AMC 10 · 2020 · #2

Grade 5 geometry-3d

Archetype Arithmetic Expression Decomposition └ Sub-archetype Weighted Group Aggregation

exponentsvolume-rectangular-prism identify-subproblemsdimensional-analysis ↑ Prerequisites: exponentsmulti-digit-arithmetic

📏 Short solution 💡 1 insight

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Problem

Carl owns 5 unit cubes (side 1) and Kate owns 5 cubes of side 2. Find the combined volume of all 10 cubes.

Pick an answer.

(A)

(B)

(C)

(D)

(E)

AMC 10 2020 problem © Mathematical Association of America (MAA AMC). Reproduced for educational use.

How to solve

Strategy Identify Subproblems

Tool #7 (Subproblems): split the question into two simple sub-jobs — find Carl's total, find Kate's total, then add. Tool #10 (Physical): if a student can't picture it, stacking unit cubes makes it obvious that a side-2 cube holds 8 unit cubes. Tool #8 (Units): the answer should be in cubic units; tracking units keeps us from confusing side length with volume.

1STEP 1

Subproblem 1 — Carl: each side-1 cube is 1 × 1 × 1 = 1, so five give 5 × 1 = 5.

5 × (1 × 1 × 1) = 5 × 1 = 5

2STEP 2

Subproblem 2 — Kate: each side-2 cube is 2 × 2 × 2 = 8, so five give 5 × 8 = 40.

5 × (2 × 2 × 2) = 5 × 8 = 40

3STEP 3

Add the two subtotals for the combined volume: 5 + 40 = 45 cubic units → (E).

5 + 40 = 45 → (E)

4STEP 4

Unit check: side length in units makes each s³ cubic units, and cubic + cubic stays cubic units, so 45 is a valid volume.

units³ + units³ = units³

Answer

Kate's cubes are 8 times bigger by volume than Carl's, so they should dominate the total — and indeed 40 (Kate) vs 5 (Carl) shows that. Total 45 is the only answer choice large enough to fit, supporting (E).

💡Key takeaway

This AMC 10 problem only needs Grade 5 “volume of a cube = side × side × side” you already know — 5(1) + 5(8) = 45.

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