Averages , Ratio and Proportion Question 6 - CAT 2021
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:
Step-by-Step Explanation
Let the fixed cost components be F and the variable cost per boarder be v. The total cost equation is:
Total Cost = F + vn
Profit = Total Revenue − Total Cost = 1600n − (F + vn) = (1600 − v)n − F
- Set up the linear equations based on boarder counts:
- For n = 50, profit per boarder is ₹200:
Total Profit = 50 × 200 = ₹10,000
50(1600) − 50v − F = 10,000
80,000 − 50v − F = 10,000
F + 50v = 70,000 --- (1)
- For n = 75, profit per boarder is ₹250:
Total Profit = 75 × 250 = ₹18,750
75(1600) − 75v − F = 18,750
120,000 − 75v − F = 18,750
F + 75v = 101,250 --- (2)
- Solve the system:
Subtract equation (1) from equation (2):
25v = 31,250
v = 1,250
Substitute v = 1,250 back into equation (1):
F + 50(1,250) = 70,000
F + 62,500 = 70,000
F = 7,500
- Calculate the total profit for 80 boarders:
Profit for 80 = 1600(80) − 1250(80) − 7,500 = 350(80) − 7,500 = 28,000 − 7,500 = ₹20,500 Correct Option: B