A 6th-Grade Math Problem: Price Up 20%, Then Down 20% — What Was It Originally?
I came across this problem — it looks simple, but I actually had to think about it. I’m publishing it because understanding it is genuinely useful in real life: people get it wrong both in shops and with investments.
Problem:
A product’s price first went up by 20% and then went down by 20%, ending at $320. What were the original and the intermediate prices?
The first instinct is “up 20%, down 20%, so the price is back where it started — $320”. That’s wrong. Try to solve it yourself before reading on.
Solution
Let the original price be x.
1. A 20% increase. The new (intermediate) price is y = x + 0.2x = 1.2x.
2. A 20% decrease. Now the 20% is taken from the new, higher price: 0.2 × 1.2x = 0.24x. The final price is 1.2x − 0.24x = 0.96x.
The short way to write the same thing: x × 1.2 × 0.8 = 0.96x.
3. Set up the equation:
0.96x = 320
x = 320 / 0.96 = 333.33…
Answer
- Original price: $333.33 (exactly 1000/3, which is why you get a repeating decimal; round to cents).
- Intermediate price: 1.2 × 1000/3 = exactly $400. If you plug in the already-rounded 333.33, you get 399.996 — that’s where the “ugly” number in a rough draft comes from.
Check: $400 − 20% = 400 − 80 = $320. It matches the problem, so the answer is correct.
The key takeaway: +20% and −20% don’t cancel out
After an increase and an equal decrease, the price ends up 4% lower than the original. That’s because the percentages are taken from different amounts: you add 20% of the smaller price and subtract 20% of the larger one.
The order doesn’t matter: if the price first drops 20% and then rises 20%, the result is the same — 96% of the original ($1000 → $800 → $960).
The general rule: after a change of +p% and −p%, you’re left with (1 − p²) of the original amount.
| Change | Loss after +p% and −p% | To recover from a p% drop, you need a rise of |
|---|---|---|
| 10% | 1% | 11.1% |
| 20% | 4% | 25% |
| 25% | 6.25% | 33.3% |
| 30% | 9% | 42.9% |
| 50% | 25% | 100% |
Where this matters in real life
- Discounts. “Price raised by 30%, then 30% off” isn’t the old price — it’s 9% lower. But it can work the other way too: sometimes the price is raised by more than the later discount, so compare with the price before the “sale”.
- Investments and exchange rates. If a stock falls 50%, it needs to rise not 50% but 100% to get back to where it was. That’s why limiting losses matters so much in trading.
- Salaries and prices. If your salary is cut by 20% and then raised by 20%, you still earn 4% less than before.
In short
The original price is $333.33 and the intermediate price is $400. The main thing to remember: percentages “up” and “down” are taken from different amounts, so +20% and −20% don’t give zero — they give minus 4%.
