The volume of rectangular prism A is 30 cubic inches. What is the volume, in cubic inches, of rectangular prism...

1. TRANSLATE the problem information

• Given information:
  • Volume of prism A = 30 cubic inches
  • Prism B has half the length of A → \(\mathrm{length = l/2}\)
  • Prism B has three times the width of A → \(\mathrm{width = 3w}\)
  • Prism B has twice the height of A → \(\mathrm{height = 2h}\)

2. INFER the strategic approach

• Key insight: We don't need to find the actual dimensions of prism A
• We can work directly with the volume relationship since \(\mathrm{V = lwh}\) for both prisms
• The scaling factors will multiply together to give us the final volume ratio

3. SIMPLIFY the volume calculation for prism B

• Volume of B = (new length) × (new width) × (new height)
• \(\mathrm{Volume\ of\ B = (l/2) \times (3w) \times (2h)}\)
• \(\mathrm{Volume\ of\ B = (1/2) \times 3 \times 2 \times l \times w \times h}\)
• \(\mathrm{Volume\ of\ B = 3 \times (l \times w \times h)}\)
• Since \(\mathrm{lwh = 30}\) for prism A: \(\mathrm{Volume\ of\ B = 3 \times 30 = 90}\)

Answer: C) 90

Why Students Usually Falter on This Problem

Most Common Error Path:

Weak TRANSLATE skill: Students misinterpret "half the length" as \(\mathrm{2l}\) instead of \(\mathrm{l/2}\), thinking "half" means "multiply by 2" rather than "divide by 2."

When they calculate \(\mathrm{(2l)(3w)(2h) = 12lwh = 12(30) = 360}\), they get confused because 360 isn't among the answer choices, leading them to guess or abandon their systematic approach.

Second Most Common Error:

Poor INFER reasoning: Students try to work backwards to find the individual dimensions of prism A first, getting bogged down in unnecessary calculations like trying to solve for l, w, and h separately when \(\mathrm{lwh = 30}\).

This approach leads to confusion with multiple unknowns and may cause them to select Choice A (30), thinking the volumes should be the same, or to give up and guess randomly.

The Bottom Line:

The key insight is recognizing that scaling factors multiply together - you don't need the individual dimensions, just their product. Students who miss this strategic approach often get lost in unnecessary complexity or make translation errors with the fractional scaling.