Percentages, and the four traps in them
Why a 20% fall needs a 25% rise to recover, how percentage points differ from percent, and why reversing a discount is a division.
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In short
- A 20% fall needs a 25% rise to get back. Percentage changes are not symmetric, because each applies to a different base.
- A rate moving from 4% to 5% is one percentage point and twenty-five percent. Both are correct and they mean very different things.
- Reversing a discount is a division. ₹800 after 20% off was ₹1,000, not ₹960.
- Stacked discounts multiply. 30% then a further 20% is 44%, not 50%.
- Averaging percentages without weighting them is wrong whenever the groups differ in size.
A percentage is a fraction with a fixed denominator of a hundred. That is the whole idea, and almost every mistake made with them comes from losing track of what the hundred refers to.
The five calculations
Nearly every percentage question is one of these:
What is A% of B? Divide the percentage by a hundred and multiply. 15% of 2,500 is 0.15 × 2,500 = 375.
A is what percent of B? Divide and multiply by a hundred. 45 out of 250 is 45 ÷ 250 × 100 = 18%.
Percentage change from A to B. The difference divided by the starting value. From 4,500 to 5,850 is 1,350 ÷ 4,500 × 100 = 30%.
Add or subtract a percentage. 2,000 plus 12% is 2,000 × 1.12 = 2,240; minus 12% is 2,000 × 0.88 = 1,760.
Reverse a percentage. Given the result, find the original. This is a division, and it is the one people get wrong.
Trap one: changes are not symmetric
A 20% fall followed by a 20% rise does not return you to where you started.
Start at 100. Fall 20% and you are at 80. Rise 20% of 80 and you gain 16, reaching 96. You are 4% down after two supposedly equal moves.
The two percentages apply to different bases: the fall to 100, the rise to 80. To recover a fall you need a larger percentage than the one that caused it:
| Fall | Rise needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 33% | 50% |
| 50% | 100% |
| 80% | 400% |
| 90% | 900% |
This is why a portfolio that halves needs to double to break even, and why avoiding large losses matters more than capturing large gains.
Trap two: percent against percentage points
If an interest rate moves from 4% to 5%, has it risen by one percent or twenty-five?
Both statements are correct and they measure different things:
- One percentage point — the arithmetic difference between the two rates
- Twenty-five percent — the relative change, because 1 is a quarter of 4
The distinction matters most where the numbers are already percentages: interest rates, unemployment, market share, tax rates. A headline saying unemployment "rose 25%" describes the same event as one saying it "rose by one point", and the first sounds four times more alarming.
Use percentage points for a change in a percentage, and percent for a change relative to the starting value. If you are unsure which your reader will assume, say both.
Trap three: reversing is a division
An item cost ₹800 after a 20% discount. What was the original price?
The instinct is to add 20% back: 800 × 1.2 = 960. That is wrong.
The ₹800 is 80% of the original, so:
original = 800 ÷ 0.8 = 1,000
Check it: 20% of 1,000 is 200, and 1,000 − 200 = 800. Correct. Adding 20% to 800 gives 960, and 20% off 960 is 768, not 800.
The same applies to removing tax from an inclusive price, which is the identical problem: divide by 1 + rate, never subtract the rate.
Add a percentage: multiply. Reverse it: divide.
Trap four: stacked percentages multiply
"30% off, then an extra 20% at the till" is not 50% off.
The second discount applies to the already-reduced price:
1,000 → 30% off → 700 → 20% off 700 → 560
That is 44% off the original, not 50%. The general rule is to multiply the remaining fractions: 0.70 × 0.80 = 0.56, so 44% comes off.
| Stacked offer | Sounds like | Actually |
|---|---|---|
| 20% + 10% | 30% | 28% |
| 30% + 20% | 50% | 44% |
| 50% + 20% | 70% | 60% |
| 50% + 50% | 100% | 75% |
The last row is the clearest illustration: two 50% discounts never make anything free.
The same multiplication applies to compounding in the other direction. Three consecutive 10% rises are not 30% but 1.1³ = 1.331, a 33.1% increase.
Averaging percentages
Averaging percentages directly is wrong whenever the groups behind them differ in size.
