I was buying sneakers online and my brain did that thing where it says, “Easy! 10% off and then 10% tax should cancel, right?” But when I checked the final price, it wasn’t the same as the tag price. Now I’m stuck. It feels like walking one block forward and one block back, but somehow I’m not back where I started.
I think I’m mixing up “percent of what.” Is the 10% after the discount taken from the original price, or the new discounted price? Same with tax vs. discount-are they talking about the same base number or different ones?
For example: if something costs $50 and there’s 20% off, then later 20% added back (like a fee or tax), should I expect it to be $50 again or not? How do I reason about this properly without tying myself in knots?
I keep making the mistake of assuming the plus and minus percents “undo” each other. What’s the simple, intuitive way to track which number is the 100% at each step?
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3 Responses
I totally have that “they should cancel, right?” reflex too! But I think the sneaky bit is that the second percent is taken from a new 100%. If you take 10% off, you multiply by 0.9; then if you add 10% back, you multiply by 1.1. Multiplying gives 0.9 × 1.1 = 0.99, so you land at 99% of where you started-ah, just a hair short. Let me sanity-check with your $50 and 20%: first 20% off gives 50 × 0.8 = 40; then 20% added on gives 40 × 1.2 = 48. Yep, not quite back. So the simple tracker is: after each step, the new price becomes the new 100%, and you multiply by its factor (0.8 for 20% off, 1.2 for +20%, etc.). Percent changes don’t add; they multiply.
A tiny analogy: imagine shrinking a photo to 80% of its size, then “growing it by 20%.” That 20% boost is of the smaller photo, so you don’t re-inflate to the original-your enlarge step is literally using a smaller ruler. If you actually want to undo a discount, you need the reciprocal, not the same percent. After r% off (factor 1 − r), the exact undo is multiplying by 1/(1 − r), which is an increase of r/(1 − r). So 10% off needs about +11.111% to get back, and 20% off needs +25%. I might be over-nerding it, but the punchline is: think “multiply by factors,” and remember the base resets after each step.
You’ve got the right instinct: the trick is “percent of what.” Each step resets what 100% means. A 10% discount makes the new price 90% of the original, and then a 10% increase means 10% of that new, smaller number. It’s like shrinking a photo to 90% size, then enlarging it by 10% of its new size-the “up” step is a smaller step than the “down” step, so you don’t land back where you started. Or, since you mentioned blocks: the city shrank after your first block, so your “one block back” is a slightly shorter block.
In numbers: “10% off then 10% on” means multiply by 0.9 and then by 1.1. That’s 0.9 × 1.1 = 0.99, so you end up at 99% of the original price, not 100%. For your $50 example with 20%: first 20% off gives 50 × 0.8 = 40; then 20% on gives 40 × 1.2 = 48. In general, if you do a d% discount and then a t% increase, you multiply by (1 − d) × (1 + t). They only “cancel” if one of them is zero. To undo a discount exactly, you need a larger percentage increase: after 20% off (factor 0.8), you need a 25% increase (multiply by 1/0.8 = 1.25) to get back.
A simple way to keep your head straight is to treat the current price as the new 100% after each step. Think, “Start at 100%. After 10% off I’m at 90%. After 10% on that, I’m at 99%.” If you like more practice or visualizations, Khan Academy’s percent increase/decrease review is great: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:exponential-growth-decay/x2f8bb11595b61c86:percent-increase-decrease/a/percent-change-percentage-increase-and-decrease.
You’re right to focus on “percent of what.” Each step resets the 100%. A 10% discount takes you to 90% of the original price; then “add 10%” means 10% of that smaller, discounted price. That’s why it doesn’t cancel. For example, with $50 and a 20% discount: $50 → $40. Adding 20% after that means 20% of $40 (which is $8), so you end at $48, not $50. It feels like +20% and −20% should balance, but they act on different bases. Sometimes people also see a small extra mismatch from rounding at each step.
To get back to the original after going down d%, you need a larger increase: up by d/(1−d)%. So after 20% down, you’d need 25% up to return exactly. If a store quotes “10% off, then +10% tax,” they only cancel if that tax is calculated from the original tag price; more commonly the tax is taken on the discounted price, so it won’t balance. And if you reverse the order-add first, then take off-you can end up slightly different again. A short refresher on successive percent changes is here: https://www.khanacademy.org/math/pre-algebra/percent/percent-word-problems/v/percent-word-problems-1
Hope this helps!