There are 320 students in a hostel. If the number of students decreases by 10, then the expenses of the mess decreased by Rs. 58 per day while the average expenditure per head increases by 20 paise. Find the original expenditure of the mess.

Answer

To solve the problem step by step, we need to define some variables and use the information given in the question.

### Step 1: Define Variables
Let:
- \( x \) = original average expenditure per head (in Rs)
- \( E \) = original total expenditure of the mess (in Rs)

### Step 2: Express Total Expenditure
Since there are 320 students, the original total expenditure can be expressed as:
\[ E = 320x \]

### Step 3: New Conditions After Decrease
When the number of students decreases by 10, the new number of students becomes:
\[ 320 - 10 = 310 \]

The new average expenditure per head increases by 20 paise (which is Rs. 0.20), so the new average expenditure per head is:
\[ x + 0.20 \]

### Step 4: New Total Expenditure
The new total expenditure after the decrease in the number of students is:
\[ E' = 310(x + 0.20) \]

### Step 5: Relate the Expenditures
According to the problem, the total expenditure decreases by Rs. 58:
\[ E' = E - 58 \]

Substituting the expressions for \( E \) and \( E' \):
\[ 310(x + 0.20) = 320x - 58 \]

### Step 6: Expand and Simplify the Equation
Expanding the left side:
\[ 310x + 62 = 320x - 58 \]

Now, rearranging the equation:
\[ 62 + 58 = 320x - 310x \]
\[ 120 = 10x \]

### Step 7: Solve for \( x \)
Dividing both sides by 10:
\[ x = 12 \]

### Step 8: Find Original Expenditure
Now, substituting \( x \) back to find the original total expenditure \( E \):
\[ E = 320x = 320 \times 12 = 3840 \]

### Final Answer
The original expenditure of the mess is Rs. 3840.

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