There are 60 students in a hostel. If the number of students increases by 20, then the expenses of the mess increased by Rs. 120 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 changes in the number of students and the expenses. Let's break it down step by step.

### Step 1: Define Variables
Let the original average expenditure per head be \( x \) rupees.

### Step 2: Calculate Original Expenditure
With 60 students, the total original expenditure of the mess can be calculated as:
\[ \text{Original Expenditure} = 60 \times x \]

### Step 3: New Number of Students
If the number of students increases by 20, the new number of students becomes:
\[ 60 + 20 = 80 \text{ students} \]

### Step 4: New Average Expenditure per Head
According to the problem, the average expenditure per head diminishes by Re. 1, so the new average expenditure per head becomes:
\[ x - 1 \]

### Step 5: Calculate New Total Expenditure
The total expenditure for 80 students at the new average expenditure is:
\[ \text{New Expenditure} = 80 \times (x - 1) \]

### Step 6: Set Up the Equation
We know that the new expenditure is also equal to the original expenditure plus the increase in expenses:
\[ 80(x - 1) = 60x + 120 \]

### Step 7: Simplify the Equation
Expanding the left side:
\[ 80x - 80 = 60x + 120 \]

### Step 8: Rearranging the Equation
Now, let's move all terms involving \( x \) to one side and the constant terms to the other side:
\[ 80x - 60x = 120 + 80 \]
\[ 20x = 200 \]

### Step 9: Solve for \( x \)
Dividing both sides by 20 gives:
\[ x = 10 \]

### Step 10: Calculate Original Expenditure
Now, substituting \( x \) back into the original expenditure formula:
\[ \text{Original Expenditure} = 60 \times 10 = 600 \text{ rupees} \]

Thus, the original expenditure of the mess is **Rs. 600**.

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