An auditorium has 300 seats. A premium ticket costs Rs. 500 and a regular ticket costs Rs. 200, and at most 100 premium tickets can be sold for a show. If all the tickets sold are paid for, the maximum amount that can be collected from one show is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → Rs. 90000
Let x premium and y regular tickets be sold. Then
Maximise \(Z = 500x + 200y\)
subject to \(x + y \le 300\), \(x \le 100\), \(x \ge 0\), \(y \ge 0\)
The line \(x = 100\) meets \(x + y = 300\) at (100, 200). The corner points are (0, 0), (100, 0), (100, 200) and (0, 300). Values of Z:
\(Z(0, 0) = 0\)
\(Z(100, 0) = 50000\)
\(Z(100, 200) = 50000 + 40000 = 90000\)
\(Z(0, 300) = 60000\)
So the maximum collection is Rs. 90000, with 100 premium and 200 regular tickets, which fills all the seats.
Why the other options are incorrect
Option (1) Rs. 50000 — Incorrect. This is from 100 premium tickets alone, leaving 200 seats empty.
Option (2) Rs. 60000 — Incorrect. This is from 300 regular tickets, which is less than Z at (100, 200).
Option (4) Rs. 150000 — Incorrect. This needs 300 premium tickets, but at most 100 can be sold.