Concert Target Feasibility
A school charity concert committee has booked a hall with a strict capacity of 240 people. Tickets cost £8 for adults and £5 for students; children under 12 enter free but still count towards capacity. Fixed costs are £450 for the band, £180 for hall hire and £75 for insurance. The committee expects to sell at least 60 adult tickets and at least 90 student tickets, and it wants to raise at least £1,200 after costs. A parent volunteers to sponsor £100 only if the committee can show the target is achievable without relying on free-child places being sold as paid tickets. The chair also insists that no more than 240 tickets/places are allocated and that the expected minimum sales are met. Before advertising, the committee needs a quick numerical check of whether the £1,200 target is realistic, and if not, what the best achievable profit is under the stated expectations.
1. Let A be adult tickets, S student tickets and C free under-12 places. Write down the profit expression and the constraints, then decide whether the £1,200 target is achievable under the committee's expectations. Show the maximum-revenue check.
Read the worked answer
Let A, S and C be adult, student and free child places. Revenue is 8A + 5S. Total fixed costs are 450 + 180 + 75 = £705, so profit is 8A + 5S − 705. The target needs 8A + 5S − 705 ≥ 1200, i.e. 8A + 5S ≥ 1905. Constraints: A ≥ 60, S ≥ 90, A + S + C ≤ 240, C ≥ 0. To maximise revenue, set C = 0 and use the capacity on the higher-priced adult tickets. With S = 90, A ≤ 150. Maximum revenue = 8(150) + 5(90) = 1200 + 450 = £1650. Maximum profit = 1650 − 705 = £945, which is below £1200. Adding free child places would only reduce the number of paid places, so the target is not achievable under the stated expectations.
2. The chair proposes raising the adult price to £9. Recalculate the maximum profit and state whether the £100 sponsor's condition can now be met.
Read the worked answer
With adult price £9 and the same constraints, maximum revenue occurs at A = 150, S = 90: 9(150) + 5(90) = 1350 + 450 = £1800. Profit = 1800 − 705 = £1095. The £100 sponsor would bring this to £1195, still £5 short of £1200. So the sponsor's condition is still not met.
3. Keeping the original ticket prices, what is the minimum reduction in total costs needed to make the £1,200 target achievable?
Read the worked answer
Maximum revenue is fixed at £1650 by the ticket constraints. To make profit at least £1200, costs must satisfy 1650 − costs ≥ 1200, so costs ≤ £450. Current costs are £705, so the committee needs a reduction of at least 705 − 450 = £255. A smaller reduction leaves the profit below target; a reduction of exactly £255 gives profit £1200.