A music teacher schedules x 30-minute individual lessons and y 45-minute group sessions each week. The teacher can conduct at...
GMAT Algebra : (Alg) Questions
A music teacher schedules \(\mathrm{x}\) \(\mathrm{30}\)-minute individual lessons and \(\mathrm{y}\) \(\mathrm{45}\)-minute group sessions each week. The teacher can conduct at most \(\mathrm{20}\) sessions in total and wants to provide at least \(\mathrm{12}\) hours of instruction time in that week. Which of the following systems of inequalities in \(\mathrm{x}\) and \(\mathrm{y}\) models these constraints?
1. TRANSLATE the session constraint
- Given information: "at most 20 sessions total"
- x individual lessons + y group sessions ≤ 20 sessions
- This gives us our first inequality: \(\mathrm{x + y \leq 20}\)
2. TRANSLATE the time constraint
- Given information: "at least 12 hours of instruction time"
- Convert to consistent units: 12 hours = \(\mathrm{12 \times 60 = 720}\) minutes
- Each individual lesson = 30 minutes, so x lessons contribute \(\mathrm{30x}\) minutes
- Each group session = 45 minutes, so y sessions contribute \(\mathrm{45y}\) minutes
3. INFER the time inequality setup
- Total instruction time = \(\mathrm{30x + 45y}\) minutes
- "At least 720 minutes" means the total must be greater than or equal to 720
- This gives us: \(\mathrm{30x + 45y \geq 720}\)
4. APPLY CONSTRAINTS to select the answer
- Our system: \(\mathrm{x + y \leq 20}\) and \(\mathrm{30x + 45y \geq 720}\)
- This matches choice (A) exactly
Answer: A
Why Students Usually Falter on This Problem
Most Common Error Path:
Weak TRANSLATE skill: Confusing the direction of inequalities when converting "at most" and "at least" to mathematical symbols.
Students often think "at most 20" means \(\mathrm{x + y \geq 20}\) (backwards) or "at least 12 hours" means \(\mathrm{30x + 45y \leq 720}\) (backwards). This systematic reversal of inequality directions may lead them to select Choice B (\(\mathrm{x + y \geq 20}\) and \(\mathrm{30x + 45y \leq 720}\)).
Second Most Common Error:
Poor TRANSLATE reasoning: Not converting hours to minutes properly or using the wrong time value.
Some students might use 6 hours instead of 12 hours, giving them 360 minutes instead of 720 minutes. This leads them to set up \(\mathrm{30x + 45y \geq 360}\), causing them to select Choice E (\(\mathrm{x + y \leq 20}\) and \(\mathrm{30x + 45y \geq 360}\)).
The Bottom Line:
This problem tests precise translation of English constraints into mathematical inequalities. Success requires careful attention to inequality direction and consistent unit conversion.