A 50% increase followed by a 50% decrease never gets you back to where you started. You end up lower. Every time.
A jacket costs $100. The shop raises the price 50%, then later cuts it 50%. What does it cost now?
Most people's gut says $100. The gut is wrong.
Wait, Let's Actually Check
Start at $100.
Raise it 50%: $100 × 1.5 = $150.
Now cut $150 by 50%: $150 × 0.5 = $75.
Not $100. $75. The jacket is now 25% cheaper than when it started, even though it went up and down by "the same" 50%.
Why Does This Feel So Wrong?
Because our brains treat "50% up" and "50% down" as opposites that should cancel, like +5 and −5. But percentage changes aren't additions — they're multiplications, and multiplication doesn't work that way.
- +50% means multiply by 1.5.
- −50% means multiply by 0.5.
Multiply them together: 1.5 × 0.5 = 0.75. Not 1. That 0.75 is the whole story: a 25% loss, hiding inside two moves that sounded like they'd cancel.
The problem is the base changes. The 50% increase is 50% of $100. The 50% decrease is 50% of $150 — a bigger number. You lose more in the second step than you gained in the first, because you're taking half of a bigger pile.
Try It Yourself
This isn't special to 50%. Pick any percentage, go up then down by the same amount, and you always end up lower than you started:
10% up, then 10% down (×1.1, ×0.9) → −1%
20% up, then 20% down (×1.2, ×0.8) → −4%
50% up, then 50% down (×1.5, ×0.5) → −25%
90% up, then 90% down (×1.9, ×0.1) → −81%
Notice the pattern: the bigger the percentage, the bigger the gap. At 10% it's barely noticeable (−1%). At 90% it's brutal (−81%), because a 90% increase followed by a 90% decrease takes you almost back to zero.
There's a tidy formula hiding in that table: increasing by a and then decreasing by the same a always multiplies your starting number by (1 + a)(1 − a) = 1 − a². Since a² is always positive (for any real change), you always end up below 1 — below where you started. The only way to break even is if a = 0, meaning you never changed anything at all.
The OddMaths Takeaway
Percentage changes don't add — they multiply. And multiplying by 1.5 then 0.5 is not the same as doing nothing, because the second 50% is 50% of a different, bigger number than the first 50% was. "Up 50%, down 50%" sounds symmetric. It isn't.
Retailers know this trick well: a "50% off" sale on an item that was quietly marked up 50% last month isn't actually back to the original price — it's 25% cheaper than that. Still a discount, just not the one your gut assumed.
Quick Check
- A price starts at $200, goes up 50%, then down 50%. What's the final price?
- True or false: a 20% increase followed by a 20% decrease gets you back to the start.
- Which loses more value: going up 90% then down 90%, or up 10% then down 10%?
- What do you multiply by for a 50% decrease?
Answers
- $150. $200 → $300 (up 50%) → $150 (down 50% of $300).
- False. You end up at 96% of the start — a 4% loss (1.2 × 0.8 = 0.96).
- Up 90% then down 90% loses far more (−81%) than up/down 10% (−1%). Bigger swings cost more.
- Multiply by 0.5 — a decrease of a means multiplying by (1 − a), and 50% as a decimal is 0.5.