Arithmetic

Profit & Loss

Medium

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

$20200$

$20800$

$20500$

$20000$

✅ Correct Option: 3

Let's understand what we're dealing with:

Revenue per boarder = Rs. 1600 (fixed for each boarder)

Total expenses = Fixed expenses + Variable expenses per boarder × Number of boarders

Profit per boarder = Revenue per boarder - Expenses per boarder

The fundamental relationship we'll use is:

Total Expenses = Total Revenue - Total Profit

Since we know profit per boarder, we can find total expenses for each scenario.

For 50 boarders:

Profit per boarder = Rs. 200

Expenses per boarder = Rs. 1600 - Rs. 200 = Rs. 1400

Total expenses = 50 × Rs. 1400 = Rs. 70000

For 75 boarders:

Profit per boarder = Rs. 250

Expenses per boarder = Rs. 1600 - Rs. 250 = Rs. 1350

Total expenses = 75 × Rs. 1350 = Rs. 101250

Now we use the fact that Total Expenses = Fixed Expenses (F) + Variable Expenses per boarder (V) × Number of boarders (n)

This gives us two equations:

$F + 50V = 70000$ ... (1)

$F + 75V = 101250$ ... (2)

Subtracting equation (1) from equation (2):

$(F + 75V) - (F + 50V) = 101250 - 70000$

$25V = 31250$

$V = Rs. 1250$

Substituting back into equation (1):

$F + 50(1250) = 70000$

$F + 62500 = 70000$

$F = Rs. 7500$

Total Expenses for 80 boarders:

$= 7500 + (80 × 1250)$

$= 7500 + 100000$

$= Rs. 107500$

Total Revenue for 80 boarders:

$= 80 × 1600 = Rs. 128000$

Total Profit for 80 boarders:

$= 128000 - 107500$

$= Rs. 20500$

Therefore, the total profit when there are 80 boarders is Rs. 20500.

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