A factory produces metal rods, and the length of each rod, in centimeters, is measured and then divided by 3...
GMAT Algebra : (Alg) Questions
A factory produces metal rods, and the length of each rod, in centimeters, is measured and then divided by 3 to determine the number of equal sections that can be cut from each rod. The minimum number of sections recorded was 4, and the maximum number of sections recorded was 8. Which inequality represents all possible values of \(\mathrm{L}\), where \(\mathrm{L}\) is the original length, in centimeters, of a rod produced by this factory?
- \(4 \leq \mathrm{L} \leq 8\)
- \(12 \leq \mathrm{L} \leq 24\)
- \(\frac{\mathrm{L}}{3} \geq 8\)
- \(\mathrm{L} \geq 12\)
1. TRANSLATE the problem information
- Given information:
- Metal rods have length L (in centimeters)
- Length is divided by 3 to get number of equal sections
- Minimum sections recorded = 4
- Maximum sections recorded = 8
- What this tells us: Number of sections = \(\mathrm{L/3}\)
2. TRANSLATE the constraints into mathematical form
- Since minimum sections = 4 and maximum sections = 8:
\(\mathrm{4 \leq (number\ of\ sections) \leq 8}\) - Substituting our relationship:
\(\mathrm{4 \leq L/3 \leq 8}\)
3. INFER the solution approach
- To find the range of L, we need to isolate L in the inequality
- Since L is divided by 3, we multiply all parts by 3 to undo this operation
- Multiplying by positive 3 preserves the inequality directions
4. SIMPLIFY to find L
- Multiply all parts of \(\mathrm{4 \leq L/3 \leq 8}\) by 3:
\(\mathrm{4 \times 3 \leq L \leq 8 \times 3}\)
\(\mathrm{12 \leq L \leq 24}\)
Answer: (B) \(\mathrm{12 \leq L \leq 24}\)
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE reasoning: Students may confuse which variable represents what, thinking the inequality should be directly about L rather than \(\mathrm{L/3}\).
They might misread and think "minimum length is 4, maximum length is 8" instead of understanding that these numbers refer to sections after dividing by 3. This leads them to incorrectly conclude that \(\mathrm{4 \leq L \leq 8}\).
This may lead them to select Choice (A) (\(\mathrm{4 \leq L \leq 8}\)).
Second Most Common Error:
Inadequate INFER skill: Students correctly set up \(\mathrm{4 \leq L/3 \leq 8}\) but then don't recognize they need to multiply through by 3 to solve for L.
They might think the answer is \(\mathrm{L/3 \geq 4}\) or get confused about how to manipulate compound inequalities, leading them to focus on partial information like just the minimum constraint.
This may lead them to select Choice (D) (\(\mathrm{L \geq 12}\)) because they only consider the lower bound.
The Bottom Line:
This problem requires careful translation of the multi-step process (length → divide by 3 → sections) and then working backwards through algebraic manipulation to find the original length constraints.