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2020 AMC 10B Problem 2

2020 AMC 10B Problem 2AMC 10 Step by Step

Name: Date: September 13, 2026

Solid Geometry · difficulty 1 of 5 · about 1 min · position P1-10

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?

A)
$24$
B)
$25$
C)
$28$
D)
$40$
E)
$45$

Answer

Key insight

A cube of side 2 has volume 8, not 2 or 4; the total is 5 times 1 plus 5 times 8.

Solution

A cube of side $1$ has volume $1^3 = 1$ , so Carl's five cubes contribute $5 \cdot 1 = 5$ .

A cube of side $2$ has volume $2^3 = 8$ , so Kate's five cubes contribute $5 \cdot 8 = 40$ .

The total volume is $5 + 40 = 45$ .

The answer is $\boxed{\textbf{(E)}\ 45}$ .

Why this works

Volume scales with the cube of the side length, so doubling the side multiplies the volume by $8$ . Whenever a problem mixes "side length" and "volume," convert every object to the quantity actually asked for before adding.

The trap

Using side length instead of volume for Kate's cubes (5 + 10 = 15) or squaring instead of cubing (5 + 20 = 25).

Common mistakes

Using side length instead of volume for Kate's cubes (5 + 10 = 15) or squaring instead of cubing (5 + 20 = 25).
Reading "5 cubes of side 2" as a single $2 \times 2 \times 5$ block or forgetting Carl's cubes and answering $40$ .

Techniques

Set up the equation/formula and compute; no special trick needed