The expense of a boarding house is partly fixed and partly varies with the number of boarders. What is the profit per head per month when there are 80 boarders?

Which one of the following is correct in respect of the Question and the Statements given below?

Statement 1: Each boarder pays $780 a month.

Statement 2: The profit is $108 per head per month when there are 50 boarders.

Statement 3: The profit is $128 per head per month when there are 60 boarders.

Correct option is C

Given:

Find profit per head per month when there are 80 boarders.

Expenses = fixed cost F + variable cost per boarder v × n

Revenue = 780 per boarder (from Statement 1)

So,

Profit per head = $$\frac{780n - (F + vn)}{n} = 780 - \frac{F}{n} - v$$​​

Statement 1:

Each boarder pays $780/month → revenue per head = 780
This is useful, but not sufficient alone since we don't know cost structure.

Statement 2:

Profit per head = 108 when n = 50

$$108 = 780 - \frac{F}{50} - v \quad$$​Equation 1

Statement 3:

Profit per head = 128 when n = 60

$$128 = 780 - \frac{F}{60} - v \quad$$​Equation 2

Solving Equations 1 & 2

Equation 1:

108 = 780 - $$\frac{F}{50} - v$$​​

​$$\Rightarrow \frac{F}{50} + v = 780 - 108 = 672$$..........(i)​

Equation 2:

​$$128 = 780 - \frac{F}{60} - v$$​​

​$$\Rightarrow \frac{F}{60} + v = 780 - 128 = 652$$ ..........(ii)​

Subtracting Eq(2) from Eq(1):

=>$$F=\left( \frac{F}{50} - \frac{F}{60} \right) = 672 - 652 = 20$$​​

​$$\Rightarrow F \left( \frac{1}{50} - \frac{1}{60} \right) = 20$$​​

​$$\Rightarrow F \left( \frac{6 - 5}{300} \right) = 20$$​​

​$$\Rightarrow \frac{F}{300} = 20$$​​

​$$\Rightarrow F = 6000$$​​

Substitute F = 6000 into Eq(1):

​$$\frac{6000}{50} + v = 672$$​​

​$$\Rightarrow 120 + v = 672$$​​

​$$\Rightarrow v = 552$$​​

Now, calculating profit per head when n = 80:

Profit/head = $$780 - \frac{6000}{80} - 552 = 780 - 75 - 552 = {153}$$​​

Statements 2 and 3 are sufficient to compute F and v, and hence profit for any number of boarders.

Statement 1 (revenue per head) is helpful but not required, since it was already used implicitly in the equations.

Thus, the correct option is (c) All the three statements together are sufficient. 

Alternate method:

Concept Used: 

Total expense = x + xy

x = Fixed cost

y = variable 

Solution:

Statement 1: Each boarder pays $780 a month.

780 x → Total received

Statement 2: The profit is $108 per head per month when there are 50 boarders.

Profit = $ 180

=> Total profit = $$108 \times 50$$​​
Expense = x + 50 y

780 $$\times$$ 50 - (x + 50 y) = 108$$\times$$ 50

39000 - x - 50 xy  = 5400

x + 50 xy = 33600......(i)

Statement 3: The profit is $128 per head per month when there are 60 boarders.

x + 60 y = 780 $$\times$$ 60 - 128 $$\times$$ 60

=> x + 60 y = 39120......(ii)

So, we can find the value of x and y by solving the equations and statement I used in the making of equations

So, All the three statements together are sufficient to give the answer.