Problem 14 · 2002 AMC 8
Hard
Fractions, Decimals & Percents
percent-multiplier
A merchant offers a large group of items at 30% off. Later, the merchant takes 20% off these sale prices and claims that the final price of these items is 50% off the original price. The total discount is
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Answer: B — 44%.
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Hint 1 of 2
The merchant's "30 + 20 = 50% off" is the trap. The second discount comes off the *already-shrunk* price, not the original — so the cuts compound, they don't add.
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Hint 2 of 2
Flip from "how much off" to "how much you still pay." After 30% off you pay 0.70; the next 20% off leaves 0.80 *of that*. Chain the survivors by multiplying.
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Approach: multiply the fractions of price still paid
- Track what *survives* each cut, not what's removed. 30% off leaves you paying 0.70 of the original; another 20% off leaves 0.80 of *that*.
- Multiply the survivors: 0.70 × 0.80 = 0.56. You pay 56%, so the real discount is 100% − 56% = 44% — less than the claimed 50%.
- *Why this transfers:* successive percent changes *multiply* their keep-factors; they never add. The second 20% is only 20% of the smaller 70%, which is why two discounts always total *less* than their sum.
Another way — add the pieces removed:
- First cut removes 30%, leaving 70%. The second cut removes 20% of that 70%, i.e. 0.20 × 70% = 14%.
- Total removed: 30% + 14% = 44%.
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