I keep tripping over percentage increases when they happen more than once. If something goes up 10% and then another 10%, is that just a 20% total increase, or do I have to handle it differently? My brain wants to just add them, but I know that can be wrong.
What’s the simplest rule I can use in my head for repeated increases, without doing a bunch of steps? Example: start at 100, increase by 10%, then increase by 10% again – what’s the overall percentage increase?
Also, what’s the quick way to reverse it? Like if the final price is 115 after a 15% increase, how do I get back to the original fast without a calculator marathon?
I’m after a clean, practical shortcut here, not a lecture.
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3 Responses
I always want to just add them too, but the quick rule is multiply the factors-so 10% then 10% is 1.1×1.1 ≈ 1.21, which is basically a 20% rise (actually 21%). To undo a +15%, divide by 1.15-115/1.15 ≈ 100-or if I’m being rough I just knock 15% off the final, and this explains why: https://www.khanacademy.org/economics-finance-domain/core-finance/interest-tutorial/compound-interest-tutorial/v/introduction-to-compound-interest
Use the multiply rule: each +p% step multiplies by (1 + p). So two +10% steps are 1.10 × 1.10 = 1.21, meaning a 21% overall increase (not 20%). A tidy head shortcut for two steps is: add the percents and then add “percent of a percent”: a% then b% gives a + b + (a×b)/100. Example: 10% then 10% → 10 + 10 + (10×10)/100 = 20 + 1 = 21%, so 100 → 121. Decreases work the same with (1 − p), e.g., two 10% drops: 0.90 × 0.90 = 0.81 → a 19% total decrease. To reverse, divide by the same factor you multiplied by: after a +15% increase, the original is final/1.15 (since the final is 115% of the original). Example: final 115 after +15% → 115/1.15 = 100. Quick memory: forward = multiply by 1 ± p; backward = divide by 1 ± p.
Think of a percentage increase like a multiplier you stick on the price. “Up 10%” means “multiply by 1.10.” Do it twice and you multiply twice: 1.10 × 1.10 = 1.21, so that’s a 21% total increase, not 20%. It’s like a snowball rolling downhill-the second 10% is taken on the already-bigger amount. Quick rule you can run in your head: repeated increases multiply, not add. If it’s the same percent twice, use the tidy shortcut (1 + x)² = 1 + 2x + x². For x = 0.10, that’s 1 + 0.20 + 0.01 = 1.21 → 21%. More generally, stack factors: (1 + r1)(1 + r2)… and the overall percent is “product − 1.”
To undo an increase, just divide by the same factor you used to go up. If a price is “after a 15% increase,” it’s “original × 1.15,” so original = final ÷ 1.15. Example: 115 after +15% means 115 ÷ 1.15 = 100. If you had two bumps, say +10% then +10%, you can undo by dividing twice: 121 ÷ 1.10 ÷ 1.10 = 100 (equivalently, divide once by 1.21). That’s the whole mental toolkit: go up by multiplying 1 + percent, come back down by dividing by that same 1 + percent.