A shop sells 10 items on Monday with a 90% success rate, and 1,000 items on Tuesday with a 50% rate. The simple average is 70%. The true rate is:
(9 + 500) ÷ 1,010 = 50.4%
Nowhere near 70%. The correct method is to weight each percentage by the size of its group, or better, to go back to the underlying counts and recompute.
This error is common in reporting on regional or departmental figures, where a small unit with an extreme percentage can swing a naive average dramatically.
Percentage of what?
Two apparently equivalent phrasings mean different things:
- "A is 25% more than B" — A = B × 1.25
- "B is 25% less than A" — B = A × 0.75
These are not the same relationship. If A is 25% more than B, then B is 20% less than A, not 25%. Whenever a comparison matters, name the base explicitly: "25% more than last year" rather than "25% more".
The same ambiguity appears in "increased by a factor of three" (three times as large) against "increased threefold" (usually the same, but read as an increase of three times by some), and "300% more" (four times as large) against "300% of" (three times as large). Plain numbers are safer than percentages when the multiple is large.
Percentage change over several periods
An average percentage change across periods is not the simple mean of the changes, for the same reason that stacked discounts multiply.
An investment that gains 50% one year and loses 50% the next has an arithmetic mean change of zero and is down 25%. Start at 100, rise to 150, fall by half to 75.
The correct figure is the geometric mean: multiply the growth factors and take
the nth root. Here 1.5 × 0.5 = 0.75, and the square root of 0.75 is 0.866 — an
average of −13.4% a year, which compounds to the −25% actually observed.
This is why compound annual growth rate exists, and why any average return quoted as an arithmetic mean overstates what an investor actually received. The gap widens with volatility: two portfolios with the same arithmetic mean return but different swings end at different values, and the steadier one ends higher.
When a percentage hides the number
A percentage without its base can conceal almost anything. "Sales up 200%" is impressive until the base is three units. "A 50% reduction in risk" means nothing without knowing whether the risk went from 2% to 1% or from 80% to 40% — the relative reduction is identical and the practical significance is not.
This is the difference between relative and absolute risk, and it is the most common way statistics mislead without stating anything false. A treatment that halves a one-in-ten-thousand risk has a 50% relative reduction and a 0.005% absolute one. Both are true. Only one is worth acting on.
Whenever a percentage is doing persuasive work, ask for the two numbers it came from.
A useful shortcut
X% of Y always equals Y% of X, because both are X × Y ÷ 100.
This makes awkward calculations easy. 4% of 75 is difficult; 75% of 4 is obviously 3. 18% of 50 is 50% of 18, which is 9. The two are the same computation with the operands swapped, and one of the two is usually far easier to do in your head.
Marks percentage across subjects
A marksheet percentage is the total marks obtained divided by the total maximum marks, multiplied by 100. It is one division over everything, not an average of the subject percentages.
| Subject | Marks | Out of | Percentage |
|---|---|---|---|
| Maths | 86 | 100 | 86% |
| Science | 78 | 100 | 78% |
| English | 91 | 100 | 91% |
| Hindi | 74 | 100 | 74% |
| Social Science | 83 | 100 | 83% |
412 out of 500 is 82.4%.
The two methods part company as soon as papers carry different maximums. Scoring 45 out of 50 in one paper and 60 out of 100 in another gives subject percentages of 90% and 60%, which average to 75%. The marks actually earned are 105 out of 150, which is 70%. The larger paper carries more of the result, and only the pooled figure reflects that.
Which papers count is a rule, not arithmetic. Some admissions use the best five subjects, some add practical marks in and some report them separately, so enter the subjects your board or college actually counts. Where a marksheet prints a percentage, that printed figure is the official one: grace marks and moderation are applied before it is issued, and a total worked out from answer books can differ from it.
What this calculator assumes
- Percentages are given as numbers rather than fractions, so 15 means 15%.
- Percentage change is measured against the first value, using its absolute value so a negative starting point still gives a sensible sign.
- Reversing a percentage divides by
1 ± rate, so a 100% discount has no recoverable original and is reported as such. - A whole of zero is undefined rather than infinite, and is reported as zero.