Percentage Calculator

Three questions, one page. Enter two numbers and get all the percentage relationships between them.

The three percentage questions

Almost every percentage problem is one of three forms, and confusing them is where errors come from.

The word “of” always means multiply. “25 % of 180” is 0.25 × 180 = 45. The word “is” marks the equals sign: “45 is what percent of 180” becomes 45 ÷ 180 × 100 = 25 %.

Mental shortcuts worth knowing

• Percentages are commutative. X % of Y equals Y % of X. 4 % of 75 is awkward; 75 % of 4 is obviously 3. This single fact solves a surprising number of problems.
• Build from 10 %. Move the decimal point one place left. 10 % of 340 is 34, so 30 % is 102 and 5 % is 17.
• 1 % is two places left. 1 % of 340 is 3.40, so 3 % is 10.20.
• Combine. 17.5 % = 10 % + 5 % + 2.5 %, each derived by halving the one before.

Where people go wrong

Percentage points versus percent. A rate rising from 4 % to 6 % has risen by two percentage points, or by 50 %. Both are correct and they describe the same event; using the wrong one is how statistics get misreported.

Increases and decreases do not cancel. Add 20 % to 100 and you get 120. Take 20 % off 120 and you get 96, not 100 — because the second percentage was applied to a larger base. To reverse a 20 % increase you must divide by 1.2, not subtract 20 %.

Averaging percentages. A 50 % success rate on 10 attempts and 90 % on 100 attempts is not a 70 % average. It is (5 + 90) ÷ 110 = 86.4 %. Percentages must be weighted by the size of their base.