You've seen it in a checkout flow: a 20% off sale, then a coupon code for another 20% off on top. It feels like you just saved 40%. You didn't -- you saved 36%. That eight-point gap isn't a rounding error, and it's not an accident either. It's just how percentages of percentages work, and it's worth understanding because it applies to a lot more than shopping carts.
The math, worked out
Say an item costs $100. Take 20% off and it drops to $80. Now apply a second 20% off -- but that 20% is calculated on $80, not on the original $100. Twenty percent of $80 is $16, so the price drops to $64. You paid $64 on a $100 item, which is a 36% discount overall, not 40%.
The reason is that each discount is multiplicative, not additive. Taking 20% off means multiplying by 0.8. Two stacked 20% discounts mean multiplying by 0.8 twice: 0.8 x 0.8 = 0.64, meaning you pay 64% of the original price. The bigger the discounts, the wider this gap gets -- two 50%-off discounts stacked don't make something free, they make it 25% of the original price (0.5 x 0.5 = 0.25), a 75% total discount, not 100%.
Order doesn't matter, but the base does
One thing that doesn't change the outcome: the order you apply the discounts in. 20% off then another 20% off gives the same $64 regardless of which one is applied first, because multiplication is commutative. What does matter is that every discount after the first is calculated on an already-reduced price, which is exactly why stacked discounts always undershoot the naive sum.
Where this actually bites people
Beyond shopping, this same logic shows up anywhere percentages compound: a salary cut followed by an equal percentage raise doesn't get you back to where you started (a 20% pay cut needs a 25% raise to recover, not 20%). Investment losses work the same way -- losing 50% of a portfolio requires a 100% gain just to break even. The pattern is identical: percentage changes stack multiplicatively against whatever the current value is, not the original one.
Working it out for a real purchase
If you're trying to figure out what something will actually cost after one discount, or comparing two differently-structured deals (say, 30% off vs. 20% off plus an extra 10% off a coupon), the safest approach is to calculate each step in order rather than mentally adding percentages -- as shown above, they don't combine the way addition suggests.
The Discount Calculator does this for you: enter the original price and the percentage off to get the discounted price and total savings without doing the multiplication by hand.