Proportion Calculator
A proportion states that two ratios are equal. Fill in any three of the four values, leave one blank, and the calculator finds the missing value, showing every step.
How to use this calculator
Type three of the four numbers in the proportion a/b = c/d and leave the fourth box empty. Press Calculate and you get the missing value plus the cross-multiplication steps used to find it. Use Reset to clear the form and try another.
Tip: it does not matter which position is blank, the calculator solves for a, b, c, or d.
How it works
A proportion is solved by cross-multiplication: multiply the numerator of each side by the denominator of the other. If a/b = c/d, then a × d = b × c. From there it is one division to isolate the unknown.
The calculator also runs a check at the end, substituting the answer back to confirm both ratios match.
Worked example
Example: 3/4 = 9/x
Step 1, Cross-multiply: 3 × x = 4 × 9, so 3x = 36.
Step 2, Divide: x = 36 ÷ 3 = 12.
Step 3, Check: 3/4 = 0.75 and 9/12 = 0.75. They match, so x = 12 is correct.
Common mistakes
- Forgetting which terms multiply: cross products pair each numerator with the opposite denominator (a with d, b with c), not the one beneath it.
- Mixing up the positions: 3/4 = 9/12 is true, but 3/4 = 12/9 is not. Keep numerators on top on both sides.
- Zero in a denominator: if the unknown sits under a division by zero, the proportion has no valid solution, the calculator will tell you.
When to use it
Proportions appear anywhere quantities scale together: converting recipes, map scales, unit conversions, similar triangles, and “if 3 workers finish in 4 hours…” style rate problems. Whenever two ratios must stay equal while the numbers change, this is the tool.
Frequently asked questions
A proportion is an equation stating that two ratios are equal, written a/b = c/d. It is the formal way of saying “these two fractions represent the same relationship.”
Cross-multiply to get a single equation, then divide. For 3/4 = 9/x: cross-multiplying gives 3x = 36, so x = 12. The calculator above does this and shows each step.
No, one proportion equation can only determine one unknown. With two unknowns there are infinitely many solutions, so you need a second equation (a system) instead.
That means the equation is an identity: every valid value works. It usually happens when both sides are the same ratio written differently, like x/4 = 2x/8.
More worked examples
The example above uses small whole numbers. Real proportions rarely stay that tidy, so here are five more, each with different numbers and a different shape: whole numbers, decimals, fractions, and two real world cases. Every one is solved the same way: cross-multiply, divide once, then check by substituting the answer back.
Example: whole numbers, 5/8 = 15/x
Step 1, Cross-multiply: 5 × x = 8 × 15, so 5x = 120.
Step 2, Divide: x = 120 ÷ 5 = 24.
Step 3, Check: 5/8 = 0.625 and 15/24 = 0.625. They match, so x = 24 is correct.
Example: decimals, 2.5/4 = x/12
Step 1, Cross-multiply: 2.5 × 12 = 4 × x, so 30 = 4x.
Step 2, Divide: x = 30 ÷ 4 = 7.5.
Step 3, Check: 2.5/4 = 0.625 and 7.5/12 = 0.625. They match, so x = 7.5 is correct. Decimals change nothing about the method: cross-multiply first, then divide.
Example: fractions, (1/2)/(3/4) = x/9
Step 1, Cross-multiply: (1/2) × 9 = (3/4) × x, so 9/2 = (3/4)x.
Step 2, Divide by the fraction: x = (9/2) ÷ (3/4). Dividing by a fraction means multiplying by its reciprocal: x = (9/2) × (4/3) = 36/6 = 6.
Step 3, Check: (1/2)/(3/4) = (1/2) × (4/3) = 2/3, and 6/9 = 2/3. They match, so x = 6 is correct.
Example: recipe scaling, 350 g of flour for 4 people
A recipe for 4 people uses 350 g of flour. How much flour do you need for 6 people? The flour per person stays the same, so set up the proportion with flour over people on both sides: 350/4 = x/6.
Step 1, Cross-multiply: 350 × 6 = 4 × x, so 2100 = 4x.
Step 2, Divide: x = 2100 ÷ 4 = 525. You need 525 g of flour.
