Question:A cube has edge length a, so its volume is a^3 cubic units.A collection consists of 11 identical cubes of...
GMAT Advanced Math : (Adv_Math) Questions
- A cube has edge length \(\mathrm{a}\), so its volume is \(\mathrm{a}^3\) cubic units.
- A collection consists of 11 identical cubes of edge length \(\mathrm{a}\).
- If 5 cubes are removed from the collection, which expression represents the total volume of the remaining cubes?
1. TRANSLATE the problem information
- Given information:
- Each cube has edge length \(\mathrm{a}\) and volume \(\mathrm{a^3}\)
- Collection starts with 11 identical cubes
- 5 cubes are removed
- Need to find volume of remaining cubes
2. INFER the solution approach
- "Remaining cubes" means we subtract removed cubes from original total
- Total volume = (number of remaining cubes) × (volume per cube)
- We need to find how many cubes are left, then multiply by \(\mathrm{a^3}\)
3. SIMPLIFY to find the remaining cubes
- Remaining cubes = \(\mathrm{11 - 5 = 6}\) cubes
4. SIMPLIFY to find total remaining volume
- Total volume = \(\mathrm{6}\) cubes × \(\mathrm{a^3}\) per cube = \(\mathrm{6a^3}\)
Answer: B (\(\mathrm{6a^3}\))
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misinterpret what the problem is asking for or get confused by the multiple pieces of information presented.
They might focus on the wrong numbers (like using 5 or 11 instead of 6) or forget that each cube has volume \(\mathrm{a^3}\), not just \(\mathrm{a}\). This confusion about which numbers to use and what operations to perform may lead them to select Choice A (\(\mathrm{6a}\)) by multiplying the correct number of remaining cubes (6) by the edge length (\(\mathrm{a}\)) instead of the volume (\(\mathrm{a^3}\)).
Second Most Common Error:
Poor SIMPLIFY execution: Students correctly identify they need 6 cubes but make errors with the exponent notation.
They might think that 6 cubes with volume \(\mathrm{a^3}\) each gives \(\mathrm{6a^6}\) (incorrectly adding exponents instead of keeping \(\mathrm{a^3}\) for each cube), or they get confused about how exponents work in multiplication. This may lead them to select Choice D (\(\mathrm{6a^6}\)).
The Bottom Line:
This problem tests whether students can carefully track multiple quantities (original amount, amount removed, volume per object) and correctly apply the volume formula while handling algebraic notation with exponents.