Question

There were 35 students in a hostel. Due to admission of 7 new students, the expenses of the mess were increased by Rs. 42 per day, while the average expenditure per head diminished by Rs. 1. What was the original expenditure of the mess?
A. 425
B. 410
C. 420
D. 430

Answer 1

Hint: We take the original expenditure of the mess as a variable. We find the total expenditure for a day. From old and new expenditure, we find a linear equation of x. we solve it to find the solution of the problem.

Complete step by step answer:
Let the original expenditure of the mess was Rs. x.
There were 35 students at the start.
So, the average expenditure per head will be a division of x by 35 which is $\dfrac{x}{35}$ Rs.
Due to admission of 7 new students, the expenses of the mess were increased by Rs. 42 per day, while the average expenditure per head diminished by Rs. 1.
So, the average expenditure per head will be 1 less than $\dfrac{x}{35}$ Rs which is $\left( \dfrac{x}{35}-1 \right)$ Rs.
After the admission of 7 new students the total count became $35+7=42$.
So, 42 students have an average expenditure per head of $\left( \dfrac{x}{35}-1 \right)$ Rs.
So, the total expenditure of 1 day for 42 students will be $42\times \left( \dfrac{x}{35}-1 \right)$ Rs.
It’s also given that the expenses of the mess were increased by Rs. 42 per day.
This means the present expenditure in 1 day is 42 Rs. extra than the previous one.
We put this notion into a linear equation to find
$x+42=42\times \left( \dfrac{x}{35}-1 \right)$.
The addition or increment of Rs. 42 to the previous x is equal to the new one which is $42\times \left( \dfrac{x}{35}-1 \right)$.
We need to solve the equation and get
$\begin{align}
  & x+42=42\times \left( \dfrac{x}{35}-1 \right) \\
 & \Rightarrow x+42=\dfrac{42x}{35}-42 \\
 & \Rightarrow 84=\dfrac{42x}{35}-x \\
\end{align}$
We solve the L.C.M to find the value of x.
$\begin{align}
  & \dfrac{42x}{35}-x=\dfrac{42x-35x}{35}=84 \\
 & \Rightarrow \dfrac{7x}{35}=84 \\
 & \therefore x=\dfrac{84\times 35}{7}=84\times 5=420 \\
\end{align}$
So, the original expenditure of the mess was Rs. 420.

The correct option is (C).
Note:
We need to remember there are two parameters which are daily expenditure and per head basis expenditure. We need to be careful about the relation which is based on what per capita. The total number of total strengths is crucial in finding the equation.