Proportion Calculator
A proportion says two ratios are equal: a/b = c/d. From that one sentence comes the whole craft of finding a missing quantity, from recipe scaling to map reading to mixing concrete. The calculator below solves a/b = c/d for any missing term, checks whether two ratios truly form a proportion, simplifies ratios, and handles direct and inverse proportion word problems plus recipe scaling, with the full working shown every time.
Cross-multiplication is not a trick: multiply both sides of a/b = c/d by b and then by d, and the denominators cancel, leaving a × d = b × c. The steps box shows that working for every calculation.
Quick Answer
Solve 3/4 = x/20 and x = 15. Cross-multiply to 3 × 20 = 4x, so 60 = 4x and x = 15. That is all this calculator does in six modes: find any missing term in a/b = c/d, check whether two ratios truly match, simplify a ratio to lowest terms, solve direct and inverse proportion word problems, and scale a recipe between serving counts. The classic recipe version, 3/4 = 15/20, is pre-filled above; leave d blank, click Calculate, and watch the working unfold.
How to use the proportion calculator
Choose the mode first, then fill in what you know. The result box stays hidden until you calculate, and every error message names the exact problem instead of flashing a generic alert.
- Solve a/b = c/d: enter any three of a, b, c, d and leave the fourth blank. The calculator cross-multiplies, divides by the number paired with the unknown, and prints every step plus a cross-product check.
- Check: enter all four numbers to get a TRUE/FALSE verdict on whether the proportion holds, with the two decimal ratios and both cross products side by side.
- Simplify a ratio a:b: enter two whole numbers to reduce them by their greatest common divisor. Decimals are rejected honestly here, because the tool's GCD arithmetic needs integers (see the honest-limits note below).
- Direct / inverse word problems: enter the known pair and the new quantity. The labels change to items and cost, or workers and days, and the steps name the logic, so a direct problem can never silently be solved as an inverse one.
- Scale a recipe: enter original servings, desired servings and one ingredient amount. The tool prints the scale factor and the new amount in the same units.
Why cross-multiplication works
Cross-multiplication is one legitimate algebraic step, not a trick. Start with a/b = c/d and multiply both sides by b, then by d: the denominators cancel, and a × d = b × c survives. Everything the solver does is a rearrangement of that single equation:
Solve for a: a = (b × c) ÷ d | Solve for b: b = (a × d) ÷ c
Solve for c: c = (a × d) ÷ b | Solve for d: d = (b × c) ÷ a
The denominators b and d can never be zero, which is why the tool refuses them and says which one is the problem.
The OpenStax prealgebra chapter solving proportions and their applications teaches the same derivation through the least common denominator, and Math is Fun's proportions page phrases it as "multiply across the known corners, then divide by the third number". Same arithmetic, same answer.
Worked examples, one per mode
Every row below is a real run of the calculator in that mode, so you can reproduce each figure exactly by typing it in.
| Mode | Input | Result |
|---|---|---|
| Solve | 3/4 = 15/d | d = 20 (3 × d = 4 × 15 = 60) |
| Check | 3/4 and 15/20 | TRUE: both 0.75, cross products 60 = 60 |
| Check | 3/4 and 15/19 | FALSE: 0.75 vs 0.7895, cross products 57 ≠ 60 |
| Simplify | 15:20 | 3:4 (GCD 5) |
| Direct | 3 items cost $12; 7 items? | $28 ((12 × 7) ÷ 3) |
| Inverse | 4 workers, 6 days; 3 workers? | 8 days ((4 × 6) ÷ 3) |
| Scale | 4 servings → 6; 500 g flour | 750 g (factor 1.5) |
Two honest limits are built into the results rather than hidden. First, the check mode uses a 0.0001 tolerance when comparing decimal ratios, and the steps box always prints the exact cross products beside it, so near-misses like 15/19 are judged on the integers, not on rounded decimals. Second, simplify mode only accepts whole numbers: a decimal like 1.5 gets "Ratio simplification needs whole numbers." Some ratio tools clear decimals exactly (CalculatorX does this well, see below); this tool states the boundary up front instead of rounding silently.
