Math · 4 min read
Four Percentage Calculations People Get Wrong
Percentages are taught at primary school and misapplied in annual reports. These four errors account for most of it.
Every one of these errors comes from the same root cause: a percentage is meaningless without knowing what it is a percentage of. Change the base and the same percentage describes a different quantity.
1. Losses and gains are not symmetric
An investment falls 50%. How much must it rise to break even?
Not 50%. £100 falls to £50. A 50% gain on £50 is £75. You need a 100% gain to get back to £100.
The recovery required grows sharply with the size of the loss:
| Loss | Gain needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25.0% |
| 33% | 50.0% |
| 50% | 100.0% |
| 75% | 300.0% |
| 90% | 900.0% |
The practical implication is that avoiding large drawdowns matters disproportionately. A portfolio that never falls more than 20% needs only ordinary returns to recover; one that falls 75% needs to quadruple.
The same asymmetry applies to a sequence of changes. Up 10% then down 10% does not return you to the start: 100 → 110 → 99. You lose 1% every time, regardless of order, because 1.1 × 0.9 = 0.99.
2. Percentage points are a different unit
An interest rate moves from 4% to 5%.
- It rose by 1 percentage point.
- It rose by 25 percent.
Both statements are correct. They are not interchangeable, and the choice between them changes how large the change sounds by a factor of 25.
This is exploited routinely. A tax rate rising from 20% to 22% can be reported as "a 2 point increase" or "a 10% increase in tax". A vaccine trial reporting risk falling from 2% to 1% can say "50% relative risk reduction" or "1 percentage point absolute reduction". The first is standard in press releases, the second is what tells you how many people are affected.
When you see a percentage change applied to something that is itself a percentage, check which one is meant. If the source does not say, assume it chose the more impressive framing.
3. Discounts do not add
A £200 jacket, 20% off, then an extra 10% at the till. That is not 30% off.
- £200 − 20% = £160
- £160 − 10% = £144
£144 is 28% off, not 30%. The second discount applies to the reduced price.
The general rule for stacked reductions is to multiply the remaining fractions:
So 20% and 10% give 0.8 × 0.9 = 0.72, meaning 28% off. Three stacked 10% discounts give 0.9³ = 0.729, or 27.1% off, not 30%.
Order does not matter, since multiplication commutes. What does matter is whether a discount is stacked or additive; "30% off, plus a further 10% off the original price" is a different offer from "30% off, then 10% off".
Tax works the same way in reverse. Adding 20% VAT then removing a 20% discount does not return the original price.
4. Working backwards from a total
An item costs £120 including 20% VAT. What is the price before tax?
It is not £96. Taking 20% off £120 removes 20% of £120, but the VAT was 20% of the pre-tax price, which is smaller.
£120 ÷ 1.2 = £100. Check: £100 + 20% = £120. Correct.
The £96 answer is wrong by £4, and the error grows with the rate. At 25% the naive method is out by 6.25% of the pre-tax price.
Same structure for reversing a discount. If a sale price of £84 is after 30% off, the original was 84 ÷ 0.7 = £120, not £109.20.
A bonus: averaging percentages
You cannot usually average percentages directly, because they may have different bases.
A shop converts 10% of 1,000 visitors on Monday and 20% of 100 visitors on Tuesday. The average conversion rate is not 15%.
- Monday: 100 conversions from 1,000
- Tuesday: 20 conversions from 100
- Total: 120 from 1,100 = 10.9%
The correct approach is to recombine the underlying counts. Averaging the rates treats a day with 100 visitors as equal in weight to one with 1,000.
This is the mechanism behind Simpson’s paradox, where a treatment can appear better in every subgroup and worse overall, purely because of how group sizes are distributed. If you ever see aggregate and subgroup results pointing in opposite directions, unequal bases are usually why.
Three habits that catch most of it
- Name the base. Say "20% of the pre-tax price", not "20%". Most of these errors cannot survive being stated precisely.
- Reverse the calculation. If you removed 20% VAT to get £100, add it back and confirm you land on £120.
- Estimate first. Two stacked discounts must be less than their sum. A recovery from a large loss must be a large gain. Knowing the direction of the answer catches most slips.
Common questions
Why is a 50% loss not cancelled by a 50% gain?
Because the two percentages are taken from different bases. £100 falling 50% gives £50. A 50% gain on £50 is £25, taking you to £75, not £100. The loss was measured against 100 and the gain against 50. To recover you need a 100% gain on the reduced figure.
What is the difference between percent and percentage points?
If a rate rises from 4% to 5%, that is a rise of one percentage point and a rise of 25 percent. Both are true and they describe the same change. Reporting the larger number without saying which unit it is in is one of the more common ways statistics get misrepresented.
Does the order of two discounts matter?
No: multiplication is commutative, so 20% then 10% gives the same result as 10% then 20%. What does matter is that stacked discounts never sum: 20% and 10% together give 28% off, not 30%, because the second is applied to the already-reduced price.