I keep getting tangled up with percentage increases, and I think it’s because I’m not sure which number I’m supposed to compare to. Like, my gym membership went from $40 to $50. Part of my brain says that’s a 20% increase, but another part says it’s 25%, and both feel weirdly reasonable depending on which number I treat as the “starting point.” I run into the same thing in the wild all the time – coffee price jumps, app subscription hikes – and I can’t tell which percentage is the “official” increase you’re meant to report. Also, when something gets bumped up twice (say 10% this month and 15% next month), is that just a 25% total increase, or is it something else because the second increase is on the new price? I think I’m confused because percentages feel like they should be symmetric, but they clearly aren’t, and ads/news headlines don’t always say what base they used. How do I decide the correct base for a percentage increase, and how should I think about back-to-back increases? Any help appreciated!
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3 Responses
Percentages love to play “gotcha,” so you’re not alone! The usual rule for an increase is: use the original amount as the base, so going from $40 to $50 is a (50−40)/40 = 25% increase; if you divide by the new price instead ((50−40)/50), you get 20%, which is the bump as a share of the final price-some headlines do that too, so there isn’t really one “official” base everyone follows, even though math and finance typically mean the original. For back-to-back bumps, adding the percents feels natural (10% + 15% = 25%), and if you think of both as being measured on the same base that sounds exactly right, but because the second increase is applied after the first, the true combined effect is multiplicative: 1.10 × 1.15 = 1.265, i.e., a 26.5% total increase. Tiny tangent: that’s why a 25% increase followed by a 25% decrease doesn’t bring you back to square one-you end up lower than you started, which always feels unfair until you remember the base switched! Rule of thumb I use: unless someone clearly says otherwise, treat “increase” as relative to the starting value, and for successive changes multiply the growth factors (1 + p1)(1 + p2)… and subtract 1 at the end to get the overall percent. If you want a friendly walkthrough, Khan Academy’s percent change article is great: https://www.khanacademy.org/math/percents/percent-word-problems/percent-change/a/percent-change. What kinds of examples have been tripping you up lately-do they actually name the base, or are they leaving you to infer it from context?
Totally fair question-percentages feel slippery until you pin down the “what are we measuring against?” rule. For ordinary percentage change, the base is the starting value: percent change = (new − old) / old × 100%. So from $40 to $50, the increase is (50−40)/40 = 25%. If you go the other way (from $50 down to $40), the decrease is (50−40)/50 = 20%. That asymmetry is normal: you’re measuring the change in “units of the old price,” and the old price is different in each direction. Think of the old price as your measuring cup-how many old-price “cups” does the change equal? When increases happen back-to-back, you multiply the growth factors: a 10% bump then 15% is 1.10 × 1.15 = 1.265, i.e., a 26.5% total increase (not 25%). Likewise, up 20% then down 20% gives 1.2 × 0.8 = 0.96, a net 4% drop-they don’t cancel. Headlines sometimes switch the base to make numbers sound nicer, so a good sanity check is to rewrite claims as “new = old × (1 ± p)” and see which value they used as “old.”
I trip over this all the time too-my brain also wants percentages to be “symmetric,” but they aren’t. The rule of thumb: a percentage increase is always measured relative to the original amount. So from $40 to $50, the increase is (50 − 40)/40 = 10/40 = 25%. If you go the other way (from $50 down to $40), that’s a decrease of (50 − 40)/50 = 10/50 = 20%. That’s why both 25% and 20% can feel “right,” but they answer different questions. For back-to-back changes, you multiply the factors: a 10% increase then a 15% increase gives 1.10 × 1.15 = 1.265, so a total 26.5% increase-not 25%-because the second bump is on the already-increased price. When reading ads or headlines, I always ask “percent of what?”; unless stated otherwise, “percent increase” usually means percent of the original. Khan Academy has a nice walkthrough on percent change if you want more examples: https://www.khanacademy.org/math/pre-algebra/rates-and-ratios/percent-word-problems/a/percent-change. Hope this helps!