Why a 50% Loss Needs a 100% Gain: Percentage Recovery Explained
Why a 20% fall needs a 25% rise to recover, the formula for the gain needed after any loss, a recovery chart, and how to check it with a percentage calculator.
The short answer
A percentage loss and the gain needed to undo it are not the same size, because each is measured against a different starting number. If ₹1,000 falls 20% to ₹800, it needs a 25% rise to get back to ₹1,000. A 50% fall needs a 100% rise. The formula is gain needed = loss ÷ (100 − loss) × 100. You can check any pair of numbers with the "% change from X to Y" mode of the Percentage Calculator.
Why the numbers are different
A percentage change always uses the starting value as its base.
- The fall: from ₹1,000 to ₹800 is a change of ₹200. ₹200 out of a starting ₹1,000 is 20%.
- The recovery: from ₹800 back to ₹1,000 is the same ₹200. But now the starting value is ₹800, and ₹200 out of ₹800 is 25%.
The rupee amount is identical. The base is smaller on the way back, so the same rupee change is a bigger percentage.
The recovery formula
If something falls by L%, the gain needed to return to where it started is:
Gain needed (%) = L ÷ (100 − L) × 100
Example for a 30% loss: 30 ÷ 70 × 100 = 42.86%. Check it with money: ₹1,00,000 falls 30% to ₹70,000. To get back to ₹1,00,000 you need ₹30,000 more, and ₹30,000 ÷ ₹70,000 = 42.86%.
Recovery chart
| Loss | Gain needed to break even |
|---|---|
| 10% | 11.11% |
| 20% | 25% |
| 25% | 33.33% |
| 33.33% | 50% |
| 50% | 100% |
| 60% | 150% |
| 75% | 300% |
| 80% | 400% |
| 90% | 900% |
Small losses need only a slightly bigger gain. Large losses need very large gains: after a 90% fall, the value must grow tenfold just to get back to the start.
Up then down (or down then up) never cancels out
The same effect explains why equal percentage moves in opposite directions leave you behind:
| Sequence | Start | After step 1 | After step 2 | Net change |
|---|---|---|---|---|
| +10%, then −10% | 100 | 110 | 99 | −1% |
| −10%, then +10% | 100 | 90 | 99 | −1% |
| −20%, then +20% | 100 | 80 | 96 | −4% |
| +50%, then −50% | 100 | 150 | 75 | −25% |
The order does not matter. Multiplying by 1.10 and by 0.90 gives 0.99 either way. The bigger the swings, the bigger the shortfall.
The "average return" trap
An investment that falls 50% one year and rises 50% the next has an average yearly return of 0%, but ₹100 has become ₹75. The simple average hides the loss. When you look at returns over several years, work out what the money actually grew to rather than averaging the percentages.
How to check a recovery in the Percentage Calculator
The Subtract X% from Y and Add X% to Y modes are useful for testing a sequence: subtract 20% from 1000 to get 800, then add 20% to 800 to see that you only reach 960.
Where this matters in real life
- Investments. A portfolio that drops 30% needs a 42.86% rise to recover. At an assumed 8% a year, that takes about 4.6 years of growth, not 30 ÷ 8 ≈ 3.75 years.
- Prices and discounts. A shop that cuts a price by 25% and later "raises it back by 25%" ends up 6.25% below the original: ₹120 × 0.75 = ₹90, then ₹90 × 1.25 = ₹112.50.
- Sales and traffic. If monthly sales fall from 1,000 to 800, getting back needs 25% growth, not 20%. Set the target in units, then convert to a percentage.
- Salary changes. A 10% pay cut followed by a 10% raise still leaves you 1% below where you were.
For combining several discounts at once, see 50% + 20% Off Is Not 70%: How to Calculate Stacked Discounts.
Repeated losses
Losses also compound. Three successive 10% falls do not make a 30% fall:
- 100 × 0.9 = 90
- 90 × 0.9 = 81
- 81 × 0.9 = 72.9
That is a 27.1% total fall, which then needs a gain of about 37.2% to recover (27.1 ÷ 72.9 × 100).
Common mistakes
- Assuming a fall and an equal rise cancel. They never do, unless the change is zero.
- Adding percentages across steps. −10% three times is −27.1%, not −30%.
- Using the wrong base. Always divide the change by the value you are starting from at that step.
- Confusing percent with percentage points. A rate going from 10% to 8% is a fall of 2 percentage points but a 20% decrease. See Percentage Points vs Percent: What's the Difference?.
Quick reference
- Gain needed after a loss: L ÷ (100 − L) × 100
- Loss that wipes out a gain: G ÷ (100 + G) × 100. A 25% rise is undone by a 20% fall.
- Net effect of +a% then −b%: multiply (1 + a/100) × (1 − b/100), then subtract 1.
For the five everyday percentage formulas in one place, read Percentage Formulas Explained: 5 Calculations You Use Every Week.
Frequently asked questions
Why does a 50% loss need a 100% gain to recover?
Because the gain is calculated on the smaller amount left after the loss. ₹100 falls 50% to ₹50, and getting from ₹50 back to ₹100 means adding ₹50, which is 100% of ₹50.
What is the formula for the gain needed to recover a loss?
Gain needed (%) = loss % ÷ (100 − loss %) × 100. For a 20% loss: 20 ÷ 80 × 100 = 25%.
If something goes up 10% and then down 10%, is it back where it started?
No. 100 rises 10% to 110, then falls 10% to 99, a net loss of 1%. The fall is calculated on the larger number, so it removes more than the rise added.
How do I check a recovery with the percentage calculator?
Use the % change from X to Y mode. Enter the lower value as From and the original value as To. From 800 to 1,000 shows a 25% change, the gain needed after a 20% fall.
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