Two Halls, One Licence: Choosing a Venue
The Hillview Youth Project needs a venue for its games night. The Community Centre charges a fixed €60 booking fee plus €12 for every hour the hall is open, including setup time. The Parish Hall charges a fixed €30 plus €18 per hour. The programme itself runs for 3 hours, the volunteers need 4 hours beforehand to set up, and 1 hour afterwards to clean and lock up. The Parish Hall is licensed only until 10 p.m. and the group cannot start before 4 p.m., so no booking there can run beyond 6 hours. Refreshments cost a flat €40 whichever venue is used. The group expects 60 paying attendees and the treasurer has a budget of €200 for the whole night, refreshments included. A committee member has looked only at the fixed fees and argued that the Parish Hall is obviously the cheaper option, while another has looked only at the hourly rates and argued the opposite.
1. Write a cost function for each hall in terms of the number of hours h, find the break-even number of hours, and explain what it means.
Read the worked answer
Step 1 – Build the functions. Community Centre: C = 60 + 12h. Parish Hall: C = 30 + 18h. In each case the constant is the fixed fee (intercept) and the coefficient of h is the hourly rate (slope).
Step 2 – Set equal. 60 + 12h = 30 + 18h → 60 − 30 = 18h − 12h → 30 = 6h → h = 5.
Step 3 – Check. At h = 5: Community Centre 60 + 60 = €120; Parish Hall 30 + 90 = €120. Equal, as required.
Step 4 – Interpret. The Parish Hall saves €30 up front but costs €6 more per hour, so it only wins for short bookings. For h < 5 the Parish Hall is cheaper; for h > 5 the Community Centre is cheaper; at exactly 5 hours they cost the same. The break-even point is where the extra fixed cost of the Community Centre is exactly repaid by its lower hourly charge.
2. Using the actual programme (4 hours setup, 3 hours activity, 1 hour clear-up), decide which hall should be booked and show the full cost including refreshments against the €200 budget.
Read the worked answer
Step 1 – Hours required. h = 4 + 3 + 1 = 8 hours.
Step 2 – Check the licence constraint before costing. The Parish Hall allows a maximum of 6 hours (4 p.m. to 10 p.m.). An 8-hour booking is therefore not possible there, regardless of price. A cost comparison that ignores this is answering a question the group cannot actually choose.
Step 3 – Cost the feasible option. Community Centre: 60 + 12 × 8 = 60 + 96 = €156. Add refreshments €40 → €196.
Step 4 – Compare with budget. €200 − €196 = €4 spare. It fits, but only just.
Step 5 – Sensitivity. If setup could be compressed to 2 hours, h = 6: Community Centre 60 + 72 = €132; Parish Hall 30 + 108 = €138. The Community Centre is still cheaper, because 6 > 5, the break-even point. So the decision is robust to the setup-time uncertainty, but the Parish Hall only becomes the cheaper choice if the whole evening could be squeezed into under 5 hours - which the 3-hour programme plus clear-up makes impossible.
3. The treasurer wants a door charge that covers the night. Calculate it, and state one risk to the plan.
Read the worked answer
Step 1 – Total to recover. Venue €156 + refreshments €40 = €196.
Step 2 – Divide by attendees. 196 ÷ 60 = €3.267 per head.
Step 3 – Round sensibly. You cannot charge €3.27 at a door, and rounding down to €3.00 gives 60 × 3 = €180, a €16 shortfall. Round up to €3.50: 60 × 3.50 = €210, a surplus of €14. Half-euro steps are practical for a youth club; €4.00 would give €240 but may deter attendance.
Step 4 – Risk. The whole calculation assumes 60 paying attendees. If only 45 come, revenue is 45 × 3.50 = €157.50 against €196 of cost - a €38.50 loss, and the venue cost is fixed regardless of numbers. Mitigations: sell tickets in advance so numbers are known before the hall is confirmed, or recruit a free setup crew to cut h from 8 to 6 and save €24, or ask a local shop to sponsor the refreshments. The key modelling point is that one side of the equation is fixed and the other is per-head, so attendance is the variable that carries the risk.