Step 3, Check: 350 ÷ 4 = 87.5 g per person, and 87.5 × 6 = 525 g. The per-person amount is unchanged, so the scaling is correct.
Example: map scale, 1:25,000 and 4 cm measured
On a map with a scale of 1:25,000, 1 cm on the map stands for 25,000 cm in reality. You measure 4 cm between two villages. Set up the proportion with map distance over real distance: 1/25,000 = 4/x.
Step 1, Cross-multiply: 1 × x = 25,000 × 4, so x = 100,000 cm.
Step 2, Convert the units: 100,000 cm ÷ 100 = 1,000 m, and 1,000 m ÷ 1,000 = 1 km. The villages are 1 km apart.
Step 3, Check: 4 × 25,000 = 100,000, and 1 × 100,000 = 100,000. The cross-products match, so the distance is correct.
Direct proportion vs inverse proportion
Everything on this page so far is a direct proportion: when one quantity grows, the other grows with it, and their ratio stays constant. Double the people, double the flour. Double the map distance, double the real distance. The calculator above solves direct proportions of the form a/b = c/d.
An inverse proportion works the opposite way: when one quantity grows, the other shrinks, and their product stays constant instead of their ratio. The classic case is workers and time. If 4 workers take 6 days to finish a job, the total work is 4 × 6 = 24 worker-days. With 3 workers, the same 24 worker-days must be spread over fewer people, so the job takes longer: 24 ÷ 3 = 8 days. Check it: 3 workers × 8 days = 24 worker-days, the same total as before.
Knowing which kind you have matters, because the algebra is different. For a direct proportion, set the two ratios equal and cross-multiply. For an inverse proportion, set the two products equal: a × b = c × d. A quick test: ask whether the answer should go up or down when the known quantity goes up. If the answer goes up, it is direct. If it goes down, it is inverse, and the calculator above is the wrong tool for it. Use the product rule instead.
Where proportions show up in real life
Proportions are easy to miss because they hide inside ordinary tasks. Here are the places they appear most often, and what the proportion looks like in each case.
- Scaling recipes: ingredients over servings stays constant. A recipe for 4 scaled to 6 is flour over people on both sides, as in the worked example above.
- Maps and models: the scale is a ratio by definition. Map distance over real distance on one side, the scale on the other.
- Unit conversion: every conversion factor is a proportion. 1 inch is 2.54 cm, so x inches is 2.54x cm.
- Similar triangles and shapes: corresponding sides stay in the same ratio, so a missing side is one proportion away.
- Medicine dosing: dose over body weight, or milligrams over millilitres, stays constant. If 250 mg comes in 5 mL, then 400 mg needs 8 mL: 250/5 = 400/x gives x = 8.
- Currency and unit pricing: price over quantity stays constant, which is how you compare value across different pack sizes.
- Fuel and speed: litres over kilometres at a steady speed is a proportion, and so is distance over time.
- Photo and image scaling: width over height stays constant when you resize. A photo 1200 by 800 pixels resized to 600 pixels wide keeps the ratio 1200/800 = 600/x, so the new height is 400 pixels.
How to set up a proportion from a word problem
Most proportion mistakes happen before any arithmetic, in the setup. Use these three steps and the numbers will take care of themselves.
Step 1, Name the two quantities. Read the problem and find the pair that changes together: litres and kilometres, grams and people, centimetres and kilometres. Those are your two quantities.
Step 2, Put matching quantities in matching positions. If the left ratio is litres over kilometres, the right ratio must be litres over kilometres too. Keep the same quantity on top on both sides and the same quantity underneath on both sides. Mixing the order is the most common mistake and it gives the wrong answer even though the arithmetic is perfect.
Step 3, Put the unknown in the matching position and solve. Write the known ratio, write the second ratio with x where the missing value belongs, then cross-multiply and divide as usual.
Example: setting it up, fuel use
A car uses 6 litres of fuel per 100 km. How much fuel does it need for 350 km? The quantities are litres and kilometres, so litres goes on top on both sides: 6/100 = x/350.
Step 1, Cross-multiply: 6 × 350 = 100 × x, so 2100 = 100x.
