Problem 10 · 2023 AMC 8
Medium
Fractions, Decimals & Percents
fraction-to-decimalpercent-multiplier
Harold made a plum pie to take on a picnic. He was able to eat only 14 of the pie, and he left the rest for his friends. A moose came by and ate 13 of what Harold left behind. After that, a porcupine ate 13 of what the moose left behind. How much of the original pie still remained after the porcupine left?
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Answer: D — 1/3.
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Hint 1 of 2
Don't track what each animal ate — track what each one left behind. Then the leftovers chain together.
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Hint 2 of 2
Each ‘a fraction of what's left’ means multiply. Harold leaves 3/4, the moose leaves 2/3 of that, the porcupine leaves 2/3 of that — so multiply 3/4 × 2/3 × 2/3.
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Approach: multiply the 'leftover' fractions
- The trap is to subtract each bite from the whole pie — but the moose eats a third of what's left, not a third of the whole pie. So work with what each animal leaves, and chain those leftovers by multiplying (‘of’ means ×).
- Harold leaves 34; the moose leaves 23 of that; the porcupine leaves 23 of that.
- 34 × 23 × 23 = 1236 = 13. This transfers: repeated ‘a fraction of what remains’ (discounts on discounts, evaporation, decay) always multiplies the survival fractions.
Another way — twelve slices (MAA):
- Cut the pie into 12 equal slices. Harold eats 3, leaving 9. Moose eats 13 of 9 = 3, leaving 6. Porcupine eats 13 of 6 = 2, leaving 4.
- 4 of 12 = 1/3.
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