Problem 13 · 2002 AMC 8
Hard
Geometry & Measurement
volume-scalingspatial-reasoning
For his birthday, Bert gets a box that holds 125 jellybeans when filled to capacity. A few weeks later, Carrie gets a larger box full of jellybeans. Her box is twice as high, twice as wide, and twice as long as Bert's. Approximately how many jellybeans did Carrie get?
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Answer: E — About 1000.
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Hint 1 of 2
"Twice as big" tricks the eye — picture how many of Bert's small boxes actually pack inside Carrie's. Imagine a 2-by-2-by-2 stack of them.
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Hint 2 of 2
That stack is 2 across, 2 deep, 2 tall: the volume (and the jellybeans) multiplies by 2 × 2 × 2, not by 2.
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Approach: scaling all three dimensions cubes the factor
- Visualize Carrie's box as a 2×2×2 stack of Bert's boxes — 8 of them fit inside. Doubling all three dimensions multiplies volume by 2 × 2 × 2 = 8.
- So Carrie gets about 8 × 125 = 1000 jellybeans.
- *Why this transfers:* when every length scales by k, area scales by k² and volume by k³. "Double" never means double for area or volume — and a quick gut-check, 250 (×2), would be the classic wrong trap.
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