An aeroplane can carry a maximum of 200 passengers. A profit of Rs. 400 is made on each first class ticket and a profit of Rs. 300 is made on each second class ticket. The airline reserves at least 20 seats for first class. However, at least four times as many passengers prefer to travel by second class then by first class. Determine how many tickets of each type must be sold to maximise profit for the airline. Form an LPP and solve it graphically.

Explanation

To solve the problem, we need to formulate a Linear Programming Problem (LPP), define the constraints, and then solve it graphically. Here’s a step-by-step solution:

### Step 1: Define the Variables
Let:
- \( x \) = number of first class tickets sold
- \( y \) = number of second class tickets sold

### Step 2: Formulate the Objective Function
The profit from selling first class tickets is Rs. 400 per ticket, and from second class tickets, it is Rs. 300 per ticket. Therefore, the objective function to maximize profit \( Z \) is:
\[ Z = 400x + 300y \]

### Step 3: Define the Constraints
1. The total number of passengers cannot exceed 200:
\[ x + y \leq 200 \]
2. At least 20 seats must be reserved for first class:
\[ x \geq 20 \]
3. At least four times as many passengers prefer to travel by second class than by first class:
\[ y \geq 4x \]
4. Non-negativity constraints:
\[ x \geq 0, \quad y \geq 0 \]

### Step 4: Graph the Constraints
To graph the constraints, we will convert each inequality into an equation:

1. \( x + y = 200 \)
2. \( x = 20 \)
3. \( y = 4x \)

Next, we plot these equations on a graph:

- The line \( x + y = 200 \) intersects the axes at (200, 0) and (0, 200).
- The line \( x = 20 \) is a vertical line at \( x = 20 \).
- The line \( y = 4x \) passes through the origin and has a slope of 4.

### Step 5: Identify the Feasible Region
The feasible region is determined by the area that satisfies all constraints. It will be bounded by the lines we plotted, and we will find the corner points of this region.

### Step 6: Find the Corner Points
The corner points of the feasible region can be found by solving the equations of the lines:
1. Intersection of \( x + y = 200 \) and \( x = 20 \):
\[ 20 + y = 200 \implies y = 180 \quad \text{(Point A: (20, 180))} \]
2. Intersection of \( x + y = 200 \) and \( y = 4x \):
\[ x + 4x = 200 \implies 5x = 200 \implies x = 40 \implies y = 160 \quad \text{(Point B: (40, 160))} \]
3. Intersection of \( y = 4x \) and \( x = 20 \):
\[ y = 4(20) = 80 \quad \text{(Point C: (20, 80))} \]

### Step 7: Evaluate the Objective Function at Each Corner Point
Now we evaluate \( Z = 400x + 300y \) at each corner point:
1. At Point A (20, 180):
\[ Z = 400(20) + 300(180) = 8000 + 54000 = 62000 \]
2. At Point B (40, 160):
\[ Z = 400(40) + 300(160) = 16000 + 48000 = 64000 \]
3. At Point C (20, 80):
\[ Z = 400(20) + 300(80) = 8000 + 24000 = 32000 \]

### Step 8: Determine the Maximum Profit
The maximum profit occurs at Point B (40, 160) with a profit of Rs. 64000.

### Conclusion
To maximize profit, the airline must sell:
- 40 first class tickets
- 160 second class tickets