Question

Company ABC starts an educational program in collaboration with Institute XYZ. As per the agreement, ABC and XYZ will share profit in $60:40$ ratio. The initial investment of rupee 100{,}000 on infrastructure is borne entirely by ABC whereas the running cost of \rupee 400 per student is borne by XYZ. If each student pays rupee 2000 for the program, find the minimum number of students required to make the program profitable, assuming ABC wants to recover its investment in the very first year and the program has no seat limits.

• A. 63
• B. 84
• C. 105
• D. 157
• E. 167

Hint:

At the {program} break-even point, the profit-sharing ratio (60:40) is irrelevant—the pool is just zero. First balance total revenue against {all} costs (fixed $+$ variable) to get $n=\lceil \tfrac{\text{fixed}}{\text{per-student net}} \rceil$. Here per-student net is $2000-400=1600$.

Answer 1

The Correct Option is A

Step 1: Write revenue and costs for $n$ students.
Revenue $= \rupee 2000 \times n = \rupee 2000n$.
Costs $= \rupee 100{,}000$ (infrastructure, by ABC) $+\ \rupee 400 \times n$ (running, by XYZ).
Thus, total cost $= \rupee(100{,}000 + 400n)$. Step 2: Profit of the {program (before 60:40 split).}
\[ \text{Profit} \;=\; \text{Revenue} - \text{Total Cost} = 2000n - (100{,}000 + 400n) = 1600n - 100{,}000. \] Step 3: Break-even / profitability condition.
For the program to be profitable (and hence ABC’s initial \rupee 100{,}000 to be recovered within the year from the joint profit), \[ 1600n - 100{,}000 \;\ge\; 0 \;\Rightarrow\; n \;\ge\; \frac{100{,}000}{1600} \;=\; 62.5. \] Minimum integer $n=\boxed{63}$.