There were 42 students in a hostel. Due to the admission of 13 new students, the expenses of mess increase by Rs 31 per day while the average expenditure per head diminished by Rs 3. What was the original expenditure of the mess ?

Options

A. 633
B. 583
C. 623
D. 533

Answer

To solve the problem, let's break it down step by step.

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

### Step 2: Calculate the Original Average Expenditure
The original number of students is 42. Therefore, the average expenditure per head can be expressed as:
\[ A = \frac{E}{42} \]

### Step 3: Determine the New Situation
After admitting 13 new students, the total number of students becomes:
\[ 42 + 13 = 55 \]

### Step 4: Calculate the New Expenditure
According to the problem, the expenses of the mess increase by Rs 31 per day. Therefore, the new expenditure can be expressed as:
\[ E + 31 \]

### Step 5: Calculate the New Average Expenditure
The new average expenditure per head after the increase in students and expenses is:
\[ \text{New Average} = \frac{E + 31}{55} \]

### Step 6: Set Up the Equation for Average Expenditure
We know that the average expenditure per head decreased by Rs 3. Thus, we can set up the equation:
\[ \frac{E + 31}{55} = A - 3 \]

### Step 7: Substitute for \( A \)
From Step 2, we know \( A = \frac{E}{42} \). Substitute this into the equation:
\[ \frac{E + 31}{55} = \frac{E}{42} - 3 \]

### Step 8: Clear the Fractions
To eliminate the fractions, we can multiply through by \( 2310 \) (the least common multiple of 55 and 42):
\[ 2310 \left(\frac{E + 31}{55}\right) = 2310 \left(\frac{E}{42} - 3\right) \]

This simplifies to:
\[ 42(E + 31) = 55E - 6930 \]

### Step 9: Expand and Rearrange
Expanding the left side:
\[ 42E + 1302 = 55E - 6930 \]
Rearranging gives:
\[ 6930 + 1302 = 55E - 42E \]
\[ 8232 = 13E \]

### Step 10: Solve for \( E \)
Now, divide both sides by 13:
\[ E = \frac{8232}{13} = 634 \]

### Conclusion
The original expenditure of the mess was Rs 634.

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