Proportion Calculator
Solve any proportion of the form a/b = c/d for any one of the four unknown terms. Choose which position holds the unknown, pick direct or inverse proportion, and the calculator cross-multiplies and shows every step with your actual numbers.
Formula
Worked example
Direct: 3/4 = 9/d. Cross-multiply: 3d = 36, so d = 12. Check: 3/4 = 0.75 and 9/12 = 0.75. Inverse: 3 x 4 = 6 x d. Product = 12, so d = 2. Check: 3 x 4 = 12 = 6 x 2.
What a proportion is and how cross-multiplication solves it
A proportion is a statement that two ratios are equal: a/b = c/d. Any one of the four terms can be unknown. Cross-multiplication eliminates both denominators in one step by multiplying each numerator by the opposite denominator, giving a x d = b x c. Dividing both sides by the coefficient of the unknown isolates it. For example, if d is unknown: d = (b x c) / a. The only restriction is that you cannot divide by zero, so the term you divide by must be non-zero. This calculator handles all four positions and shows every arithmetic step so you can reproduce the work by hand.
Direct proportions vs. inverse proportions
In a direct proportion, both quantities increase or decrease together at a fixed ratio: a/b = c/d. Doubling a recipe, reading a scale map, and converting currencies are all direct proportions. In an inverse proportion, one quantity increases as the other decreases, and their products are equal: a x b = c x d. Travel time and speed at a fixed distance is the classic example: drive twice as fast and the trip takes half as long. Selecting the right type before solving is important because the algebra, and the answer, differ. The "Proportion type" option on this calculator handles both cases.
The scale factor and how to use it
The scale factor is the ratio c / a, which tells you how many times larger (or smaller) the right-hand pair is compared to the left-hand pair. A scale factor of 2 means you doubled the numerator and the denominator together; 0.5 means you halved them. Scale factors are especially useful when planning: if you know the scale factor of a map (1:50000 means the factor is 50000) you can find any real distance from a map measurement, or vice versa. The "Ratio difference" output confirms how close the two sides are after solving: it should be zero (or very close to it due to floating-point rounding) whenever the proportion is balanced.
Real-world applications: recipes, maps, and unit conversions
Toggle the "Show real-world application" option to model a practical scenario with your numbers. For recipe scaling, enter the ingredient amount and the base serving count; the calculator finds how much of that ingredient you need for your target serving count. For map or scale-model problems, enter the map measurement and the map scale as a ratio; the calculator returns the actual distance. For currency or unit conversions, enter the exchange rate as a known ratio and the amount you want to convert. All three are direct proportions under the hood: the math is identical, only the labels change.
Proportion formulas for each unknown position
| Solve for | Direct proportion formula | Inverse proportion formula |
|---|---|---|
| a | a = (b x c) / d | a = (c x d) / b |
| b | b = (a x d) / c | b = (c x d) / a |
| c | c = (a x d) / b | c = (a x b) / d |
| d | d = (b x c) / a | d = (a x b) / c |
Choose the row matching the term you want to find. These are all derived from a x d = b x c (direct) or a x b = c x d (inverse).
Frequently asked questions
How do I solve a proportion for any of the four terms?
Use the "Solve for" dropdown to select the unknown position (a, b, c, or d). Then enter your three known values in the a, b, and c input fields. The calculator applies cross-multiplication: a x d = b x c, then divides by the coefficient of the unknown to isolate it.
What is the difference between a direct and an inverse proportion?
In a direct proportion (a/b = c/d), both sides share the same ratio and the quantities grow together. In an inverse proportion (a x b = c x d), the products of each pair are equal and the quantities move in opposite directions. Choose "Inverse proportion" in the Proportion type selector to switch the formula.
Why can the denominator not be zero?
Division by zero is mathematically undefined. If a denominator is zero, the ratio has no finite value and the proportion equation has no solution. The calculator returns a blank result in that case rather than an error.
What does the scale factor tell me?
The scale factor (c / a) is the multiplier between the numerators of the two ratios. A scale factor of 3 means the right-hand numerator is three times the left-hand numerator, and the denominator must also be three times as large for the proportion to hold. It is the same number as the constant of proportionality k in the equation a/b = k.
How can I use this calculator for recipe scaling?
Turn on the "Show real-world application" toggle and select "Recipe scaling". Enter the ingredient amount and the base serving count (as the known quantity and known total), then enter how many servings you want to make (as the target total). The calculator applies the proportion a/b = c/d to find exactly how much of that ingredient you need for the new batch.
How do I verify my answer after solving?
Substitute the solved value back into both sides of the proportion and check that the two decimals match. You can also cross-multiply the complete proportion: the product of the outer terms (a and d) must equal the product of the inner terms (b and c). The "Ratio difference" output will read 0% when the proportion is perfectly balanced.