Concert ticket sales and an unrealistic target
A school hall has 320 seats for a charity concert. Tickets cost €8 for adults and €5 for students. Before the final week, 260 tickets have been sold and the takings are €1,690. The organisers need to know how many adult and student tickets were sold in advance, because they must print a summary for the sponsor. They also have a target of €2,300 total income. All remaining seats could still be sold, but the organisers are unsure whether to advertise mainly to adults or to students. The hall manager warns that no more than 60 tickets remain and that all seats are in one price band only for the final sales push. The sponsor will ask whether the target is realistic under the current prices, and if not, what the maximum possible income would be.
1. Using simultaneous equations, find how many adult and student tickets were sold in advance. Show the equations and check your answer.
Read the worked answer
Let a be the number of adult tickets and s the number of student tickets sold in advance.
Equations: a + s = 260 8a + 5s = 1690
Multiply the first equation by 5: 5a + 5s = 1300
Subtract this from the second equation: (8a + 5s) - (5a + 5s) = 1690 - 1300 3a = 390 a = 130
Then s = 260 - 130 = 130.
Check: 8(130) + 5(130) = 1040 + 650 = 1690. This matches the takings, so 130 adult and 130 student tickets were sold in advance.
2. All 60 remaining seats are sold. Determine the maximum possible total income and whether the €2,300 target can be reached. Give a recommendation.
Read the worked answer
Let x be the number of remaining seats sold to adults. Then 60 - x are sold to students.
Total income = 1690 + 8x + 5(60 - x) = 1690 + 8x + 300 - 5x = 1990 + 3x
To maximise income, make x as large as possible: x = 60. Then total income = 1990 + 3(60) = 1990 + 180 = €2,170.
Even with all remaining seats sold to adults, the income is €2,170, which is €130 short of €2,300. The target cannot be reached under the current prices and seat limit.
Recommendation: either lower the target to about €2,170, raise ticket prices, or find additional sponsorship. Advertising mainly to adults helps, but it cannot close the gap on its own.