A price goes up 20%, then the new price goes down 20%. Most people say it's back where it started. It isn't. It ends up 4% lower, and the SAT asks versions of this all the time.
Start at 100
When a question is about percent change and doesn't give you a starting number, start at $100. Percents of 100 are easy to see.
- Up 20%: $100 becomes $120.
- Down 20% of the new price: 20% of $120 is $24, so you land on $96.
- $96 is 4% less than $100.
The drop is bigger than the raise because it's taken from a bigger number. That's the whole trick.
The faster way: multiply
Going up 20% is the same as multiplying by 1.2. Going down 20% is multiplying by 0.8. Do both: 1.2 × 0.8 = 0.96. That means you keep 96% of the original, so it's 4% lower. In Desmos, type 100*1.2*0.8.
This works for any chain of changes. Up 10% then up 10% is 1.1 × 1.1 = 1.21, a 21% increase, not 20%.
The pattern behind it
Up by any percent and then down by the same percent always ends lower. Going up and down 10% leaves you 1% lower, 30% leaves you 9% lower, and 50% leaves you 25% lower. The amount lost is the percent times itself.
Try these
- A jacket's price is cut by 25%, then raised by 25%. How does the final price compare to the original?
- A town's population grows 10% one year and 10% the next. What's the total percent increase?
- A stock falls 50%, then rises 50%. Is it above or below where it started, and by how much?
Answers: 1. 0.75 × 1.25 = 0.9375, so it's 6.25% lower. 2. 1.1 × 1.1 = 1.21, a 21% increase. 3. 0.5 × 1.5 = 0.75, so it's 25% below where it started.
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