I’m preparing for a test and I keep getting tripped up by successive percentage changes. For example: if a price goes up by 25% and then down by 25%, does it go back to the original price?
My attempt: I tried using multipliers. For a 25% increase I wrote 1.25, and for a 25% decrease I wrote 0.75. So the combined effect should be 1.25 × 0.75. I think that’s the right setup, but I’m not fully confident about how to interpret this product as the “overall percent change” without plugging in a specific starting price.
– Is my multiplier approach correct here?
– How do I clearly express the overall percentage change from start to finish?
– Why doesn’t adding 25% and then subtracting 25% cancel out? Is it just that the base changes, and how should I think about that step by step?
A concise way to reason through this (and a general rule I can remember under test conditions) would help a lot. I want to avoid guessing and be sure about the base I’m using at each step.
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3 Responses
Lovely question! Your multiplier idea is exactly the right magic wand: successive percentage changes multiply, not add. Up 25% means multiply by 1.25, then down 25% means multiply by 0.75, so the overall factor is 1.25 × 0.75 = 0.9375. That says you end with 93.75% of the start, i.e., an overall 6.25% decrease. Simple worked example: start at $100 → after +25% you have $125 → then −25% of $125 is $31.25, so you land at $93.75. Why don’t +25% and −25% cancel? Because the second percent is taken on a different base: the decrease bites a chunk out of the larger $125, so it removes more than you added. General rule to stash in your brain-pocket: if you go up p% and then down q%, the overall multiplier is (1 + p/100)(1 − q/100), and the overall percent change is [(1 + p/100)(1 − q/100) − 1] × 100% = p − q − pq/100. Special memory gem: up p% then down p% always leaves you down by p^2/100 percent (for 25%, that’s 625/100 = 6.25%). Multiply the multipliers, not the percents-like fastening both seatbelts on the percentage roller coaster!
You’re absolutely on the right track with multipliers-yay for 1.25 and 0.75! Multiply them: 1.25 × 0.75 = 0.9375, which means the combined effect is 93.75% of the original, i.e., an overall change of (0.9375 − 1) × 100% = −6.25%. So it doesn’t bounce back; it ends 6.25% lower. The reason “+25% then −25%” doesn’t cancel is that the second 25% is taken on a different base: after the increase you’re at 125% of the original, and 25% of that is 31.25% of the original, so you remove more than you added. General rule to remember under test-pressure: successive percentage changes multiply, not add; combine them as (1 + p1)(1 + p2) …, and convert back to a percent by subtracting 1. Bonus memory nugget: to exactly undo a p% increase, you need a decrease of p/(1 + p), not p itself (e.g., up 25% needs down 20% to return). Little analogy: stretch a rubber band by a quarter, then snip off a quarter of its new length-you end up shorter. Nice walkthrough: https://www.mathsisfun.com/percentage-change.html
Your multiplier approach is spot on! A 25% increase is 1.25 and a 25% decrease is 0.75, so the combined multiplier is 1.25 × 0.75 = 0.9375. That means the final price is 93.75% of the original, i.e., an overall decrease of 6.25%. In general: convert each percentage change to its multiplier, multiply them all, then subtract 1 to get the overall percent change (and convert to a percentage). For a quick refresher on this style, see Math Is Fun’s page on percentage change: https://www.mathsisfun.com/percentage-change.html
Why don’t “+25%” and “−25%” cancel? Because the base changes between steps. If you start at 100, going up 25% gets you to 125; going down 25% then takes 25% of 125 (which is 31.25), landing at 93.75-not back at 100. A small analogy: inflate a balloon by 25%, then deflate it by 25%. The second 25% is taken from a bigger balloon, so you remove more air than you added, and end up smaller than you started.
Handy general rules to remember: successive percent changes multiply, not add. If you increase by a (as a decimal), then decrease by the same a, the net multiplier is (1 + a)(1 − a) = 1 − a², so you always lose a² overall. For a = 0.25, that’s a 0.0625 (6.25%) drop. And if you ever want to exactly undo a +p% change, you need a decrease of p/(1 + p); for example, after a +25% increase, you’d need a 20% decrease to get back to the original, since 1.25 × 0.8 = 1.