Proportion Calculator
Three numbers in, the fourth solved — cross-multiplication does the rest.
Solve any proportion a/b = c/d for the missing value. Pick which term is unknown, enter the other three, and the calculator cross-multiplies to solve for x, shows both cross products, and confirms the equation balances. Switch to Standard for direct (y = kx) and inverse (y = k/x) proportion, Detailed for percent problems via is/of = %/100, or Advanced to scale a recipe or plan to a new size.
Exact arithmetic on the numbers you enter — nothing estimated. Denominators can't be zero; impossible setups are flagged instead of guessed.
How to solve a proportion by cross-multiplication
Three steps, any missing term
A proportion a/b = c/d hides one solvable equation: a × d = b × c — the product of the extremes equals the product of the means. To solve 3/4 = x/20: cross-multiply diagonally to get 4 × x = 3 × 20, simplify to 4x = 60, and divide both sides by 4: x = 15. Check the answer the same way you found it — 3 × 20 = 60 and 4 × 15 = 60, so the proportion balances.
The formula for each position
| Missing term | Formula | Example with 3/4 = 15/20 |
|---|---|---|
| a (first numerator) | a = b × c ÷ d | 4 × 15 ÷ 20 = 3 |
| b (first denominator) | b = a × d ÷ c | 3 × 20 ÷ 15 = 4 |
| c (second numerator) | c = a × d ÷ b | 3 × 20 ÷ 4 = 15 |
| d (second denominator) | d = b × c ÷ a | 4 × 15 ÷ 3 = 20 |
Why it works. Multiply both sides of a/b = c/d by b × d. The b cancels on the left, the d cancels on the right, and what's left is a × d = b × c — no fractions, one unknown, one division to finish. That's all cross-multiplication is: clearing both denominators in a single move.
Direct vs inverse proportion — finding k and predicting
Two opposite relationships
| Direct proportion | Inverse proportion | |
|---|---|---|
| Equation | y = k × x | y = k ÷ x |
| What stays constant | The ratio y ÷ x = k | The product x × y = k |
| Double x and… | y doubles | y halves |
| Everyday example | Hours worked → pay earned | Speed → travel time |
Worked both ways from the pair (4, 10)
Direct: k = y ÷ x = 10 ÷ 4 = 2.5, so at x = 6, y = 2.5 × 6 = 15. Inverse: k = x × y = 4 × 10 = 40, so at x = 6, y = 40 ÷ 6 ≈ 6.67. Same known pair, opposite predictions — which is why naming the relationship correctly matters more than the arithmetic.
The classic inverse case is speed and time over a fixed distance: a 240-mile trip takes 240 ÷ 60 = 4 hours at 60 mph and 240 ÷ 80 = 3 hours at 80 mph — the product of speed and time is pinned at 240 whatever you drive. A quick test for your own data: if y ÷ x is (roughly) constant across pairs, it's direct; if x × y is constant, it's inverse; if neither, it isn't a proportion at all.
Percent problems, recipe scaling and map distances
One template for every percent question
The percent proportion — part/whole = percent/100, taught in US classrooms as is/of = %/100 — turns all three percent question types into the same cross-multiplication:
| Question | Setup | Solve |
|---|---|---|
| What percent is 45 of 60? | 45/60 = p/100 | p = 4,500 ÷ 60 = 75% |
| What is 30% of 250? | x/250 = 30/100 | x = 7,500 ÷ 100 = 75 |
| 18 is 24% of what? | 18/x = 24/100 | x = 1,800 ÷ 24 = 75 |
Scaling a recipe: 4 servings to 7
Every ingredient scales by the ratio of the serving counts: 7/4 = ×1.75. So 2.5 cups of flour becomes 2.5 × 7 ÷ 4 = 4.375 cups (4 cups plus 6 tablespoons), 3 eggs becomes 3 × 1.75 = 5.25 — round to 5 and hold back a little liquid. The Advanced level runs this for any quantity; only seasoning breaks the rule of three, since salt, spice and leavening usually want less than a full linear scale-up.
Map scales are proportions too
A 1:250,000 scale means 1 inch of map is 250,000 inches of ground. Two towns 3.2 inches apart: 1/250,000 = 3.2/x, so x = 800,000 inches ÷ 63,360 inches per mile ≈ 12.6 miles. Word scales skip the conversion: at 1 inch = 5 miles, a 3.5-inch route is 17.5 miles. Keep like quantities in like positions — map on top on both sides, ground on the bottom — and the units sort themselves out.
