x gives x = 4. GCD simplification, on the other hand, only makes sense for positive integers, because the GCD is defined for whole numbers.
Why can’t I divide by zero?
If the term acting as the divisor is zero, the formula has no solution: division by zero is undefined. The calculator detects this and shows a notice instead of an infinite or meaningless result.
Does the order of the terms matter?
Yes. A:B is not the same as B:A: 3:4 equals 0.75 while 4:3 equals 1.3333. Always keep the same order on both sides of a proportion so the comparison stays valid.