In this article
A share is priced at ₹100. It jumps 50 percent to ₹150. Next month it drops 50 percent. Where is it now? Most people say ₹100. The honest answer is ₹75, and the missing ₹25 is the whole lesson of this article.
Percentages feel like a fixed unit, like centimetres. They are not. A percentage is always a share of something, and the something changes every time the number moves. Once you see that, a handful of strange results stop being strange.
Why the second 50 percent is bigger
The first 50 percent is taken from ₹100, so it is worth ₹50. The second 50 percent is taken from ₹150, so it is worth ₹75. You gained a little and then gave back a lot, because the thing you were taking half of had become larger.
The end point is 25 percent below where you started, even though the two moves look like mirror images.
Mathematically it is a product, not a sum. Going up 50 percent multiplies by 1.5. Going down 50 percent multiplies by 0.5. And 1.5 × 0.5 is 0.75. Percent changes multiply, they do not cancel.
How much you need to climb back
This is the one to remember. If something falls by a percentage, the gain needed to return to the start is always bigger than the fall. The formula is a fall of x needs a rise of x ÷ (100 − x).
| You lose | Left with | Gain needed to recover |
|---|---|---|
| 10% | ₹90 of ₹100 | 11.1% |
| 20% | ₹80 of ₹100 | 25% |
| 30% | ₹70 of ₹100 | 42.9% |
| 50% | ₹50 of ₹100 | 100% |
| 75% | ₹25 of ₹100 | 300% |
| 90% | ₹10 of ₹100 | 900% |
A small fall is nearly symmetrical. A big fall is brutally one-sided. Halving your money means you must double what is left.
The bars grow much faster than the losses do. This is why avoiding big drops matters more than chasing big rises.
Percent versus percentage points
A bank raises its rate from 6 percent to 8 percent. News says “up 2 percent”. That is wrong in a way that matters. The rate went up by 2 percentage points. In relative terms it rose by one third, which is 33 percent.
- Simple subtraction: 8 − 6
- Use for rates, shares and polls
- (8 − 6) ÷ 6 × 100
- Use for prices, amounts and sizes
Both are true. Trouble starts when a headline says “percent” and means points, or the reverse.
Sale signs and stacked discounts
“Extra 20 percent off” on top of “30 percent off” does not make 50 percent off. Each discount applies to the price left by the one before. 100 becomes 70, then 70 × 0.8 = 56. You pay ₹56, which is 44 percent off.
| Offer | You pay on ₹1,000 | Real discount |
|---|---|---|
| 50% off | ₹500 | 50% |
| 30% off, then 20% off | ₹560 | 44% |
| 20% off, then 30% off | ₹560 | 44% |
| 10% off, then 10% off | ₹810 | 19% |
The order does not matter, because multiplication does not care. But the total is always less than the two numbers added together.
The same trap works in the other direction. Raise a price by 20 percent and later cut it by 20 percent, and the item costs 4 percent less than it began: 100 × 1.2 × 0.8 = 96.
The formula, once and for all
- Percent change has a direction. From 80 to 100 is +25 percent. From 100 to 80 is −20 percent.
- Percent difference between two values with no “before” divides by their average, so it is the same either way round. Use it when neither number is the starting point.
- Percent of a number is simply multiplication. 18 percent of 2,500 is 2,500 × 0.18 = 450.
Questions people ask
If something rises 50 percent and then falls 50 percent, where does it end?
At 75 percent of the start. Percent changes multiply: 1.5 × 0.5 = 0.75, a net loss of 25 percent.
How do I calculate percentage change?
Subtract the old value from the new one, divide by the old value, and multiply by 100. A negative answer means a decrease.
What gain do I need to recover a 20 percent loss?
25 percent. After a 20 percent fall you hold 80 percent of the original, and 100 ÷ 80 is 1.25.
What is the difference between percent and percentage points?
Percentage points are plain subtraction between two percentages, like 6 percent to 8 percent being 2 points. Percent change compares them proportionally, which here is 33 percent.
Do two successive discounts add up?
No. They multiply. 30 percent off then 20 percent off leaves 0.7 × 0.8 = 0.56 of the price, which is 44 percent off, not 50.