Step 1:

Let the first class air tickets and second class tickets sold be $x$ and $y$

Now as the seating capacity of the aeroplane is 200,so $x+y \leq 200$

As 20 tickets for first class are to be reserved ,So we have $x\geq 20$

And as the number of tickets of second class should be at least 4 times that of first class $y \geq 4x$

Profit on sale of $x$ tickets of first class and $y$ tickets of second class $Z=1000x+600y$

Therefore LPP is (i.e) maximize $Z=1000x+600y$ subject to constraints $x+y \leq 200,x \geq 20,y\geq 4x$ and $x,y\geq 0$

Step 2:

Now let us plot the lines on the graph .

$x=y=200,x=20$ and $y=4x$

The region satisfying the inequalities $x+y\leq 200,x\geq 20$ and $y\geq 4x$ is ABC and it is shown in the figure as the shaded portion.

Step 3:

$Z=100x+600y$

The corner points of the feasible region $A(20,180),B(40,160),C(20,80)$

The values of the objective function at these points are as follows:

At the Points $(x,y)$ the value of the objective function subject to $z=1000x+600y$

At $A(20,180)$,value of the objective function $Z=1000x+600y\Rightarrow 1000\times 20+600\times 180=20000+108000=128000$

At $A(40,160)$,value of the objective function $Z=1000x+600y\Rightarrow 1000\times 40+600\times 160=40000+96000=136000$

At $A(20,80)$,value of the objective function $Z=1000x+600y\Rightarrow 1000\times 20+600\times 80=20000+48000=68000$

Step 4:

It is clear that at $B(40,160)$ $Z$ has the maximum value.

Hence $x=40,y=160$

This implies 40 tickets of first class and 160 of second class should be sold to get the maximum profit of Rs.136000.