A part of monthly hostel charges in a colege are fixed and the remaining depend on the number of days one has taken food in the mess. When a student A takes food for 15 days, he has to pay Rs 1200 as hostel charges whereas a student B, who takes food for 24 days, pays Rs 1560 as hostel charges. Find the fixed charges and the cost of food per day.

Answer

To solve the problem, we will set up a system of linear equations based on the information provided and then solve for the fixed charges and the cost of food per day.

### Step 1: Define Variables
Let:
- \( y \) = fixed charges (in Rs)
- \( x \) = cost of food per day (in Rs)

### Step 2: Set Up Equations
From the problem, we can create two equations based on the information given for students A and B.

1. For student A (who takes food for 15 days and pays Rs 1200):
\[ y + 15x = 1200 \quad \text{(Equation 1)} \]

2. For student B (who takes food for 24 days and pays Rs 1560):
\[ y + 24x = 1560 \quad \text{(Equation 2)} \]

### Step 3: Subtract the Equations
Next, we will subtract Equation 1 from Equation 2 to eliminate \( y \):
\[ (y + 24x) - (y + 15x) = 1560 - 1200 \]
This simplifies to:
\[ 24x - 15x = 360 \]
\[ 9x = 360 \]

### Step 4: Solve for \( x \)
Now, divide both sides by 9:
\[ x = \frac{360}{9} = 40 \]
Thus, the cost of food per day is Rs 40.

### Step 5: Substitute \( x \) back to find \( y \)
Now, we will substitute \( x = 40 \) back into Equation 1 to find \( y \):
\[ y + 15(40) = 1200 \]
\[ y + 600 = 1200 \]
Subtracting 600 from both sides gives:
\[ y = 1200 - 600 = 600 \]
Thus, the fixed charges are Rs 600.

### Final Answer
- Fixed charges \( y = 600 \) Rs
- Cost of food per day \( x = 40 \) Rs