Problem 8 · 2012 AMC 8
Easy
Fractions, Decimals & Percents
successive-percentages
A shop advertises everything is "half price in today's sale." In addition, a coupon gives a 20% discount on sale prices. Using the coupon, the price today represents what percentage off the original price?
Show answer
Answer: D — 60%.
Show hints
Hint 1 of 2
Don't add 50% + 20% — the 20% comes off the already-halved price, not the original. Track what survives each step, not what's taken off.
Still stuck? Show hint 2 →
Hint 2 of 2
This is chaining percentages by multiplying: a discount of d leaves the fraction (1 − d), and successive discounts multiply. Always work with the surviving fraction.
Show solution
Approach: multiply the surviving fractions, then subtract from 1
- Half price means you still pay 1/2 of the original. The 20%-off coupon then leaves 80% = 0.8 of that, so discounts chain by multiplying.
- What you actually pay: 0.5 × 0.8 = 0.4 of the original price.
- So the total discount is 1 − 0.4 = 60% — not the 70% you'd get by wrongly adding 50 + 20.
- Why multiply, not add: each percent acts on the price left after the previous one. Multiplying the "keep" fractions is the safe path for any stacked discount or tax.
Mark:
· log in to save