Percentage Calculator

Almost every percentage problem is one of three shapes, and naming the shape is most of the work. “What is X% of Y?” gives you a part when you know the rate and the total — a tip, a commission, a discount amount, a tax charge. “X is what percent of Y?” gives you a rate when you know a part and a total — a test score, a market share, a conversion rate. “X is Y% of what number?” gives you the total when you know a part and its rate — recovering a pre-tax price, or working out a target from the fraction you have already reached.

Getting the wrong one is not a rounding problem, it produces a completely different number. If 36 is 15% of 240, then asking “36 is what percent of 240?” gives 15%, but asking “what is 36% of 240?” gives 86.4. Same two figures, three legitimate answers depending on the question.

A percentage is a fraction whose denominator is always 100 — the word comes from the Latin per centum, “by the hundred”. So 15% is exactly 15/100, which is exactly 0.15. Every percentage calculation reduces to converting to that decimal and then doing ordinary multiplication or division.

Understanding this makes several awkward cases obvious. Percentages above 100 are fine: 150% of 80 is 1.5 × 80 = 120, which is simply more than the original. Percentages below 1 are fine too: 0.5% of 200 is 0.005 × 200 = 1. And a percentage of a percentage is ordinary multiplication — 50% of 20% is 0.5 × 0.2 = 0.1, which is 10%.

Percentages are commutative in a way most people never notice: X% of Y always equals Y% of X. That turns awkward sums into easy ones. 4% of 75 looks unpleasant; 75% of 4 is obviously 3. 18% of 50 is the same as 50% of 18, which is 9.

The other reliable technique is decomposition. 10% is a decimal-point shift, and everything else is built from it: 5% is half of 10%, 1% is a tenth of it, 20% is double. So 17% of 340 becomes 34 (10%) + 17 (5%) + 6.8 (2%) = 57.8. That is enough precision to catch a calculator typo, which is really what mental arithmetic is for.

In retail, percentages drive discounts, markups and sales tax, and the order of operations matters — a 20% discount applied before tax and after tax give different final prices. In finance they express interest rates, returns and allocations, usually annualised, which is why a "5% return" always needs a period attached to be meaningful.

In statistics and reporting they compress large numbers into comparable ones, which is exactly where they mislead: a percentage without its base ("sales up 300%") hides whether the underlying number moved from 1 to 4 or from a million to four million. Whenever you publish a percentage, publish the denominator with it.