Percentages are not symmetric. Up 50% then down 50% leaves you 25% down, and that asymmetry causes more errors than any other arithmetic in ordinary use.
The formula and the trap
Percentage change is (new − old) ÷ old × 100. Straightforward, and the whole difficulty is in what counts as "old".
Rise from 80 to 100: (100 − 80) ÷ 80 = 25% increase. Fall from 100 to 80: (80 − 100) ÷ 100 = 20% decrease. The same absolute change, two different percentages, because the base changed.
This is why a 50% rise followed by a 50% fall does not return you to the start. £100 up 50% is £150; £150 down 50% is £75. To reverse a 50% fall you need a 100% rise. The direction is not symmetric and never was.
Reversing a change
| Change | To reverse it, apply |
|---|---|
| −10% | +11.1% |
| −20% | +25% |
| −25% | +33.3% |
| −50% | +100% |
| −75% | +300% |
| −90% | +900% |
The pattern: to undo a fall of x%, you need a rise of x ÷ (100 − x) × 100. This is why an investment that halves needs to double to recover, and why a 90% drawdown is close to unrecoverable — it is not "90% back", it is nine times the remaining value.
Percent versus percentage points
These get used interchangeably in reporting and they are different quantities.
If an interest rate moves from 4% to 6%, that is a rise of 2 percentage points. It is also a 50% increase in the rate. Both are true and they describe different things.
The ambiguity is exploited routinely. "Fees rise by 1%" sounds trivial; if the fee was 1% and is now 2%, your cost doubled. Whenever a percentage is itself the thing changing, ask which is meant — and when writing, say percentage points if that is what you mean.
Removing a percentage that was added
The most common practical error in this whole subject, and it appears on invoices constantly.
To remove 20% that was added, divide by 1.2 — do not subtract 20%. £120 including 20% is £100 plus £20. Subtracting 20% of £120 gives £96, which is wrong by £4.
The reason is the base again: the 20% was calculated on £100, not on £120. As a proportion of the gross, the tax is 16.67%. The full working, with the shortcuts for each common rate, is in how to calculate VAT backwards.
Sales tax and VAT are not the same shape
Both are consumption taxes and they behave differently in ways that affect the arithmetic.
VAT is charged at every stage of the supply chain, with businesses reclaiming what they paid on inputs. Consumer prices in the UK and EU are shown inclusive, so the displayed price is what you pay.
US sales tax applies only at the final sale, varies by state, county and city, and is normally excluded from the shelf price — so the total at the till is higher than the label.
That last difference is why "how much is this" has a straightforward answer in Europe and does not in the US. It also means a US shelf price and a European shelf price are not comparable figures without knowing the local rate.
Compounding percentages
Successive percentage changes multiply rather than add, which is another place intuition fails.
Three consecutive 10% rises are not 30%. They are 1.1 × 1.1 × 1.1 = 1.331, so 33.1%. Three 10% falls are 0.9³ = 0.729, so a 27.1% decline rather than 30%.
Over longer runs the gap widens considerably, which is the same mechanism that drives compound interest — see compound interest explained. Any time you are adding percentage changes together, check whether they should be multiplied instead.
Frequently asked questions
How do I calculate percentage increase?
(new − old) ÷ old × 100. The base is the original value, which is what makes increases and decreases asymmetric — going 80 to 100 is a 25% rise, but 100 to 80 is a 20% fall, despite being the same absolute change.
Why does a 50% rise then a 50% fall not break even?
Because the second percentage is calculated on a larger base. £100 up 50% is £150; 50% of £150 is £75, not £50. To reverse a 50% fall you need a 100% rise — the general rule is x ÷ (100 − x) × 100.
What is the difference between percent and percentage points?
A move from 4% to 6% is 2 percentage points and also a 50% increase. Both are correct and describe different things. The ambiguity gets exploited — "fees rise by 1%" sounds small, but if the fee was 1% and is now 2%, your cost doubled.
How do I remove a percentage that was added?
Divide, do not subtract. To remove 20%, divide by 1.2. £120 including 20% is £100 net — subtracting 20% gives £96 and is wrong by £4, because the 20% was calculated on £100 rather than on £120.
Do successive percentage changes add up?
No, they multiply. Three 10% rises give 1.1³ = 33.1%, not 30%. Three 10% falls give 0.9³, a 27.1% decline rather than 30%. Any time you find yourself adding percentage changes, check whether they should be multiplied.
Why is US sales tax added at the till?
Because it applies only at the final sale and varies by state, county and city, so a single displayed price cannot cover every jurisdiction. VAT is charged throughout the chain and shown inclusive in the UK and EU, so the label is what you pay.