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Question:

One part of a hostel’s monthly expenses is fixed, and the other part is proportional to the number of its boarders. The hostel collects ₹ 1600 per month from each boarder. When the number of boarders is 50, the profit of the hostel is ₹ 200 per boarder, and when the number of boarders is 75, the profit of the hostel is ₹ 250 per boarder. When the number of boarders is 80, the total profit of the hostel, in INR, will be 

Explanation:

Total cost = Fixed cost + Variable cost

Variable cost = k × b
where, b = number of boarders and k = variable cost per boarder.

Total cost = FC + kb

Total collection = 1600b

Case 1: Total profit = 200 × 50 = 1600 × 50 – (FC + k × 50)   …(1)
Case 2: Total profit = 250 × 75 = 1600 × 75 – (FC + k × 75)   …(2)

(2) - (1), we get
18750 – 10000 = 1600 × 25 - 25k
⇒ k = 1250 and FC = 7,500 

Now, total profit when there are 80 boarders = 1600 × 80 – (7500 + 1250 × 80) = 20,500

Alternately,
Total cost when there are 50 students = (1600 – 200) × 50 = 70,000
Total cost when there are 75 students = (1600 – 250) × 75 = 1,01,250

Due to 25 students cost increases by (101250 - 70000) = 31250
∴ Due to 5 students cost increases by = 31250/5 = 6250

∴ Total cost when there are 80 students = 1,01,250 + 6250 = 1,07,500
Total revenue for 80 students = 1600 × 80 = 1,28,000

∴ Profit = 1,28,000 – 1,07,500 = 20,500.

Hence, option (d).