Do two discounts just add up?

I’m cramming for a test and my brain is doing cartwheels: if a $100 price gets cut by 20% and then by another 10%, is that just 30% off, or is there a sneaky twist I keep forgetting?

3 Responses

  1. I’d treat successive discounts by multiplying the “what remains” factors. Starting at 100, a 20% cut leaves 80 (that’s 100 × 0.8). Then a further 10% cut applies to 80, leaving 72 (that’s 80 × 0.9). So the final price is 72, which is a total reduction of 28%, not 30%. The sneaky twist is that the second discount is taken on the already-reduced price, not on the original. In general, two discounts a and b combine to a + b − ab (here 0.20 + 0.10 − 0.02 = 0.28). The order doesn’t change the result because 0.8 × 0.9 = 0.9 × 0.8. Unless there’s some extra fee or tax lurking, I’m pretty sure this is the whole story.

  2. Nope, they don’t just add. The second discount hits the already-discounted price. Start with $100: 20% off gives $80, then 10% off that is $8, so you end up at $72. That’s a total discount of $28, i.e., 28% off, not 30%. Shortcut: combined discount = a + b − ab (treat a and b as decimals), so 0.20 + 0.10 − 0.20×0.10 = 0.28 = 28%. The order doesn’t matter (0.8×0.9 = 0.72 either way), but they never just add unless one of them is 0. Hope this helps!

  3. Not quite 30%. Start at $100: a 20% cut takes it to $80, and then another 10% off gives $72. That’s a total discount of 28%. In general, two successive discounts a and b combine to 1 − (1 − a)(1 − b) = a + b − ab, so you subtract the product too. Here, 0.20 + 0.10 − (0.20)(0.10) = 0.28.

    The order doesn’t matter (multiplication commutes), but simply adding the percentages slightly overstates the discount by the product ab. For small percentages this difference is small, but it’s still there-two 50% discounts, for instance, give 75% off, not 100%. As a quick check: adding would predict 30% off, but the exact is 28%, a 2-point gap.

    As a follow-up, what if you had 20% off followed by a 10% increase-what would the net percentage change be?

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