Do two ratios form a proportion? Three tests
| Test | 6/8 vs 9/12 | 5/7 vs 7/10 |
|---|---|---|
| Cross products equal? | 6 × 12 = 72 = 8 × 9 ✓ | 5 × 10 = 50 ≠ 7 × 7 = 49 ✗ |
| Same lowest terms? | Both reduce to 3/4 ✓ | 5/7 vs 7/10 — already reduced, different ✗ |
| Same decimal? | Both 0.75 ✓ | 0.714… vs 0.7 ✗ |
All three tests always agree, but cross-multiplying is the safest: it uses whole-number arithmetic only, so there's no rounding to blur a near-miss like 5/7 vs 7/10, which differ by barely 0.014 in decimal form yet clearly fail the 50-vs-49 cross-product check.
Means, extremes and the old notation
Written the older way — a : b :: c : d — the outer terms a and d are the extremes and the inner terms b and c the means. The rule "the product of the means equals the product of the extremes" is exactly the cross-product test, three centuries older. When the two means are equal (a : m :: m : d), m is the geometric mean of a and d: for 4 : 6 :: 6 : 9, check 4 × 9 = 36 = 6 × 6.
Ratio vs proportion
A ratio is a comparison of two quantities (3 : 4); a proportion is an equation claiming two ratios are equal — which is why it can be solved. Simplifying 45 : 60 to 3 : 4, splitting $200 in a 3 : 5 ratio, or converting to 1 : n form are ratio jobs — our ratio calculator handles those. This page is for the equation: finding the missing term that makes two ratios match.
❓ Frequently asked Frequently asked questions
What is a proportion in math?
A proportion is an equation stating that two ratios are equal — written a/b = c/d, or a : b = c : d. It says the two fractions have the same value: 3/4 = 15/20 is a proportion because both sides equal 0.75. The outer terms (a and d) are called the extremes and the inner terms (b and c) the means, and in every true proportion the product of the means equals the product of the extremes: 4 × 15 = 60 and 3 × 20 = 60. That single fact — cross products are equal — is what lets you solve for any one missing value.
How do you solve a proportion by cross-multiplying?
Multiply diagonally across the equals sign and set the two products equal, then divide. To solve 3/4 = x/20: cross-multiply to get 4 × x = 3 × 20, so 4x = 60, and divide both sides by 4 to get x = 15. It works because multiplying both sides of a/b = c/d by b × d clears both denominators, leaving a × d = b × c. The same three steps solve for a missing value in any position — numerator or denominator, left side or right.
How can I check whether two ratios form a proportion?
Cross-multiply and compare. The ratios 6/8 and 9/12 form a proportion because 6 × 12 = 72 and 8 × 9 = 72 — equal cross products. The ratios 5/7 and 7/10 do not, because 5 × 10 = 50 while 7 × 7 = 49. Equivalently, reduce both fractions to lowest terms (6/8 and 9/12 both simplify to 3/4) or convert to decimals (both 0.75) — any of the three tests gives the same verdict, but cross-multiplying avoids rounding error entirely.
What is the difference between direct and inverse proportion?
Two quantities are directly proportional when they rise and fall together at a constant ratio: y = kx, so doubling x doubles y. Hours worked and pay earned behave this way. They are inversely proportional when one rises as the other falls so their product stays constant: y = k/x. Speed and travel time are the classic case: a 240-mile trip takes 4 hours at 60 mph and 3 hours at 80 mph — the product (the distance) stays fixed at 240. The Standard level of this calculator finds k from one known pair and predicts new values in either mode.
How do you solve percent problems with a proportion?
Use the percent proportion: part/whole = percent/100 (often taught as is/of = %/100). To find what percent 45 is of 60, write 45/60 = p/100, cross-multiply to get 60p = 4,500, and divide: p = 75, so 45 is 75% of 60. The same setup finds a missing part (what is 30% of 250? x/250 = 30/100 → x = 75) or a missing whole (18 is 24% of what? 18/x = 24/100 → x = 1,800 ÷ 24 = 75). One template covers all three percent question types.
How do I use a proportion to read a map scale?
A map scale is a ratio between map distance and real distance, so a proportion converts one to the other. On a 1:250,000 map, 1 inch on paper represents 250,000 inches of ground. If two towns sit 3.2 inches apart, set up 1/250,000 = 3.2/x and cross-multiply: x = 3.2 × 250,000 = 800,000 inches. Divide by 63,360 (inches per mile) to get about 12.6 miles. Scales given in words work the same way: at 1 inch = 5 miles, a 3.5-inch route is 3.5 × 5 = 17.5 miles.
