Go from 120 back down to 100 and most people assume that's a 20% drop, since going the other way — 100 up to 120 — was a 20% rise. It isn't. It's 16.7%. That gap between what feels symmetric and what's actually true is where the majority of real-world percentage mistakes happen, and it comes down to one rule almost nobody states explicitly: the base you're measuring against changes depending on which direction you're going.
The three questions hiding behind "percentage"
Nearly every percentage problem people actually search for is one of three distinct questions: what's X% of a number, what percent is one number of another, or what's the percentage change between two numbers. Each has its own formula. Reaching for the wrong one is the most common source of errors, more so than any arithmetic slip.
Finding X% of a number
The most common case by far — "what's 20% of 150?" Convert the percentage to a decimal (divide by 100), then multiply: 150 × 0.20 = 30. Same logic at any scale: a 15% tip on a $60 bill is 60 × 0.15 = $9.
Finding what percentage one number is of another
This flips the question around — "36 is what percent of 240?" Divide the part by the whole, multiply by 100: (36 ÷ 240) × 100 = 15%. This is the exact formula behind "what share of my income goes to rent" or "what percentage of students passed."
Percentage change, done correctly
(New value minus old value) divided by old value, times 100 — and the old value is always the denominator, never the new one, never whichever number happens to be bigger. 100 to 120: (120−100)÷100×100 = a genuine 20% increase. Reverse it, 120 back to 100: (100−120)÷120×100 = −16.7%, not −20%. The percentage needed to undo an increase is always smaller than the increase itself, precisely because the base shifted underneath the calculation.
Percentage points are not percent change
News reporting mangles this distinction constantly. An interest rate moving from 4% to 5% is a 1 percentage-point increase — and, separately, a genuine 25% relative increase, since that single point represents a quarter of the original 4%. Both descriptions are technically correct. They tell very different stories, and which one gets used in a headline is rarely an accident.
Reversing sales tax
An $80 item with 7.5% sales tax: tax owed is 80 × 0.075 = $6, total $86. Working backward from just the total — say you only know the $86 final price — divide by 1 plus the tax rate: 86 ÷ 1.075 = $80 exactly. The same divide-by-(1+rate) move works for stripping out VAT, undoing a markup, or reversing any percentage that was layered on top of a base price. The VAT Calculator walks through exactly this in more depth.
Run any of these calculations instantly with the Percentage Calculator, which handles both "X% of Y" and percentage change in one tool.
A few things people actually ask
If something drops 20%, does raising it 20% get back to the original?
No. A 20% increase followed by a 20% decrease (or vice versa) never returns to the starting value, because each percentage is calculated against a different base — the increase uses the original number, the decrease uses the already-changed number.
What's the difference between a percentage point and a percent change?
A percentage point measures the raw arithmetic difference between two percentages (4% to 5% is 1 point), while percent change measures the relative size of that difference against the original (that same 1 point is a 25% relative increase). Both are correct; they answer different questions.
How do I find the original price before tax was added?
Divide the total by 1 plus the tax rate as a decimal — an $86 total at 7.5% tax divides by 1.075 to give the exact $80 pre-tax price. The same technique reverses VAT or any percentage markup.