A geologist estimates that the volume of a slab of granite is greater than 12.7 cubic feet but less than...
GMAT Algebra : (Alg) Questions
A geologist estimates that the volume of a slab of granite is greater than \(12.7\) cubic feet but less than \(15.7\) cubic feet. The geologist also estimates that the slab of granite weighs \(165\) pounds per cubic foot of volume. Which inequality represents this situation, where \(\mathrm{w}\) represents the estimated total weight, in pounds, of the slab of granite?
1. TRANSLATE the problem information
- Given information:
- Volume is between 12.7 and 15.7 cubic feet: \(12.7 \lt \mathrm{V} \lt 15.7\)
- Weight per cubic foot = 165 pounds
- Need to find inequality for total weight w
- What this tells us: Total weight depends on volume multiplied by weight per cubic foot
2. INFER the mathematical relationship
- To find total weight: multiply volume by weight per cubic foot
- Formula: \(\mathrm{w} = \mathrm{V} \times 165\)
- Since we have bounds on V, we need to apply the same bounds to w
3. TRANSLATE the inequality relationship
- If \(12.7 \lt \mathrm{V} \lt 15.7\)
- Then \(12.7 \times 165 \lt \mathrm{V} \times 165 \lt 15.7 \times 165\)
- Which means: \(165(12.7) \lt \mathrm{w} \lt 165(15.7)\)
Answer: C. \(165(12.7) \lt \mathrm{w} \lt 165(15.7)\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Students misunderstand the relationship between volume, weight per cubic foot, and total weight. They might think that weight per cubic foot should be added to or subtracted from the volume bounds, rather than multiplied.
This conceptual confusion leads them to think: "If volume is between 12.7 and 15.7, and weight per cubic foot is 165, then total weight is between 165 + 12.7 and 165 + 15.7." This may lead them to select Choice B (\(165 + 12.7 \lt \mathrm{w} \lt 165 + 15.7\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Students recognize they need to do something with 165 and the volume bounds, but get confused about which operation to use. They might try division, thinking they need to find how many "units of 165" fit into the volume range.
This leads to backwards thinking and may cause them to select Choice D (\(165/15.7 \lt \mathrm{w} \lt 165/12.7\)).
The Bottom Line:
This problem tests whether students understand that "pounds per cubic foot" means you multiply volume by that rate to get total pounds. The word "per" indicates a rate that requires multiplication, not addition or division.