The three percentage formulas
- Percent of a number: (percentage × value) ÷ 100. Example: 20% of 150 = 30.
- Percent change: ((new − original) ÷ original) × 100. Example: 80 to 100 is a 25% increase.
- Reverse percentage: finds the original value before a percentage was applied. Example: if 80 is the price after a 20% discount, the original price was 100.
Percentage vs. percentage point
These get mixed up constantly. If interest rates rise from 5% to 7%, that's a 2 percentage point increase, but a 40% percentage increase (2 ÷ 5 × 100). Both numbers are correct — they're just answering different questions. Percentage point differences compare the raw numbers directly; percentage change compares the size of the move relative to the starting value.
Common real-world uses
Discounts and sales tax use the "percent of a number" formula in opposite directions — a discount subtracts a percentage, tax adds one. Tips are the same calculation as a discount, just added rather than subtracted. Tracking growth (revenue, followers, weight, anything measured over time) uses percent change, and working out what something cost before a markup or before tax was added uses the reverse percentage formula.
Two mistakes worth watching for
First: "what percentage is 40 of 200" and "what percentage is 200 of 40" are completely different questions (20% vs 500%) — always be clear which number is the part and which is the whole before you calculate. Second: a 25% increase followed by a 25% decrease does not return you to the original number, because each percentage is calculated against a different base. 100 increased by 25% is 125; 125 decreased by 25% is 93.75, not 100. Percentages don't cancel out just because the numbers look symmetrical.
Percentages over multiple periods don't add up the simple way
If something grows 10% one year and 10% the next, it's tempting to add the two and call it a 20% total gain — but the real total is 21%, because the second 10% is calculated on the already-grown value, not the original one. The gap is even more important on the downside: a 50% loss followed by a 50% gain does not bring you back to even. Losing half your starting value and then gaining half of the new, smaller value leaves you at a 25% loss overall, not break-even. This is the same underlying issue as the 25%-up-25%-down example above, just compounded over more than two steps.
A few reference points to check your own numbers against
These are here as a sanity check, not a substitute for the calculator above — if your own result on a similar-sized number looks wildly different from the matching reference here, it's worth double-checking which formula (percent of, percent change, or reverse percentage) actually matches the question you're asking.