The Overcrowded Coaching Batch
Rafiq runs a coaching centre in Mirpur preparing students for the NU Honours admission test. He has one classroom that seats 60 students and one senior teacher, Mr. Hasan, who can teach at most 4 batches per week. Each batch meets twice weekly. Demand this year is 150 students, and Rafiq has promised every enrolled student a seat. He can hire a second teacher, Ms. Nusrat, who is available for 3 batches per week, but her salary is 18,000 taka monthly against Mr. Hasan's 22,000. Rafiq's total monthly budget for teacher salaries is 35,000 taka. Fees are 1,200 taka per student per month. Students have complained that batches of more than 45 feel crowded and reduce attention.
1. Can Rafiq seat all 150 students within the room's 60-seat limit and the teachers' combined 7-batch capacity? Show the arithmetic and state the binding constraint.
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Each batch can hold at most 60 students. With 7 batches total (4 from Mr. Hasan + 3 from Ms. Nusrat), the theoretical capacity is 7 × 60 = 420 seats, far above 150. But each batch meets twice weekly and a student attends one batch only, so batch count is what matters, not meeting frequency. To seat 150 students at 45 per batch (the comfort threshold), Rafiq needs 150 ÷ 45 ≈ 3.34, i.e. 4 batches. That is well within 7. So the room and teacher capacity are not binding - the binding constraint is the salary budget, examined next.
2. Given the salary budget of 35,000 taka, which teachers can Rafiq actually employ, and what is the maximum number of students he can serve while respecting the 45-student comfort limit?
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Mr. Hasan costs 22,000 and Ms. Nusrat costs 18,000; together they cost 40,000, which exceeds the 35,000 budget by 5,000. So Rafiq can afford only one teacher. With Mr. Hasan alone (4 batches), at 45 per batch he serves 4 × 45 = 180 students - enough for 150. With Ms. Nusrat alone (3 batches), he serves 3 × 45 = 135 - short of 150. Therefore he must retain Mr. Hasan and cannot hire Ms. Nusrat. Maximum servable under the comfort limit is 180, comfortably covering demand. The salary budget, not classroom size or teacher hours, is the binding constraint.
3. Rafiq considers raising fees to 1,500 taka to fund both teachers. Evaluate whether this solves his problem, and identify one risk.
Read the worked answer
At 1,500 taka × 150 students, monthly revenue is 225,000 taka, easily covering the 40,000 combined salary. So the budget constraint is relieved and both teachers can be hired, giving 7 batches and up to 315 seats at 45 each. However, raising fees by 25% may deter price-sensitive students; if enrolment falls below roughly 27 students (40,000 ÷ 1,500), revenue no longer covers both salaries, so the plan depends on holding demand. The judgement call is whether demand is elastic enough to absorb the hike - a decision the exam expects students to reason about, not recall.