Percentage Calculator

Because percentages taken from different bases don't add directly — they compound. The second 40% is applied to a smaller number than the first, so the two reductions aren't equivalent. Start with 100 items and remove 40% (40 items), leaving 60. Now remove 40% of the remaining 60, which is 24 items, leaving 36. The total reduction is 64%, not 80%. The shortcut is to multiply the surviving fractions: 0.60 × 0.60 = 0.36, meaning 36% remains and 64% is gone. The same logic applies to stacked discounts and successive growth rates: '40% off, then 40% off' is never 80% off, because each step resets the base the next percentage acts on.

Percentage change has a direction and a fixed baseline; percentage difference has neither. Percentage change compares a new value to an original one using (new − old) ÷ old × 100, so it can be positive (increase) or negative (decrease) and the old value is always the reference. Percentage difference compares any two values symmetrically, dividing the gap by their average: |A − B| ÷ ((A + B) ÷ 2) × 100. For example, the difference between 40 and 60 is 20 ÷ 50 × 100 = 40%, with no implied direction. Use change when one value comes before the other in time — last year's revenue versus this year's — and difference when comparing two things side by side with no natural baseline, such as two competing measurements.

Anchor everything to 10%, which you get by moving the decimal one place left. On a $47.50 bill, 10% is $4.75. From there, 20% is simply double: $9.50. For 15%, take the 10% figure and add half of it: $4.75 + $2.38 = $7.13. For 18%, start from 20% ($9.50) and trim a little, since 18% is slightly less. The trick works on any amount because percentages scale linearly — once you have 10%, every common tip is a quick multiple or combination. Rounding the bill first ($47.50 up to $48) makes the mental math cleaner, and rounding the final tip up is an easy way to be generous without recalculating.

Because the gain and the loss are calculated on different dollar amounts, so they don't cancel out. Starting at $100, a 50% gain adds $50, taking you to $150. The following 50% loss is then applied to $150, removing $75 and leaving you with $75 — a net 25% loss, not break-even. The second percentage acts on a larger base, so it subtracts more than the first one added. This asymmetry is why recovering from a loss always takes a bigger percentage gain than the loss itself: a 50% loss requires a 100% gain to get back to even. In general, recovering from an X% loss needs X ÷ (100 − X) × 100% growth.

A percentage point is the arithmetic gap between two percentages, while a percent change describes that gap relative to the starting value. If an interest rate rises from 5% to 7%, that is a 2 percentage-point increase — you simply subtract. But measured as a percent change, it is a 40% increase, because the 2-point rise is 2/5 = 40% of the original 5%. The two figures describe the same move but answer different questions, and conflating them can make a change sound far larger or smaller than it is. News reports often say 'points' to avoid this ambiguity, but not always, so check whether a figure is an absolute point difference or a relative percent change before drawing conclusions.