The volumes of two cubes are in a ratio of 1:27. If the smaller cube has a surface area of...

1. TRANSLATE the problem information

• Given information:
  • Volume ratio of two cubes = \(1:27\)
  • Surface area of smaller cube = \(24\) square inches
• What we need: Surface area of larger cube

2. INFER the scaling relationship

• Key insight: For similar 3D shapes, different measurements scale differently
• Volume scales by \(\mathrm{k}^3\), but surface area scales by \(\mathrm{k}^2\)
• If volume ratio is \(1:27\), then side length ratio is \(\sqrt[3]{1} : \sqrt[3]{27} = 1:3\)

3. SIMPLIFY to find the surface area ratio

• Since side length ratio is \(1:3\)
• Surface area ratio = \((1)^2 : (3)^2 = 1:9\)
• This means the larger cube has 9 times the surface area

4. SIMPLIFY to calculate final answer

• Surface area of larger cube = \(9 \times 24 = 216\) square inches

Answer: 216 (Choice C)

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak INFER skill: Students don't recognize the scaling relationships between volume and surface area. They might think that if volumes are in ratio \(1:27\), then surface areas are also in ratio \(1:27\).

Following this incorrect reasoning: Surface area = \(27 \times 24 = 648\)
This may lead them to select Choice D (648)

Second Most Common Error:

Poor TRANSLATE reasoning: Students confuse which scaling factor to use. They might correctly find that side length ratio is \(1:3\), but then incorrectly apply this ratio directly to surface area instead of squaring it.

Following this logic: Surface area = \(3 \times 24 = 72\)
This may lead them to select Choice A (72)

The Bottom Line:

This problem requires understanding that different geometric measurements scale differently. Volume changes by the cube of the scaling factor, while surface area changes by the square of the scaling factor. Missing this relationship makes the problem nearly impossible to solve correctly.