Direct vs inverse proportion: the one question that decides
Mixing the two up is the most common error in proportion word problems. Ask what happens when one quantity doubles:
| Type | When to use | The math |
|---|---|---|
| Direct | Both quantities move together: twice the items cost twice as much | a/b = c/d; x = (known2 × new1) ÷ known1 |
| Inverse | They move in opposite directions: twice the workers take half the time | w₁ × d₁ = w₂ × d₂; x = (known1 × known2) ÷ new1 |
The calculator keeps the two modes separate with different labels and different step text, so a direct problem can never silently be solved with the inverse formula. Khan Academy's proportional relationships unit puts the same distinction on graphs: a direct proportion is a straight line through the origin, because doubling one quantity always doubles the other from a zero start.
What competing calculators miss
The proportion SERP is crowded with single-purpose tools, and the gaps fall in a pattern:
- Solvers with no steps (proportionscalculator.com, ai math proportion pages): they find the missing term, but students copying the answer learn nothing. This page prints the cross-multiplication line by line for every result, including the cross-product check at the end.
- Check-only or solve-only tools (size.ly, calculatorlib): solid at the one job, no simplify mode, no word-problem modes, no scaling. Getting all six jobs done needs six tabs open.
- Decimal simplification (CalculatorX's ratio tool): the strongest rival in one respect: it simplifies decimal ratios exactly by clearing denominators, where this tool requires whole numbers and says so. If you need 1.5:2.25 reduced to 2:3, it handles that; for everything else, this page's six modes and shown working cover the job.
- The gap this page fills: six modes in one tool, honest integer boundaries stated instead of hidden, direct and inverse kept as separate modes with labelled logic, and a recipe scaler alongside the pure algebra, with the working shown every time. No first-page rival combines all of them.
Where proportions take you next
Recipe scaling is proportion work with a shopping list: the recipe serving size calculator is the dedicated kitchen tool. Proportions also underlie percent work, since a percent is a ratio out of 100: the fraction to percent calculator converts between the two readings. The fixed multiplier inside a direct proportion has its own name and tool in the constant of proportionality calculator, and mixture problems, where concentration ratios drive the arithmetic, belong to the dilution molarity calculator.
Frequently asked questions
It is one legitimate algebraic step, not a trick. Multiply both sides of a/b = c/d by b and then by d: the denominators cancel, and a × d = b × c survives. The calculator's step box shows exactly that for every calculation, so you can follow the algebra instead of trusting a black box.
No. A zero denominator makes the ratio undefined, so the calculator refuses it and names the offending field: "b cannot be zero." or "d cannot be zero." In solve mode it also refuses setups that would divide by zero, such as solving for b when c is zero.
Ask what happens when one quantity doubles. If the other doubles too, it is direct (a/b = c/d). If the other halves, it is inverse (w1 × d1 = w2 × d2). The calculator keeps the two modes separate with different labels, so a direct problem is never solved with the inverse formula by accident.
That the two ratios are equal, which is exactly what a proportion claims. In check mode, 3/4 and 15/20 give cross products 3 × 20 = 60 and 4 × 15 = 60, so the proportion holds. With 15/19 the products are 57 and 60, so it fails.
Yes: a map scale like 1:50,000 is a ratio of map distance to real distance. Enter 1 and 50000 as one pair and your measurement with a blank as the other, and solve for the missing term. The same setup works for any scale drawing, model or unit conversion built on a fixed ratio.
A simplified ratio is easier to compare and scale. Dividing both parts by their greatest common divisor keeps the relationship while shrinking the numbers. Simplify mode works on whole numbers; decimals are rejected with an honest message rather than rounded silently.
Yes, because a percent is a ratio: 25% is 25/100. Set up part/whole = percent/100 and solve for the missing term. For 25% of 160: x/160 = 25/100 gives x = 40. Math is Fun's proportions page works through exactly this application.
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References
- OpenStax. Prealgebra 2e: Solve Proportions and their Applications, deriving cross-products via the least common denominator. Accessed October 23, 2026.
- OpenStax. Elementary Algebra: Solve Proportion and Similar Figure Applications, on matching units in numerators and denominators. Accessed October 23, 2026.
- Math is Fun. Proportions, the "multiply across the known corners" method with worked examples. Accessed October 23, 2026.
- LibreTexts (OpenStax). Solve Proportions and their Applications (Part 2), on the percent proportion, amount/base = percent/100. Accessed October 23, 2026.
- Khan Academy. Proportional Relationships: graphs of proportional relationships, on direct proportions as straight lines through the origin. Accessed October 23, 2026.