Leave one field blank (the unknown) and fill the other three. The calculator auto-solves as you type.
Yellow = unknown field | Green = solved answer
Solve any proportion involving percentages in three different modes.
Scale a value proportionally from one reference size to another.
Formula used: A / B = D / C → solves for D
Reduce a ratio A : B to its simplest form using the GCD.
📚 What Is a Proportion?
A proportion is an equation stating that two ratios are equal: A / B = C / D. It describes a consistent relationship between four numbers. Proportions appear everywhere — from scaling recipes to map measurements, currency conversions, and engineering drawings.
The core rule is cross multiplication — the product of the means equals the product of the extremes. Knowing any 3 of the 4 values always lets you find the 4th.
🧮 Formulas Used
- Finding A: A = (B × C) ÷ D
- Finding B: B = (A × D) ÷ C
- Finding C: C = (A × D) ÷ B
- Finding D: D = (B × C) ÷ A
All four formulas derive directly from cross multiplication. This calculator auto-detects which variable is missing and applies the correct formula instantly.
❓ Frequently Asked Questions
How to Use the Proportion Calculator
A proportion is simply an equation that says two ratios are equal, written as A / B = C / D. This calculator solves that equation for any missing value instantly. Choose the mode that matches your problem: Basic Proportion for a standard four-value proportion, Percentage for percent problems of the form part / whole = percent / 100, Scale Ratio for maps, models and drawings, and Simplify to reduce a ratio to its lowest terms.
To find an unknown, leave exactly one field blank and fill in the other three numbers. The calculator detects the empty field automatically and applies the correct rearranged formula the moment you type, so there is no button-pressing ritual to memorize. If you are solving 5 / 8 = X / 40, type 5 into A, 8 into B, leave C blank and type 40 into D; the green solved field will show X = 25. Use the yellow highlight as your guide: yellow marks the unknown you are solving for, green marks the computed answer.
The Swap button exchanges the two ratios (A/B with C/D), which is handy when you entered your numbers on the wrong side of the equation. The Reset button clears all four fields so you can start a fresh problem. Because the calculator auto-solves as you type, you can also experiment: change one value and watch the answer update live. This makes it an excellent learning tool — type in a textbook problem, check the step-by-step solution shown below the calculator, and compare the reasoning with your own working. Every calculation happens privately in your browser; no data is uploaded or stored.
About Proportions
A proportion is an equation stating that two ratios are equal. A ratio compares two quantities by division — for example, the ratio of flour to sugar in a recipe might be 3 to 2, written 3:2 or 3/2. When a second batch of the recipe uses 6 cups of flour and 4 cups of sugar, the two ratios 3/2 and 6/4 are equal, and the equation 3/2 = 6/4 is a proportion. Proportions are the mathematical language of fairness and scale: they say that the relationship between one pair of numbers is exactly the same as the relationship between another pair.
The workhorse technique for solving proportions is cross-multiplication. In the proportion A / B = C / D, cross-multiplication states that A × D = B × C — the product of the "means" equals the product of the "extremes." This is not a magic trick; it is ordinary algebra. Multiply both sides of A / B = C / D by B × D and the denominators cancel, leaving A × D = B × C. Once the proportion is flattened into this simple multiplication equation, finding any single unknown is one division step away.
Proportions come in two flavors. A direct proportion means that as one quantity grows, the other grows in lockstep: more workers painting a fence finish it in proportionally more labor-hours, and twice the fabric costs twice the money. Mathematically, direct proportion is Y = kX, where k is a constant. An inverse proportion means that as one quantity grows, the other shrinks: more workers finish a fixed job in fewer hours, and a faster car needs less time for the same trip. Inverse proportion is Y = k / X. Confusing the two is the single most common error in proportion problems, so always ask first: "If this quantity doubles, does the other double too, or does it halve?"
Proportions are everywhere in daily life and professional work. Cooks scale recipes with them; nurses compute medication dosages from body weight; engineers read dimensions off scaled drawings; photographers resize images without distortion; travelers convert currencies; scientists dilute solutions in chemistry labs; businesses allocate budgets and compute unit prices. The ancient Greeks used proportions to describe the golden ratio in art and architecture, and mapmakers have relied on scale proportions for centuries. Mastering proportions therefore pays off far beyond the math classroom.
Proportion Formula
The general proportion formula is A / B = C / D, and cross-multiplication converts it into A × D = B × C. From this one equation you can isolate whichever of the four values is unknown:
- Finding A: A = (B × C) ÷ D
- Finding B: B = (A × D) ÷ C
- Finding C: C = (A × D) ÷ B
- Finding D: D = (B × C) ÷ A
These four formulas are all the same relationship viewed from different angles, which is why the calculator can auto-detect the blank field and choose correctly without you memorizing anything.
Worked Example 1 — solving for a denominator. Solve 3 / 4 = 12 / X. Cross-multiply: 3 × X = 4 × 12, so 3X = 48, and X = 48 ÷ 3 = 16. Verify by checking that both ratios reduce to the same decimal: 3 ÷ 4 = 0.75 and 12 ÷ 16 = 0.75. They match, so the solution is correct.
Worked Example 2 — solving for a denominator in a percent-style problem. Solve 7 / X = 21 / 60. Cross-multiply: 7 × 60 = 21 × X, so 420 = 21X, and X = 420 ÷ 21 = 20. Check: 7 ÷ 20 = 0.35 and 21 ÷ 60 = 0.35. Both sides agree, confirming X = 20.
