Question

An aeroplane can carry a maximum of 200 passengers. $A$ profit of $₹1000$ is made on each executive class ticket and a profit of $₹600$ is made on each economy class ticket. The airline reserves atleast 20 seats for executive class. However, atleast 4 times as many passengers prefer to travel by economy class than by the executive class. Then, to maximise the profit, number of tickets of executive class and economy class are respectively

• A. 40 and 160
• B. 160 and 40
• C. 140 and 60
• D. 60 and 140

Answer 1

Answer: (A)

Let $x$ passengers travel by executive class and $y$ passengers travel by economy class. We construct the following table

Classes	Number of tickets	Profit (in ₹)
Executive	x	1000x
Economy	y	600y
Total	x+y	1000x+600y

So, our problem is to maximise
$Z=1000x+600y$ ...(i)
Subject to the constraints are
$x+y\le 200$...(ii)
$x\ge 20$...(iii)
$y-4x\ge 0\iff y\ge 4x$...(iv)
$x\ge 0,y\ge 0$...(v)
Firstly, draw the graph of the line $x+y=200$.

x	0	200
y	200	0

Putting $(0,0)$ in the inequality $x+y\le 200$, we have
$0+0\le 200\Rightarrow 0\le 200$(which is true)
So, the half plane is towards the origin.
Secondary, draw the graph of the line $y=4x$.

x	0	20
y	0	80

Putting $(10,0)$ in the inequality $y\ge 4x$, we have
$0\ge 4\times 10\Rightarrow 0\ge 40$(which is false)
So, the half plane is towards $X$-axis.

Thirdly, draw the graph of the line $x=20$.
Putting $(0,0)$ in the inequality $x\ge 20$, we have
$0\ge 20$(which is false)
So, the half plane is away from the origin.
Since, $x,y\ge 0$
So, the feasible region lies in the first quadrant.
On solving the equations, we get $A(20,80),B(40,160)$ and $C(20,180)$.
$\therefore$ Feasible region is $ABCA$.
The corner points of the teasible region are $A(20,80)$, $B(40,160)$ and $C(20,180)$.
The values of $Z$ at these points are as follows

Corner points	$1000x+600y$
$A(20,80)$	68000
$B(40,160)$	$136000\to$ Maximum
$C(20,180)$	128000

Thus, the maximum value of 7 is 136000 at $R(40,160)$. Thus, 40 tickets of executive class and 160 tickets of economy class should be sold to maximise the profit and the maximum profit is ₹ 136000 .