There are 35 students in a hostel. If the number of students increases by 7, then the expenses of the mess increased by Rs. 42 per day while the average expenditure per head diminishes by Re. 1. Find the original expenditure of the mess.

Answer

To solve the problem, we need to find the original expenditure of the mess given the conditions about the increase in students and the change in expenses. Let's break it down step by step.

### Step 1: Define Variables
Let the original expenditure of the mess be \( E \) (in Rs.).

### Step 2: Calculate Original Average Expenditure
Since there are 35 students, the average expenditure per head can be calculated as:
\[ \text{Average expenditure per head} = \frac{E}{35} \]

### Step 3: New Number of Students
If the number of students increases by 7, the new number of students becomes:
\[ 35 + 7 = 42 \]

### Step 4: New Expenditure
According to the problem, the expenses of the mess increase by Rs. 42 per day. Therefore, the new expenditure is:
\[ E + 42 \]

### Step 5: New Average Expenditure
The new average expenditure per head with 42 students is:
\[ \text{New average expenditure per head} = \frac{E + 42}{42} \]

### Step 6: Set Up the Equation
The problem states that the average expenditure per head diminishes by Rs. 1. Therefore, we can set up the equation:
\[ \frac{E + 42}{42} = \frac{E}{35} - 1 \]

### Step 7: Cross Multiply to Eliminate Fractions
To eliminate the fractions, we can cross-multiply:
\[ 35(E + 42) = 42E - 42 \times 35 \]

### Step 8: Expand and Simplify
Expanding both sides gives:
\[ 35E + 1470 = 42E - 1470 \]
Now, rearranging the equation:
\[ 1470 + 1470 = 42E - 35E \]
\[ 2940 = 7E \]

### Step 9: Solve for E
Now, divide both sides by 7 to find \( E \):
\[ E = \frac{2940}{7} = 420 \]

### Conclusion
The original expenditure of the mess is Rs. 420.

---