Down 50%, Up 50%, Down 25%
A stock falls 50%. The next day it rises 50%.
You're not back where you started. You're down 25%.
€100 → €50 → €75.
This isn't a trick. It's the most consequential property of percentages and the one most consistently misapplied: percentage changes don't add, and they don't cancel.
Why the Base Changes
A percentage is always a percentage of something. The moment a value changes, the base for the next percentage changes with it.
- The 50% fall was 50% of €100 = €50
- The 50% rise was 50% of €50 = €25
Same percentage, different base, different amount. The rise operates on a smaller number, so it recovers less than the fall took away.
This gives a general and slightly unintuitive rule: for any sequence of a gain and a loss of the same percentage, in either order, you end up below where you started. The order doesn't matter — multiplication is commutative, so 0.5 × 1.5 and 1.5 × 0.5 both give 0.75.
The Recovery Table
The percentage gain needed to recover from a loss grows non-linearly:
| Loss | Gain needed to recover |
|---|---|
| 10% | 11.1% |
| 20% | 25% |
| 25% | 33.3% |
| 33% | 50% |
| 50% | 100% |
| 60% | 150% |
| 75% | 300% |
| 90% | 900% |
The formula:
gain needed = loss / (1 − loss)
Where loss is expressed as a decimal. A 40% loss needs 0.4 / 0.6 = 66.7%.
The asymmetry accelerates. Losses below 20% are recoverable with comparable gains. Beyond 50%, the required recovery becomes very large. This is the arithmetic behind risk management advice about limiting drawdowns — not because large losses are psychologically difficult, though they are, but because the recovery mathematics get steep.
Chaining Percentage Changes
To combine sequential percentage changes, multiply the growth factors:
combined factor = (1 + r₁) × (1 + r₂) × (1 + r₃) × ...
Three consecutive years of +10%, −5%, +8%:
1.10 × 0.95 × 1.08 = 1.1286
A total change of +12.86%. Adding the percentages gives +13%, which happens to be close here — but only because the values are small. The approximation degrades rapidly with larger changes:
+50%, then −50%: add → 0%. Actual → −25%.
+100%, then −50%: add → +50%. Actual → 0%.
Never add percentage changes. Multiply the factors.
The Percentage Calculator handles individual calculations:
- Select the type of problem.
- Enter your values.
- Read the result.
For chains, calculate each step and carry the running value forward rather than trying to combine the percentages.
Percent vs Percentage Points
A distinction that changes the meaning of a claim entirely.
An interest rate moves from 4% to 6%.
- That's an increase of 2 percentage points
- That's an increase of 50 percent
Both statements are correct. They describe the same change and sound very different.
The convention: percentage points measure the arithmetic difference between two percentages. Percent measures the relative change.
Where this matters:
Interest rates. "Rates rose by 5%" is ambiguous. From 4%, five percentage points is 9% and five percent is 4.2%.
Election polling. A party moving from 20% to 24% gained four percentage points, or 20 percent of its previous share.
Medical statistics. A risk moving from 2% to 3% is a one percentage point increase and a 50 percent relative increase. Reporting the second without the first makes a small absolute change sound dramatic. This is why absolute risk should accompany relative risk in any health reporting.
Market share. Same issue. Going from 5% to 6% is "20% growth" in one framing and "one point" in another.
When you see a percentage change applied to something that is itself a percentage, ask which is meant. If the source doesn't say, the ambiguity is theirs and the figure is unreliable.
Discounts That Stack
Two successive discounts don't add.
A €200 item with 20% off, then a further 10% off at checkout:
200 × 0.80 = 160
160 × 0.90 = 144
Total discount: €56, or 28% — not 30%.
The general form for stacked discounts:
combined = 1 − (1 − d₁)(1 − d₂)
For 20% and 10%: 1 − (0.8 × 0.9) = 1 − 0.72 = 0.28.
Order doesn't matter for the final price. 10% then 20% gives the same €144.
Order does matter when tax is involved and applied at a specific point. Whether a discount is applied before or after VAT changes both the final price and the tax owed, which is why the sequence is specified in retail systems rather than left to chance.
Markup and Margin
The most commercially expensive percentage confusion.
Markup is calculated on cost. Margin is calculated on selling price.
An item costs €60 and sells for €100:
Markup = (100 − 60) / 60 = 66.7%
Margin = (100 − 60) / 100 = 40%
Same transaction, two very different numbers.
Apply a 40% markup when you meant 40% margin and you get:
60 × 1.40 = €84
Actual margin: (84 − 60) / 84 = 28.6%
You've lost more than a quarter of your intended margin on every unit.
Conversion formulas:
markup = margin / (1 − margin)
margin = markup / (1 + markup)
A 40% margin requires a 66.7% markup. A 50% margin requires a 100% markup.
The confusion is common enough that it's worth stating pricing intentions explicitly — "40% gross margin" rather than "40% profit," which could mean either.
VAT and Reverse Calculations
Adding VAT is straightforward. Removing it is where errors happen.
Add 20% VAT: net × 1.20 = gross
Remove 20% VAT: gross / 1.20 = net
The mistake is subtracting 20% from the gross figure:
Wrong: 120 − (120 × 0.20) = €96
Right: 120 / 1.20 = €100
The €4 error occurs because the 20% was calculated on the net amount, not the gross. To reverse a percentage increase you divide; you don't subtract the same percentage.
Same principle applies to any reverse percentage calculation — working back from a final price to an original, or from a gross to a net figure.
Practical Tips
Multiply growth factors, never add percentages.
Say "percentage points" when you mean the arithmetic difference between two percentages. It removes an entire class of ambiguity.
Ask "percentage of what?" whenever a percentage appears without a stated base.
Convert markup to margin explicitly rather than assuming which one a figure refers to.
Divide to reverse a percentage increase. Don't subtract.
Check the recovery table when reasoning about losses. The asymmetry is larger than intuition suggests.
Be suspicious of relative changes without absolute figures. A 50% increase in something rare is still rare.
FAQ
Why doesn't a 50% loss cancel a 50% gain? Because each percentage is applied to a different base. The gain is calculated on the reduced amount, so it recovers less than the loss removed.
How do I combine two percentage changes? Multiply the growth factors: (1 + r₁) × (1 + r₂). Adding the percentages is only a rough approximation and fails badly for large changes.
What's the difference between percent and percentage points? Percentage points are the arithmetic difference between two percentages. Percent is the relative change between them.
Do stacked discounts add up? No. 20% then 10% gives 28% total, not 30%. Multiply the remaining fractions.
What's the difference between markup and margin? Markup is calculated on cost; margin on selling price. A 40% margin requires a 66.7% markup.
How do I remove VAT from a gross price? Divide by (1 + rate). For 20% VAT, divide by 1.20. Subtracting 20% gives the wrong answer.
The Takeaway
Almost every percentage error comes from losing track of the base. Percentages don't add, don't cancel, and don't reverse by subtraction — they multiply, and each one is measured against whatever the value was at that moment. Keeping the base explicit is what makes the rest of it straightforward.
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