I keep tripping over percentage increases when there are multiple steps. For example, if an item is $120, I increase it by 25% and then take 10% off the new price. My instinct is to do 25% − 10% = 15% overall, but when I actually calculate the numbers (add 25%, then remove 10%), the final price doesn’t match a simple 15% increase on the original. I can see the mismatch in the arithmetic, but I don’t understand the deeper reason why adding and subtracting the percentages like that isn’t valid.
Could someone walk me through the right way to combine percentage changes step by step, and how to turn a two-step change into one equivalent percentage? Also, how do I reverse it: if I only know the final price and the percentage steps used, how can I get back to the original price? As another checkpoint, what should happen with something like “increase by 20% then increase by 30%”-is that the same as a 50% increase, or not? I’d really appreciate a clear way to think about this so I stop making the same mistake.
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4 Responses
Percentages aren’t little numbers you add and subtract; they’re scale factors you multiply-like stacking resizing spells, each one acts on whatever the current size is. So “up 25% then down 10%” means multiply by 1.25 and then by 0.90, giving 1.25 × 0.90 = 1.125, i.e., a net 12.5% increase (not 15%) because that 10% discount is taken on the larger, already-marked-up price. Worked example: start at $120 → after 25% increase: 120 × 1.25 = 150 → after 10% off: 150 × 0.90 = 135; compared to the original, that’s 135/120 = 1.125, or +12.5%. If you had done a single 15% increase, you’d get 120 × 1.15 = 138-too high. The general rule: a change of r1 followed by r2 gives a net factor (1 + r1)(1 + r2); subtract 1 to get the equivalent single percentage, with decreases using negative r (e.g., −0.10 for 10% off). For “up 20% then up 30%” the factor is 1.2 × 1.3 = 1.56, so a 56% increase, not 50%. To reverse the process (recover the original from the final), divide by the same product of factors: original = final / [(1 + r1)(1 + r2)⋯]; for our numbers, 135 / (1.25 × 0.90) = 135 / 1.125 = 120. If percentages were pastries, they’d be croissants-deliciously layered, and very much multiplicative. For a friendly walkthrough, see Khan Academy’s successive percent changes: https://www.khanacademy.org/math/cc-seventh-grade-math/cc-7th-ratios-proportions/cc-7th-percent-problems/v/successive-percent-changes
Think of percentage changes like resizing a photo: “up 25%” means multiply the size by 1.25, and “down 10%” means multiply by 0.90-each step uses the current size as its base, not the original. That’s why you can’t just do 25% − 10% = 15%: the 10% is taken off the already-bigger number. The clean rule is: a change of r% corresponds to a multiplier of (1 + r/100), and multiple changes multiply together. Worked example: starting at $120, go up 25% to 1.25 × 120 = $150, then 10% off gives 0.90 × 150 = $135; the combined factor is 1.25 × 0.90 = 1.125, i.e., a 12.5% overall increase (not 15%). To combine any steps, just multiply the factors and then convert back to a single percent: product − 1, expressed as a percentage. For your checkpoint, “up 20% then up 30%” is 1.20 × 1.30 = 1.56, so a 56% increase, not 50%. To reverse the process (find the original from the final), divide by the same product of factors: for the $135 above, original = 135 ÷ (1.25 × 0.90) = 135 ÷ 1.125 = $120. I love this view because it turns a tangle of percentages into simple multiplier Lego blocks you just snap together-or pull apart when you need to go backwards.
I love this question because it’s exactly where our instincts clash with how percentages really behave. Think of each percentage change as a “resize” operation: like zooming a photo. “Up 25%” means multiply by 1.25, and “down 10%” means multiply by 0.9. Resizes stack by multiplying, not by adding their percentages, because each step uses the new size as its base. That’s the whole trick: the base changes after the first step.
Let’s do your $120 example. First increase by 25%: 120 × 1.25 = 150. Then 10% off that: 150 × 0.9 = 135. The single equivalent change is the product of the factors: 1.25 × 0.9 = 1.125, i.e., a 12.5% increase overall-not 15%-because that 10% was taken off the higher price. Another checkpoint: “up 20% then up 30%” is 1.2 × 1.3 = 1.56, so that’s a 56% increase. Quick example: start at 100 → 120 → 156.
To reverse, divide by the same overall factor (or undo steps in reverse). If the final price is 135 after “up 25% then down 10%,” the overall factor is 1.25 × 0.9 = 1.125, so original = 135 ÷ 1.125 = 120. Step-by-step undo also works: 135 ÷ 0.9 = 150, then 150 ÷ 1.25 = 120. In short: p% change → multiply by (1 + p/100); combine changes by multiplying the factors; the equivalent single percentage is (product − 1) × 100%.
The trap is that each percentage is taken on whatever the current price is, not the original. Percent changes are really “multiply-by” moves. A 25% increase means “multiply by 1.25,” and a 10% discount means “multiply by 0.9.” So starting at 120: first step 120 × 1.25 = 150, second step 150 × 0.9 = 135. Overall you multiplied by 1.25 × 0.9 = 1.125, which is a 12.5% increase, not 15%. My brain still wants to subtract 10 from 25 too, but the second 10% lives on a bigger base than the original 120, so the simple subtraction doesn’t line up.
Here’s the neat general rule: chain the steps by multiplying their factors. If you change by a then by b (where a and b are written as decimals, so +25% is +0.25 and −10% is −0.10), the single equivalent change is (1 + a)(1 + b) − 1 = a + b + ab. That extra ab term is the “compounding” bit that breaks the simple add/subtract instinct. For example, +20% then +30% gives (1.2)(1.3) = 1.56, i.e., a 56% increase, not 50%. Order doesn’t matter for exact arithmetic here because multiplication commutes: you’ll get the same result whether you do the discount first or second.
To reverse a sequence, just divide by the same multipliers in reverse order. If the final price is F after “+25% then −10%,” the original is F ÷ 0.9 ÷ 1.25 = F ÷ 1.125. In our numbers, from 135 back to 120: 135 ÷ 0.9 = 150, then 150 ÷ 1.25 = 120. Does it help to think in terms of “multipliers” instead of “percentages”? And are there other combos you bump into a lot where a quick mental rule would be handy?