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 has $5$ cubes each having side length $1$ , and Kate has $5$ cubes each having side length $2$ . What is the total volume of these $10$ cubes?

Pick an answer.

(A)

(B)

(C)

(D)

(E)

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

Try it yourself first — the explanation is most useful after you’ve attempted it.

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Toolkit + CCSS Solution

Understand

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

Givens: Carl has $5$ cubes, each with side length $1$; Kate has $5$ cubes, each with side length $2$; Volume of a cube of side $s$ is $s \times s \times s = s^3$; Answer choices: (A) $24$, (B) $25$, (C) $28$, (D) $40$, (E) $45$

Unknowns: Total volume of all $10$ cubes added together

Understand

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

Plan

Primary tool: #7 Identify Subproblems

Secondary: #10 Create a Physical Representation, #8 Analyze the Units

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.

Execute — Answer: E

#7 Identify Subproblems 5.MD.C.5 Step 1

Subproblem 1 — Carl's total.
Each of his cubes has volume $1 \times 1 \times 1 = 1$ cubic unit.
Five such cubes give $5 \times 1 = 5$ cubic units.

$$5 \times (1 \times 1 \times 1) = 5 \times 1 = 5$$

💡 Volume of a unit cube is $1$ — Grade 5 “relate volume to multiplication.”

#10 Create a Physical Representation 5.MD.C.5 Step 2

Subproblem 2 — Kate's total.
Each of her cubes has volume $2 \times 2 \times 2 = 8$ cubic units (imagine stacking $8$ unit cubes into a $2 \times 2 \times 2$ block).
Five cubes give $5 \times 8 = 40$.

$$5 \times (2 \times 2 \times 2) = 5 \times 8 = 40$$

💡 Doubling each side multiplies volume by $2 \times 2 \times 2 = 8$ — building it from unit cubes makes this concrete.

#7 Identify Subproblems 4.OA.A.3 Step 3

Add the two subtotals to get the combined volume: $5 + 40 = 45$ cubic units.

$$5 + 40 = 45 \;\Rightarrow\; \textbf{(E)}$$

💡 When pieces don't overlap, total volume is just the sum — Grade 4 multi-step problem.

#8 Analyze the Units 5.MD.C.5 Step 4

Unit check: side lengths are in units, so each $s^3$ is in cubic units.
Adding cubic units to cubic units yields cubic units — the answer $45$ is dimensionally consistent.

$$\text{units}^3 + \text{units}^3 = \text{units}^3$$

💡 If our number is in cubic units, the answer is sensible as a “total volume.”

[1] #7 5.MD.C.5 Subproblem 1 — Carl's total. Each of his cubes has volume $1 \times 1 \times 1 =

[2] #10 5.MD.C.5 Subproblem 2 — Kate's total. Each of her cubes has volume $2 \times 2 \times 2 =

[3] #7 4.OA.A.3 Add the two subtotals to get the combined volume: $5 + 40 = 45$ cubic units.

[4] #8 5.MD.C.5 Unit check: side lengths are in units, so each $s^3$ is in cubic units. Adding c

Review

Reasonableness: 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).

Alternative: Tool #2 (Systematic List) — list every cube's volume: $1,1,1,1,1,8,8,8,8,8$ and add them straight: $5 \cdot 1 + 5 \cdot 8 = 45$. Same result, no formula needed.

CCSS standards used (min grade 5)

4.OA.A.3 Solve multi-step word problems using four operations with whole numbers (Combining two volume subtotals with a single addition $5 + 40 = 45$.)
5.MD.C.5 Relate volume to the operations of multiplication and addition (Using $V = s \times s \times s$ for a cube and adding non-overlapping volumes.)

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

Problem

Toolkit + CCSS Solution

Understand

Plan

Execute — Answer: E

Review

CCSS standards used (min grade 5)

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