I’m trying to wrap my head around percentage changes applied one after the other. If something goes up 25% and then down 25%, my gut says it should end up the same. But when I do a quick calc I get different results depending on how I think about it, and I can’t tell which step I’m messing up.
Say I start with x. After a 25% increase I have 1.25x. To undo that, my first thought was to take away 25% of that, so I wrote 1.25x − 0.25x = 1.0x (so, back to x). But if I instead multiply by 0.75 after the increase, I get 1.25x × 0.75, which isn’t x. Which approach is actually correct, and why do these two lines of thinking disagree? Is there a tidy rule for chaining percentage changes (like +a% then −b%) so I don’t keep tripping over this? Also, kinda related: if I only know the final price after, say, a 30% discount, what’s the clean way to get back to the original without guessing?
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5 Responses
Your gut is bumping into the “percent of what?” trap. Percent changes are always of the current amount. After a 25% increase, you’re at 1.25x. “Take away 25% of that” means subtract 0.25×(1.25x) = 0.3125x, not 0.25x. So you land at 1.25x − 0.3125x = 0.9375x. That’s the same as multiplying by 1.25 then by 0.75: 1.25×0.75 = 0.9375. Real-life vibe: if a store marks a $100 jacket up 25% to $125, then runs a 25% off sale, you’re taking 25% off $125, not off $100-so you pay $93.75, not $100. The base changed under your feet!
Tidy rule: chain percent changes by multiplying their factors. A +a% then −b% change gives a net factor (1 + a)(1 − b). In particular, +a then −a lands at 1 − a^2, so you end up below where you started. To undo a single change, divide by its factor: to reverse +25%, multiply by 1/1.25 = 0.8 (that’s a 20% drop, not 25%); to reverse −25%, multiply by 1/0.75 ≈ 1.333… (a 33⅓% increase). And to recover an original price from a final price after a 30% discount, just do original = final ÷ 0.70. For example, if you paid $70 after 30% off, the original was $70 ÷ 0.70 = $100.
The key is that each percentage change applies to the current amount, not the original, so the operations are multiplicative. Starting at x, a 25% increase gives 1.25x; a 25% decrease after that means subtracting 25% of 1.25x, i.e., 0.25·(1.25x) = 0.3125x, so you land at 1.25x − 0.3125x = 0.9375x, which is 6.25% below where you started. The step that looks like 1.25x − 0.25x = x quietly assumes you’re removing 25% of the original x rather than 25% of the new amount. A tidy rule is: chaining percentage changes multiplies factors. For +a% then −b%, multiply by (1 + a/100)(1 − b/100). In particular, equal up and down percentages don’t cancel: (1 + p)(1 − p) = 1 − p^2, so you end up below the start unless p = 0. To truly “undo” a +p% increase, you must multiply by 1/(1 + p), which corresponds to a decrease of p/(1 + p) of the increased value (for +25%, you need a 20% decrease of the new price). Likewise, to undo a −p% discount, multiply by 1/(1 − p), which corresponds to an increase of p/(1 − p) of the discounted value. If you only know the final price after a 30% discount, say final = 0.70·original, then original = final/0.70, and more generally, original = final divided by the product of all the (1 ± rate) factors. I remember getting tripped up on this during a store sale: I thought “20% off then 20% on” would return the shelf price, but a quick check on a $100 example-down to $80, then up 20% to $96-finally made the multiplicative rule click for me.
This one tripped me up for ages too-the sneaky bit is that the second 25% is taken off a different base than the first one added to. Start with x: after a 25% increase you have 1.25x. A “25% decrease” now means 25% of that new amount, so you multiply by 0.75, giving 1.25x × 0.75 = 0.9375x, i.e., you’ve actually lost 6.25% overall. The line 1.25x − 0.25x = x accidentally subtracts 25% of the original x, not 25% of 1.25x, so it mixes bases. The tidy rule is: chain percentage changes by multiplying their factors. A +a% change is a factor of (1 + a/100); a −b% change is a factor of (1 − b/100). So +a% then −b% gives x × (1 + a/100) × (1 − b/100), and the net percent change is (1 + a/100)(1 − b/100) − 1. To “undo” a change, you use the reciprocal, not the same percentage in the other direction: undoing +25% needs a factor 1/1.25 = 0.8, which is a 20% decrease; undoing a 30% discount means original = final ÷ 0.70. Want to try one: what percent decrease exactly cancels a +40% increase?
I totally get the gut feeling-if you go up 25% and then down 25%, shouldn’t you be right back where you started? The trick is that the second 25% is taken off the new, larger amount, not the original. Starting with x, a 25% increase gives 1.25x. A 25% decrease after that means you multiply by 0.75, but that’s 25% of 1.25x, so you get 1.25x × 0.75 = 0.9375x, which is 6.25% less than where you began. The line 1.25x − 0.25x = x accidentally subtracts 25% of the original x, not 25% of 1.25x. A tidy rule is: chaining percentage changes is multiplicative, not additive. If you do +a% then −b%, the overall factor is (1 + a)(1 − b). In particular, +a% then −a% gives (1 − a^2), which is always less than 1 unless a = 0. To exactly undo a change, you divide by the factor you multiplied by: undo +25% by dividing by 1.25 (i.e., multiply by 0.8), and undo −25% by dividing by 0.75 (i.e., multiply by 4/3). For your last question: if a price after a 30% discount is F, then original = F / 0.70. I remember messing this up buying a “25% off then extra 25% off” jacket in college-thought I was getting half off, but the receipt taught me fast that percentages stack by multiplying, not by just adding the percents.
This is a classic gotcha-I used to trip over it too! The key is that each percentage change is taken on whatever the current amount is, not on the original. Think of it like stretching a picture by 25% and then shrinking it by 25%: that second 25% is of the bigger picture. In math-speak, percentage changes are multipliers you chain together. A +a% change multiplies by (1 + a/100), and a −b% change multiplies by (1 − b/100). So +25% then −25% gives (1.25)(0.75) = 0.9375 of the starting value-not back to where you began. Worked example: start at 100, go up 25% to 125, then down 25% of 125 (which is 31.25) to land at 93.75. Notice how 1.25x − 0.25x = x mistakenly subtracts 25% of the original x instead of 25% of the new 1.25x. A tidy rule: chain changes by multiplying their factors; +a% then −a% equals 1 − (a/100)^2 overall, so it only returns to the start when a = 0. To undo a single change, divide by the factor that was applied (don’t add the “opposite” percent). For example, a 30% discount means final = 0.70 × original, so original = final ÷ 0.70 (e.g., if the final price is 70, the original was 100). And to exactly undo a +25% increase, you’d need a −20% decrease, because 1.25 × 0.80 = 1.