Percent Math

Percent of multiplies: X% of a value is the value × (X ÷ 100). That's the whole formula. Percent change compares two numbers against the original: (new − old) ÷ old × 100. The original value is always the yardstick, which is why the answer comes out negative for a decrease — and why the calculation is undefined when the original is zero. There is no percentage of nothing to measure change against, so this page says "undefined" rather than inventing a number.

Worked examples

12% of $260: 260 × 0.12 = $31.20. That's the tip-and-tax family of problems — a percentage applied to a known amount.

A bill goes from $48 to $60: (60 − 48) ÷ 48 × 100 = a 25% increase. Note the direction matters: $60 falling back to $48 is a 20% decrease, because now $60 is the yardstick.

Common questions

Percent vs. percentage points? If a savings rate moves from 4% to 5%, it rose one percentage point — but that's a 25% increase in the rate itself. Headlines mix these up constantly; when a change in a rate sounds dramatic, check which one is being quoted.

Why does a 50% drop need a 100% gain to recover? Because the yardstick shrinks with the loss. $100 falling 50% leaves $50 — and getting from $50 back to $100 means doubling, which is a 100% gain measured from the new, smaller base. Losses and the gains that undo them are never the same percentage.

How do I reverse a percentage? "X is 30% of what?" — divide instead of multiplying: X ÷ 0.30. If a $45 payment is 30% of some total, the total is 45 ÷ 0.30 = $150.

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