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

• A. 20800
• B. 20200
• C. 20500
• D. 20000

Answer 1

The Correct Option is C

Let the fixed cost be ₹F and the variable cost per boarder be ₹V.

Case 1: When there are 50 boarders

Given: Profit per boarder = ₹200
Fee collected per boarder = ₹1600
Hence, expenditure per boarder = ₹1600 - ₹200 = ₹1400

Total expenditure: 
\( F + 50V = 50 \times 1400 \)
\( F + 50V = 70000 \quad \text{... (1)} \)

Case 2: When there are 75 boarders

Given: Profit per boarder = ₹250
Fee collected per boarder = ₹1600
Hence, expenditure per boarder = ₹1600 - ₹250 = ₹1350

Total expenditure: 
\( F + 75V = 75 \times 1350 \)
\( F + 75V = 101250 \quad \text{... (2)} \)

Subtracting (1) from (2):

\( (F + 75V) - (F + 50V) = 101250 - 70000 \)
\( 25V = 31250 \)
\( V = \frac{31250}{25} = 1250 \)

Substitute \( V = 1250 \) in Equation (2):

\( F + 75 \times 1250 = 101250 \)
\( F + 93750 = 101250 \)
\( F = 101250 - 93750 = 7500 \)

Now, calculate total expenditure for 80 boarders:

\( \text{Expenditure} = F + 80V = 7500 + 80 \times 1250 = 7500 + 100000 = ₹107500 \)

Total revenue from 80 boarders:

\( \text{Revenue} = 80 \times 1600 = ₹128000 \)

Therefore, profit:

\( \text{Profit} = \text{Revenue} - \text{Expenditure} \)
\( = 128000 - 107500 = ₹20500 \)

Final Answer: When there are 80 boarders, the total profit is ₹20500.