Pricing the school drama tickets
A community drama club in Pokhara plans a fundraising show in a 400-seat hall. Last year tickets were NPR 200 and average attendance was 300. A quick survey suggests that for every NPR 10 increase in ticket price, attendance falls by 10. The hall and production cost NPR 25,000 plus NPR 80 per audience member for refreshments and programmes. The club wants to maximise profit, not just ticket revenue. They are considering prices of NPR 250, NPR 290 and NPR 320. A member argues that because the hall is not full, lowering the price to NPR 180 would fill more seats and raise profit. The treasurer wants an algebraic model, the break-even attendance, and a recommendation. The club also wants at least NPR 15,000 profit for a library corner.
1. Write the profit as a function of the number of NPR 10 increases, and find the price and attendance that maximise profit.
Read the worked answer
Let x be the number of NPR 10 increases.
Price = 200 + 10x; attendance = 300 − 10x.
Revenue = (200 + 10x)(300 − 10x) = 60,000 + 1,000x − 100x².
Cost = 25,000 + 80(300 − 10x) = 49,000 − 800x.
Profit P(x) = 11,000 + 1,800x − 100x².
This is a downward parabola, maximum at x = −1800 ÷ (2 × −100) = 9.
So price = NPR 290, attendance = 210, and maximum profit = 11,000 + 16,200 − 8,100 = NPR 19,100.
2. Compare the profits at NPR 250, NPR 290, NPR 320 and NPR 180. Is the member's claim about NPR 180 correct?
Read the worked answer
Use x = (price − 200) ÷ 10.
- NPR 250: x = 5, P = 11,000 + 9,000 − 2,500 = NPR 17,500.
- NPR 290: x = 9, P = NPR 19,100.
- NPR 320: x = 12, P = 11,000 + 21,600 − 14,400 = NPR 18,200.
- NPR 180: x = −2, P = 11,000 − 3,600 − 400 = NPR 7,000.
The member is wrong. Lowering the price to NPR 180 fills more seats, but the margin per ticket falls too much. The best of the listed prices is NPR 290.
3. At the recommended price, what attendance is needed to break even, and how much cushion does the model give? Does the club reach its NPR 15,000 profit target?
Read the worked answer
At NPR 290, contribution per attendee = 290 − 80 = NPR 210.
Break-even attendance = 25,000 ÷ 210 ≈ 119.05, so 120 tickets.
The model predicts 210 attendees, a cushion of about 90 tickets over break-even.
For NPR 15,000 profit, total contribution needed = 25,000 + 15,000 = NPR 40,000. Attendance needed = 40,000 ÷ 210 ≈ 190.5, so 191 tickets.
The predicted 210 clears the target, but only by about 19 tickets. If attendance is weaker than the survey suggests, the target is at risk. Recommendation: price at NPR 290 and monitor advance bookings.