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 a, so volume = \(\mathrm{a^3}\)
- Collection starts with 11 identical cubes
- 5 cubes are removed from the collection
- Need to find total volume of remaining cubes
2. INFER the approach
- To find total volume, I need: (number of remaining cubes) × (volume per cube)
- First find how many cubes remain, then multiply by \(\mathrm{a^3}\)
3. Calculate remaining cubes
- Started with: 11 cubes
- Removed: 5 cubes
- Remaining: \(11 - 5 = 6\) cubes
4. SIMPLIFY to find total volume
- Each cube has volume \(\mathrm{a^3}\)
- Total volume = \(6 \times \mathrm{a^3} = 6\mathrm{a^3}\)
Answer: (B) \(6\mathrm{a^3}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Students incorrectly handle the multiplication \(6 \times \mathrm{a^3}\)
Many students write \(6 \times \mathrm{a^3} = 6\mathrm{a}\) (forgetting the exponent) or \(6 \times \mathrm{a^3} = 6\mathrm{a^6}\) (incorrectly multiplying exponents). The key insight is that when multiplying a number by a variable with an exponent, the number becomes the coefficient while the variable and exponent remain unchanged.
This may lead them to select Choice (A) (\(6\mathrm{a}\)) or Choice (D) (\(6\mathrm{a^6}\))
Second Most Common Error:
Poor TRANSLATE reasoning: Students misinterpret "5 cubes are removed" as addition instead of subtraction
Some students think "5 cubes are removed" means 5 cubes are added to the collection, calculating \(11 + 5 = 16\) remaining cubes, leading to \(16\mathrm{a^3}\).
This may lead them to select Choice (C) (\(16\mathrm{a^3}\))
The Bottom Line:
This problem tests whether students can correctly translate a multi-step word problem and handle basic algebraic multiplication without making exponent errors. The key is staying organized: find the number of objects first, then multiply by the volume per object.