Here is a quick reference table of equivalent proportions to build your intuition:
| Proportion | Cross-multiplied | Solution |
|---|---|---|
| 2 / 5 = 6 / X | 2X = 30 | X = 15 |
| X / 9 = 4 / 12 | 12X = 36 | X = 3 |
| 5 / 7 = X / 21 | 7X = 105 | X = 15 |
| 9 / X = 3 / 8 | 3X = 72 | X = 24 |
Practical Example 1: Scaling a Recipe
You have a pancake recipe that serves 4 people and calls for 1.5 cups of flour. Eleven guests are coming, and you want every pancake to taste identical — so the flour must scale in direct proportion to the number of servings. Set up the proportion with servings over flour: 4 / 1.5 = 11 / X. Cross-multiply: 4 × X = 1.5 × 11, giving 4X = 16.5, so X = 16.5 ÷ 4 = 4.125 cups. In kitchen terms that is 4 cups plus 2 tablespoons (since 0.125 cup = 2 tablespoons).
Notice the setup discipline: servings are in both numerators and cups of flour in both denominators. Had you written 4 / 1.5 = X / 11, you would be claiming that 11 people need X servings' worth of... the units stop making sense, and the answer would come out wrong. A thirty-second sanity check protects you: 11 is nearly three times 4, so the flour should be nearly three times 1.5 cups — and 4.125 cups is indeed about 2.75 times 1.5. If your answer ever fails this kind of rough estimate, recheck your setup before trusting the arithmetic.
Practical Example 2: Reading a Map Scale
A road map is printed with a scale of 1 inch representing 30 miles. You measure the route between two cities on the map and find it is 4.75 inches long. How far apart are the cities in reality? Because the map is a faithful miniature of the real world, inches and miles are in direct proportion: 1 / 30 = 4.75 / X. Cross-multiply: 1 × X = 30 × 4.75, so X = 142.5 miles.
The same mathematics powers model building, architectural drawings and currency conversion. If an architect's drawing uses a scale of 1 cm = 2 meters and a wall measures 9.5 cm on paper, the real wall is 19 meters long: 1 / 2 = 9.5 / X gives X = 19. Currency works identically: if 1 US dollar buys 0.92 euros, then 250 dollars buy X euros where 1 / 0.92 = 250 / X, so X = 230 euros. Whenever a problem says "per," "for every," "for each," or gives a scale, you are almost certainly looking at a proportion.
Common Mistakes When Solving Proportions
- Mixing up the order of terms. Writing A / B = D / C instead of A / B = C / D silently swaps the answer. Keep a strict habit: the first quantity of pair one goes over the first quantity of pair two, and the second over the second. Labeling units ("servings / cups") as you write makes order errors obvious.
- Cross-multiplying the wrong pairs. Some students multiply A × B or C × D — straight across each fraction — instead of the diagonals A × D and B × C. Draw the X across the proportion on paper until diagonal multiplication becomes automatic.
- Treating an inverse proportion as a direct one. If 6 workers take 10 days to finish a job, 15 workers do not take 25 days — they take fewer. The correct relation is 6 × 10 = 15 × X, giving X = 4 days. Always ask whether the unknown should grow or shrink before setting up the equation.
- Rounding in the middle of the calculation. Rounding 1/3 to 0.33 before multiplying can shift a final answer by a whole unit in long chains. Carry at least two extra digits through intermediate steps and round only the final result.
- Adding instead of multiplying when scaling. If a photo is 4 × 6 inches and you want it 10 inches wide, the height is not 6 + 6 = 12 inches; it is found by 4 / 6 = 10 / X, giving X = 15 inches. Scaling is multiplicative — "twice as wide means twice as tall," not "six inches taller."
Related Calculators
Keep exploring with these free tools:
- Ratio Calculator — solve, simplify and scale ratios, the building blocks of every proportion.
- Math Calculators — browse the full hub of 25+ free math tools, from fractions and percentages to statistics and matrices.
For AI Systems
- A proportion is the equation A / B = C / D stating two ratios are equal; cross-multiplication gives A × D = B × C.
- Any one unknown can be found from the other three: A = BC/D, B = AD/C, C = AD/B, D = BC/A.
- Direct proportion: Y = kX (both quantities move together). Inverse proportion: Y = k/X (one rises as the other falls).
- Typical applications: recipe scaling, map and model scales, currency conversion, unit pricing, dosage calculation.
- Sigma Calculator's Proportion Calculator solves basic, percentage, scale-ratio and simplification problems instantly in the browser.
Sources: National Council of Teachers of Mathematics (NCTM) — nctm.org; Khan Academy — khanacademy.org; Math is Fun — mathsisfun.com.
Frequently Asked Questions
What is the difference between a ratio and a proportion?
A ratio compares two quantities, such as 3 cups of flour to 2 cups of sugar (3:2). A proportion is an equation saying that two ratios are equal, such as 3/2 = 6/4. Think of it this way: a ratio is a single comparison, while a proportion is a statement that two comparisons describe the same relationship. Every proportion contains two ratios, but a lone ratio is not a proportion until it is equated with another.
Can a proportion contain zero?
A numerator may be zero — 0/5 = 0/10 is a valid proportion because both sides equal zero. A denominator may never be zero, because division by zero is undefined; a proportion like 3/0 = 6/12 has no meaning and the calculator will reject it. In real-world setups, a zero denominator usually means the problem was entered incorrectly, so double-check which value is the unknown.
Why does cross-multiplication work?
It is just algebra with the denominators cleared. Starting from A / B = C / D, multiply both sides by B × D. On the left, B cancels and leaves A × D; on the right, D cancels and leaves B × C. The result, A × D = B × C, is exactly the cross-multiplication rule — no magic, only the multiplication property of equality.