How to calculate proportion, step by step
A proportion is two fractions set equal to each other — 3/4 = x/12 says three quarters and x twelfths are the same amount, and x is the number that makes them equal. To calculate a proportion, cross-multiply and divide: multiply the first term by the fourth term, divide by the second term, and the answer is the missing term. On the defaults — 3/4 = x/12 — the missing term is 9, because 3 times 12 is 36 and 36 divided by 4 is 9. Work it out in the proportion calculator.
The worked example: 3/4 = x/12
The worked example, line by line. Step 1 — cross-multiply: 3 × 12 = 36, and 4 × x = 4x. Step 2 — set them equal: 4x = 36. Step 3 — divide by 4: x = 36 ÷ 4 = 9. Step 4 — check: 9 ÷ 12 = 0.75, and 3 ÷ 4 = 0.75 — the same, so the proportion holds. Step 5 — the ratio: 3:4 is already in lowest terms, because 3 and 4 share no divisor but 1. Step 6 — as a percent: 3 ÷ 4 × 100 = 75%, because three quarters is three quarters of a hundred. The same six steps work for any proportion: cross-multiply, set equal, divide, check, simplify, and turn it into a percent if you need one.
What a proportion is
A proportion says two ratios are the same. 3/4 and 9/12 are a proportion because both simplify to 3:4 — the same relationship between the top and the bottom, written with different numbers. 2/3 and 6/9 are a proportion too, because 6/9 simplifies to 2/3. But 2/3 and 3/4 are not a proportion: 2/3 is about 67% and 3/4 is 75%, and no number makes them equal. The test is always the same — simplify both fractions and see whether they match, or cross-multiply and see whether the two products match. If they do, the proportion holds; if they do not, it does not.
Cross-multiplication, in plain words
Cross-multiplication is the one move that solves every proportion. In a/b = x/d, multiply the top of the left fraction by the bottom of the right — a × d — and the bottom of the left by the top of the right — b × x — and the two products are equal, because the two fractions are equal. Set them equal and divide to find x: x = a × d ÷ b. Try it on 2/3 = x/9: cross-multiply to get 2 × 9 = 18 and 3 × x = 3x, set them equal, and divide 18 by 3 to get x = 6. Check it: 6 ÷ 9 is 0.67, and 2 ÷ 3 is 0.67 — the same, so the proportion holds. The move works because multiplying both sides of an equation by the same number keeps it equal — cross-multiplication just does the two multiplications at once.
Ratios: the same proportion in lowest terms
A ratio is a proportion in lowest terms, and simplifying it is the fastest way to see whether two fractions are a proportion. 3:4 is already in lowest terms, because 3 and 4 share no divisor but 1. 6:8 simplifies to 3:4, because 6 and 8 share 2 — so 6/8 and 9/12 are both 3/4 in different clothes, and the missing term is 9 in both. 9:12 simplifies to 3:4 as well, because 9 and 12 share 3. To simplify a ratio, divide both sides by their greatest common divisor: 10:15 shares 5, so it becomes 2:3, and 12:18 shares 6, so it becomes 2:3 too. The ratio simplifier does it for any ratio.
Proportions as percentages
A proportion and a percent are the same idea with a different denominator: a percent is a proportion out of a hundred. 3/4 is 75%, because 3 ÷ 4 = 0.75 and 0.75 × 100 = 75. 4/5 is 80%, because 4 ÷ 5 = 0.8 and 0.8 × 100 = 80. And 3/5 is 60%, because 3 ÷ 5 = 0.6. To turn a proportion into a percent, solve it for x with the fourth term set to 100: 3/4 = x/100 gives x = 75, because 3 × 100 = 300 and 300 ÷ 4 = 75. The other way round works too: 80% is 80/100, which simplifies to 4/5. The fraction to percentage converter does it for any fraction.
Scaling recipes and other everyday uses
Proportions are everywhere once you look for them. A recipe that calls for 3 cups of flour to 4 cups of milk scales to 12 cups of milk with 9 cups of flour — 3/4 = x/12 gives x = 9. A map with a scale of 1 inch to 5 miles shows a town 35 miles away as 7 inches — 1/5 = x/35 gives x = 7. A shop selling 5 apples for $2 charges $8 for 20 apples — 2/5 = x/20 gives x = 8. A photo 4 inches wide and 3 inches tall, enlarged to 12 inches wide, is 9 inches tall — 3/4 = x/12 gives x = 9. The pattern is always the same: write the two ratios, set them equal, cross-multiply and divide. The proportion calculator does it for any three numbers you know.
Common mistakes
The first mistake is dividing the wrong way round — x is a × d ÷ b, not a × b ÷ d, so in 3/4 = x/12 the answer is 3 × 12 ÷ 4 = 9, not 3 × 4 ÷ 12 = 1. The second is forgetting to check — divide both fractions and compare, because 9 ÷ 12 and 3 ÷ 4 are both 0.75, and if they differ the proportion is wrong. The third is treating a ratio as a fraction of the whole — 3:4 means 3 parts to 4 parts, not 3 parts out of 4, even though 3/4 happens to be the same number here. The fourth is simplifying only one side — a proportion needs both fractions in lowest terms before you compare them, or cross-multiplication will hide the mismatch. Work it out in the proportion calculator, and check the fraction calculator and the percentage calculator before you decide what the answer means.