The basic percentage-change formula
Percentage change always divides by the ORIGINAL (starting) value, never the new one: take the new value minus the original, divide by the original, and multiply by 100. This single formula covers increases (a positive result), decreases (a negative result), and — because the original is the denominator — it means the same-sized move reads as a different percentage depending on which direction you're going.
For example, 100 rising to 125 is a 25% increase. But 125 falling back to 100 is not a 25% decrease — it's a 20% decrease, because the denominator has changed from 100 to 125.
The reverse-percentage trap
This is the mechanism behind the classic trap: a value that increases by X% and then decreases by that same X% does not return to where it started — and the same is true in the other order. Take 100, increase it by 25% to 125, then decrease that 125 by 25%. The second move applies to 125, not the original 100.
125 minus 25% of 125 is 125 − 31.25 = 93.75, not 100. The two 25% moves aren't equal in absolute terms, because each one is calculated against a different base — the first against 100, the second against the larger value of 125.
Worked example
Why the base keeps shifting
Every percentage change is only meaningful relative to whatever value it's calculated against at that moment. Once a value has grown, a percentage decrease of the same size removes a larger absolute amount than the increase added, because it's a percentage of a bigger number. Symmetrically, once a value has shrunk, a percentage increase of the same size adds back a smaller absolute amount than the decrease removed.
This is also why a series of percentage changes — three consecutive months of +10%, −5%, +8%, for instance — can't be added together (+13%) to get the total. Each period's percentage applies to that period's own starting value, so the correct total has to multiply the successive multipliers, not sum the percentages.
How much gain it actually takes to undo a loss
Because the base shrinks after a loss, recovering it always needs a LARGER percentage gain than the loss itself, and the gap widens fast as the loss grows. After any loss, the remaining value is 100 minus the loss percentage; the gain required to get back to 100 is the loss divided by what's left, not divided by the original 100.
A 20% loss leaves 80, and needs a 25% gain to get back to 100 (20 ÷ 80 × 100). A 50% loss leaves only 50, and needs a full 100% gain — the value must double. A 90% loss leaves just 10, and needs a 900% gain to recover — the value must grow to ten times what was left.
Worked example
Total change vs annualised change (CAGR)
A related mix-up shows up over multiple years: the total percentage change across a whole period is not the same number as the annual growth rate, because growth compounds. Compound Annual Growth Rate (CAGR) answers "what constant yearly rate, compounded, produces this same total result?" — a different question from "what was the total change?"
A value that goes from 100 to 200 over 5 years has a 100% total change, but that's not 20% a year. Compounding at 20% a year for 5 years would more than double the value (1.2⁵ ≈ 2.49×). The rate that actually compounds to exactly double over 5 years is about 14.87% a year.
Worked example
A related mix-up: percentage points vs percent
One more pair of numbers that looks interchangeable but isn't: percentage POINTS is the plain difference between two percentages, while relative percentage CHANGE divides that difference by the starting rate. A rate that moves from 3% to 5% has changed by 2 percentage points — but relative to the 3% starting rate, that's a 66.67% relative increase.
Saying "the rate rose 2%" is genuinely ambiguous — it could mean 2 percentage points (3% to 5%) or a 2% relative increase (3% to 3.06%), and those are very different outcomes. Saying "2 percentage points" or "a 66.67% relative increase" removes the ambiguity.
Worked example
Common mistakes
- Assuming a percentage decrease and an equal percentage increase cancel out. They don't — whichever move happens second is calculated against a different (already-changed) base, so the two moves aren't equal in absolute terms.
- Applying the same percentage to reverse a change, instead of solving for the correct reversing percentage. Undoing a 25% increase needs a 20% decrease, not another 25% move.
- Adding a sequence of percentage changes together to get the total (e.g., treating +10% then −5% as a net +5%). Each percentage applies to a different base, so the multipliers need to be multiplied together, not the percentages added.
- Confusing percentage points with percent. "The rate rose 2%" is ambiguous between a 2-point move and a 2%-relative move, which can mean very different things, especially for small starting rates.
- Treating total percentage change over several years as if it were the annual rate. Total change and CAGR answer different questions and are usually different numbers whenever there's more than one period involved.
- Underestimating how fast the recovery gain grows for a large loss. The relationship is not linear — the gain needed accelerates well past the loss percentage as the loss gets larger, which is why a severe loss is disproportionately hard to recover from.