Question:A manufacturing plant's production efficiency E (in units per hour) decreases over time according to the model E = M...
GMAT Advanced Math : (Adv_Math) Questions
A manufacturing plant's production efficiency \(\mathrm{E}\) (in units per hour) decreases over time according to the model \(\mathrm{E = M - \frac{W}{D + 3}}\), where \(\mathrm{M}\) is the maximum theoretical efficiency, \(\mathrm{W}\) is a waste factor constant, and \(\mathrm{D}\) represents the number of days since the last maintenance. Which equation correctly expresses \(\mathrm{D + 3}\) in terms of \(\mathrm{E}\), \(\mathrm{M}\), and \(\mathrm{W}\)?
- \(\mathrm{D + 3 = \frac{W}{M - E}}\)
- \(\mathrm{D + 3 = \frac{W}{M + E}}\)
- \(\mathrm{D + 3 = \frac{M - E}{W}}\)
- \(\mathrm{D + 3 = \frac{W}{M}}\)
- \(\mathrm{D + 3 = \frac{M + E}{W}}\)
1. TRANSLATE the problem requirements
- Given: \(\mathrm{E = M - \frac{W}{D + 3}}\)
- Find: An expression for \(\mathrm{D + 3}\) in terms of \(\mathrm{E, M, and W}\)
2. SIMPLIFY to isolate the fraction term
- Start with: \(\mathrm{E = M - \frac{W}{D + 3}}\)
- Subtract M from both sides: \(\mathrm{E - M = -\frac{W}{D + 3}}\)
- Multiply both sides by -1: \(\mathrm{M - E = \frac{W}{D + 3}}\)
3. SIMPLIFY using cross multiplication
- From \(\mathrm{M - E = \frac{W}{D + 3}}\)
- Cross multiply: \(\mathrm{(M - E)(D + 3) = W}\)
4. SIMPLIFY to solve for D + 3
- Divide both sides by (M - E): \(\mathrm{D + 3 = \frac{W}{M - E}}\)
- This matches choice (A)
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak SIMPLIFY execution: Sign errors when rearranging the equation. Students often struggle with \(\mathrm{E - M = -\frac{W}{D + 3}}\) and incorrectly write \(\mathrm{E - M = \frac{W}{D + 3}}\), missing the negative sign. This leads to \(\mathrm{D + 3 = \frac{W}{E - M}}\), which would be the reciprocal of choice (C) but isn't among the options.
This leads to confusion and guessing among the available choices.
Second Most Common Error:
Inadequate SIMPLIFY execution: Students correctly get to \(\mathrm{M - E = \frac{W}{D + 3}}\) but then incorrectly cross multiply or make division errors. Some might flip the fraction incorrectly and get \(\mathrm{D + 3 = \frac{M - E}{W}}\), leading them to select Choice (C).
The Bottom Line:
This problem tests careful algebraic manipulation with attention to signs and proper fraction operations. The key insight is recognizing that M - E (not E - M) should be positive in the context, and systematically applying inverse operations to isolate the desired term.