How do I scale a recipe for a different number of servings?
Set each ingredient in a proportion against the serving counts. A recipe for 4 that you're stretching to 7 people scales every ingredient by 7/4 = 1.75. For 2.5 cups of flour: 2.5/4 = x/7, so x = 2.5 × 7 ÷ 4 = 4.375 cups — that's 4 3/8 cups, or 4 cups plus 6 tablespoons. The Advanced level does this arithmetic for you: enter the original quantity and both serving counts and it returns the scale factor and the new amount. One caution from the kitchen: spices, salt, and leavening often scale less than linearly, so taste before doubling those.
What is the difference between a ratio and a proportion?
A ratio compares two quantities (3 : 4); a proportion is an equation asserting that two ratios are equal (3/4 = 15/20). A ratio is a single value you can simplify, scale, or divide; a proportion is a statement you can solve — because it's an equation, cross-multiplication applies and a missing term can be recovered. If you want to simplify a ratio, split an amount in a given ratio, or convert a ratio to 1 : n form, use our ratio calculator; this page is for solving equations between ratios.
Can a proportion contain zero or negative numbers?
Zeros in a numerator are fine — 0/5 = 0/8 is a true proportion, since both sides equal 0 — but a zero denominator makes the ratio undefined, so b and d can never be 0, and this calculator reports no solution rather than dividing by zero. Solving can also hit zero: 3/4 = x/0 has no solution. Negative numbers are legal and cross-multiplication handles them normally: −3/4 = 6/−8 is true because −3 × −8 = 24 = 4 × 6. Just track the signs — one negative term on each side, placed diagonally, keeps the proportion balanced.
Where these figures come from
Proportions are pure arithmetic — there are no rates or thresholds to go stale. The definitions, terminology and conversion constants used on this page come from these references:
- Definition of proportionality, the constant k, and direct/inverse relationships — Wolfram MathWorld — Proportional.
- Cross-multiplication and why it clears both denominators (the "rule of three") — Wikipedia — Cross-multiplication.
- Direct and inverse proportionality, means and extremes — Wikipedia — Proportionality (mathematics).
- US customary length conversions (63,360 inches per mile) used in the map-scale examples — NIST — Office of Weights and Measures, unit conversion.
Last checked: July 2026. The math on this page doesn't change with tax years or markets — the worked examples are exact and can be verified by hand with the formulas shown above.
Select the question that matches where you are right now.
The headline number is the missing term that makes your two ratios equal — the value of x that balances a/b = c/d.
With the solved value substituted, both fractions have exactly the same decimal value and the two cross products match. The breakdown shows all of it: the completed proportion, both cross products, and each side as a decimal.
Cross products are the proof, not just the method. If a × d and b × c agree, the answer is right — no rounding involved. The chart's two bars split at the same point for the same reason: equal ratios have the same shape.
A dash means your setup divides by zero — a zero denominator, or a zero in the term the formula divides by. That's mathematics refusing, not the calculator failing: 3/4 = x/0 genuinely has no answer.
Three ideas do all the work on this page: equal cross products, the multiplier between the fractions, and the constant k.
a/b = c/d exactly when a × d = b × c. Solving, checking, and testing two ratios are all the same move — multiply diagonally and compare.
In 3/4 = 15/20, the second fraction is the first scaled by ×5 (top and bottom alike). Every proportion hides one scale factor — it's the recipe factor, the map scale, and the percent all at once.
Direct proportion keeps a constant ratio (y ÷ x = k); inverse proportion keeps a constant product (x × y = k). From the same pair (4, 10), direct predicts y = 15 at x = 6 while inverse predicts 6.67 — identifying the relationship correctly matters more than the arithmetic.
Proportion errors are nearly always setup errors, not arithmetic errors.
Like quantities in like positions: map inches over ground miles on both sides, part over whole on both sides. Swapping one side's order silently inverts the answer.
More workers, less time per job is inverse; more hours, more pay is direct. Setting up an inverse situation as a direct proportion gives a confidently wrong answer — the most common proportion mistake there is.
The "of" number is the whole, and it goes under the part. "45 is what percent of 60" and "60 is what percent of 45" are different questions (75% vs 133.33%) — pin down the whole before you set anything equal.
Proportions sit between ratios and percentages — these tools cover the neighbors.
Simplify, divide an amount in a ratio, or convert to 1 : n form.
Ratio calculator →Volumes of tanks, boxes and cylinders — scaling in three dimensions.
Volume calculator →