AMC 8 · 2002 · #13

Grade 6 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Fraction / Expression Sum Evaluation

volume-rectangular-prismexponentssimilar-figuresspatial-visualization dimensional-analysisidentify-subproblems ↑ Prerequisites: multi-digit-arithmeticvolume-rectangular-prism

📏 Short solution 💡 2 insights

Problem

Bert's box holds exactly 125 jellybeans. Carrie's box has the same shape but twice the length, twice the width, and twice the height. About how many jellybeans does Carrie's box hold?

Pick an answer.

(A)

250

(B)

500

(C)

625

(D)

750

(E)

1000

AMC 8 2002 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Solve an Easier Related Problem

We are not told Bert's actual dimensions, only that the count is 125. Tool #9 (Easier Related Problem) lets us pick convenient dimensions that match: a 5 × 5 × 5 cube of jellybeans gives exactly 125. Once the easy case is concrete, Carrie's box becomes 10 × 10 × 10 and we can multiply directly. Tool #7 (Identify Subproblems) handles the three independent doublings — length doubles, width doubles, height doubles — so the volume factor is 2 × 2 × 2 = 8, which explains why 125 × 8 = 1000 no matter what shape Bert's box really is.

1STEP 1

Pick easy dimensions: since 125 = 5 × 5 × 5, model Bert's box as a 5 × 5 × 5 cube.

Bert: 5 × 5 × 5 = 125

2STEP 2

Double each side: Bert's 5 × 5 × 5 becomes a 10 × 10 × 10 box.

Carrie: (2 × 5) × (2 × 5) × (2 × 5) = 10 × 10 × 10

3STEP 3

Multiply out Carrie's box: 10 × 10 × 10 gives 1000 jellybeans.

10 × 10 × 10 = 1000 → (E)

4STEP 4

Check with scaling: doubling three sides multiplies volume by 2 × 2 × 2 = 8, so Carrie holds 8 times Bert's count.

125 × 8 = 1000

Answer

1000

Two checks agree. First, the concrete 5 × 5 × 5 → 10 × 10 × 10 model gives 1000. Second, the scaling argument gives 125 × 8 = 1000 without needing to know Bert's shape. The other choices are easy to rule out: 250 corresponds to doubling only one dimension, 500 to doubling two, and 625 = 125 × 5 has no geometric meaning here. Choice (E) 1000 is the only count consistent with doubling all three dimensions.

💡Key takeaway

Doubling one side doubles the volume, doubling two sides quadruples it, doubling all three sides multiplies it by 8. That is why Carrie's box holds 125 × 8 = 1000 jellybeans.

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