Question

There were $35$ students in a hostel. Due to the admission of $7$ new student, the expenses of the mess were increased by Rs. $42$ per day, while the average expenditure per head diminished by Re $1$. What was the original expenditure of the mess?

• A. $425$
• B. $410$
• C. $420$
• D. $430$

Answer 1

The correct option is C $420$

Let the average expenditure of $35$ students be $\overline{x}$

After adding $7$ new students, the average was decreased by $1$ rupee.

The expenses of the mess were increased by $42$ rupees.

Therefore, $\overline{x}=\frac{\text{expenses of 35 students}}{35}$

$⟹\text{expenses of 35 students}=35\overline{x}$ ------(1)

After adding $7$ new students, the average is

$\overline{x}-1=\frac{\text{expenses of 35 students}+42}{35+7}$ ------(2)

substituting (1) in (2) we get

$\overline{x}-1=\frac{35\overline{x}+42}{42}$

$⟹42\overline{x}-42=35\overline{x}+42$

$⟹7\overline{x}=84$

$⟹\overline{x}=12$ ----(3)

substituting (3) in (1) we get

$⟹\text{expenses of 35 students}=35\overline{x}=35\times 12=420$

Therefore, the original expenditure of the mess is $420$