Use the free Proportion Solver below: your data never leaves your browser — all work happens on your device, and nothing is uploaded to any server.
What Is a Proportion?
A proportion is a statement that two ratios are equal. If you write a/b = c/d, you are saying that the relationship between a and b is exactly the same as the relationship between c and d. Ratios show up in cooking, maps, scale models, medicine doses, currency exchange and building plans, so proportions are one of the most practical ideas in school mathematics. Whenever a quantity grows or shrinks in step with another quantity, a proportion is hiding underneath.
Consider a simple recipe. If 3 cups of flour make 4 servings, how much flour do you need for 12 servings? The ratio of flour to servings must stay the same, so 3/4 = x/12. Solving gives x = 9 cups. The numbers change, but the ratio does not. That is the whole idea: a proportion keeps the relationship fixed while the amounts change.
Cross-Multiplication, the Key Move
To solve a proportion, you cross-multiply. From a/b = c/d you get a × d = b × c. This single step turns a fraction equation into an ordinary multiplication equation, and from there the unknown term is one division away. If d is unknown, d = (b × c) / a. If a is unknown, a = (b × c) / d. The pattern is always the same: multiply the two numbers on one diagonal, then divide by the number left on the other diagonal.
Why does cross-multiplication work? Multiply both sides of a/b = c/d by b × d. On the left, the b cancels and you get a × d. On the right, the d cancels and you get b × c. Nothing mysterious is happening; it is just clearing denominators in one tidy step. Understanding that step matters, because it tells you when the method is safe: it is safe whenever neither denominator is zero, since division by zero is never allowed.
How to Use This Calculator
Pick which term is unknown: a, b, c or d. Enter the other three numbers, then press Solve proportion. The tool shows the missing value, the cross-multiplication step with your numbers filled in, and a check that divides the two ratios to prove they match. For the recipe example, set it up as flour/servings on both sides: 3/4 = 9/x. Choose d as unknown, enter a = 3, b = 4 and c = 9, and the tool returns d = 12 servings. The tool handles any arrangement; you only need to place your numbers in matching positions, with the same kind of quantity on top on both sides.
The check line is the part most people skip, and it is the most valuable part. It divides a by b and c by d and shows both results. If the two decimals agree, the proportion is correct. If they do not agree, something was entered in the wrong position, and the check tells you at once instead of letting a wrong answer travel into your homework or your recipe.
A Worked Example: Map Scale
Suppose a map has a scale of 1 cm to 25 km, and two towns are 8 cm apart on the map. How far apart are they in reality? Set up the proportion as map/real = 1/25 = 8/x. Cross-multiplying gives 1 × x = 25 × 8, so x = 200 km. Enter a = 1, b = 25, c = 8 with d unknown, and the tool walks through exactly those steps: 1 × d = 25 × 8, d = 200.
Map scales, model scales and photo enlargements all work this way. The scale is one ratio, the measurement is the other ratio, and the unknown is the real-world size or the drawing size. Once you can solve one proportion, you can solve all of them, because the structure never changes.
Common Mistakes
The most common mistake is mismatching the positions: putting map distances with real distances in the wrong ratio. Always label your numbers before you type them. Write “map = 1, real = 25, map = 8, real = ?” and the positions fall into place. The second mistake is forgetting that a denominator cannot be zero. A ratio with zero on the bottom has no meaning, and the tool refuses such input rather than inventing an answer.
A third mistake is cross-multiplying the wrong pair: a × b instead of a × d. Remember the diagonals. In a/b = c/d, the diagonal pairs are a with d and b with c. Multiply each diagonal pair and set the products equal. Draw the cross on paper the first few times until the pattern is automatic.
Try It Yourself
Start with the recipe numbers, then try the map example. Then invent your own: a photo 10 cm wide enlarged to 25 cm, with a detail that was 4 cm wide — how wide is the detail now? The answer is 10 cm, from 10/25 = 4/x. Proportions reward practice, because every new example looks different but solves the same way. Use the tool as often as you like; every calculation runs in your browser, and nothing you type is sent anywhere.
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