Step 2, Divide: x = 2100 ÷ 100 = 21 litres.
Step 3, Check: 6/100 = 0.06 litres per kilometre, and 21/350 = 0.06 litres per kilometre. The rate is unchanged, so 21 litres is correct.
How to check whether two ratios form a true proportion
Two ratios form a true proportion exactly when their cross-products are equal. For a/b and c/d, compute a × d and b × c. If the two products match, the ratios are equal and the proportion holds. If they differ, the ratios are not equal.
Example, true: 4/10 and 6/15. Cross-products: 4 × 15 = 60 and 10 × 6 = 60. They match, so 4/10 = 6/15 is a true proportion. Both reduce to 2/5, which is a second way to see it.
Example, false: 3/8 and 5/12. Cross-products: 3 × 12 = 36 and 8 × 5 = 40. They do not match, so 3/8 = 5/12 is not a true proportion, even though the numbers look close. This test is also how the calculator above verifies its own answers before showing them.
Proportion vs ratio vs percentage
These three ideas are related but they are not the same thing, and mixing them up is what makes word problems hard.
| idea | What it is | Written as | Example |
|---|---|---|---|
| Ratio | A comparison of two quantities by division | a:b or a/b | Flour to sugar, 2:1 |
| Proportion | A statement that two ratios are equal | a/b = c/d | 350/4 = 525/6 |
| Percentage | A ratio with 100 as the second term | x/100 | 15% means 15/100 |
A ratio on its own compares two things. A proportion takes two ratios and says they are the same relationship. A percentage is simply a ratio whose second term is fixed at 100, which is why every percentage question is secretly a proportion. “What is 15% of 240?” is the proportion x/240 = 15/100. Cross-multiply: 100x = 15 × 240 = 3600, so x = 36. If you already know which percentage question you are asking, the percentage calculator is quicker, but the proportion method works for all of them.
The unit rate is the other idea worth knowing. It is the value of one unit of the second quantity, found by dividing the first term by the second. In the recipe example, the unit rate is 350 ÷ 4 = 87.5 g of flour per person. Once you have the unit rate, any scaling is one multiplication: 87.5 × 6 = 525 g. Cross-multiplication and the unit rate are two routes to the same answer, and checking one against the other is a good way to catch setup mistakes.
More questions
In a direct proportion, both quantities rise or fall together and their ratio stays constant: double one and the other doubles too. In an inverse proportion, one quantity rises as the other falls and their product stays constant: double the workers and the time halves. Direct proportions are solved by setting the ratios equal, a/b = c/d. Inverse proportions are solved by setting the products equal, a x b = c x d. The calculator above solves direct proportions.
Compare their cross-products. For the ratios a/b and c/d, compute a x d and b x c. If the two products are equal, the ratios form a true proportion. For example, 4/10 and 6/15 give 4 x 15 = 60 and 10 x 6 = 60, so they are equal. If the products differ, as with 3/8 and 5/12 (36 against 40), the ratios are not equal and the proportion is false.
First name the two quantities that change together, such as litres and kilometres. Then write the known ratio, and write the second ratio with x in the matching position, keeping the same quantity on top on both sides and the same quantity underneath on both sides. For example, a car using 6 litres per 100 km needs x litres for 350 km, so write 6/100 = x/350 and solve: x = 21 litres. Keeping the positions matched is the step that matters most.
The unit rate is the value of one unit of the second quantity: divide the first term by the second. If a recipe uses 350 g of flour for 4 people, the unit rate is 350 / 4 = 87.5 g per person. Once you have it, any scaling is one multiplication, such as 87.5 x 6 = 525 g for 6 people. It is the same answer cross-multiplication gives, reached by a different route, so the two methods check each other.
It is not a trick, it is the denominators cancelling. Start with a/b = c/d and multiply both sides by b, then by d. On the left, b cancels and d remains; on the right, d cancels and b remains, leaving a x d = b x c. The proportion is unchanged at every step because both sides were multiplied by the same numbers, so solving a x d = b x c gives exactly the same answer as solving a/b = c/d. The only restriction is that no denominator may be zero.