The Museum Trip: Coaches, Free Places and a Budget
Year 6 is planning a museum trip. There are 87 pupils and 9 adults going. Each coach seats 52 people, costs £240 for the day, and must be booked whole - the company will not sell half a coach. Museum entry is £7.25 per pupil and £9.50 per adult. The museum's group offer gives one free adult place for every 10 paying pupils; leftover pupils in a part group of fewer than 10 do not earn a free adult. The head teacher has promised two things: the trip will not cost more than £1,200, and every person going will have a seat.
1. How many coaches must be booked? Show why both one coach and two coaches are worth considering, and why only one of them is allowed.
Read the worked answer
Total people: 87 + 9 = 96.
One coach: 52 seats < 96, so at least 44 people would have no seat. This breaks the head teacher's promise, so one coach is impossible - even though it would be cheapest. Cheapness cannot override a stated constraint.
Two coaches: 2 × 52 = 104 seats ≥ 96, so everyone is seated with 8 seats spare.
Three coaches: not needed; 104 seats already covers 96.
So the minimum is 2 coaches.
Note the trap: 96 ÷ 52 = 1.85, which rounds up to 2 - here the rounding-up instinct happens to give the right answer, but the reasoning must be 'seats ≥ people', not 'round the division up', because rounding the wrong way would leave people standing outside the museum.
2. Work out the total cost and decide whether the head teacher's £1,200 limit is met. Show clearly how many adults actually pay.
Read the worked answer
Coaches: 2 × £240 = £480
Free adult places: 87 pupils gives 87 ÷ 10 = 8 full groups of ten, so 8 adults go free. (The remaining 7 pupils earn nothing.)
Paying adults: 9 − 8 = 1, at £9.50 = £9.50
Pupils: 87 × £7.25. Break it up: 87 × 7 = 609, and 87 × 0.25 = 21.75, giving £630.75.
Total: 480 + 630.75 + 9.50 = £1,120.25
£1,120.25 < £1,200, so the limit is met with £79.75 to spare.
The judgement point is the free-place rule: a careless reading ('one free adult for every 10 pupils') can tempt you to round 87 up to 90 and free 9 adults. That would cost the school £9.50 it does not have to pay and would hide the fact that the spare money is small.
3. Five pupils drop out. Does the number of free adult places change? Is the trip still within budget, and can the school now afford a third coach?
Read the worked answer
New numbers: 82 pupils, still 9 adults, still 2 coaches (82 + 9 = 91 ≤ 104).
Free places: 82 ÷ 10 = 8 full groups, so still 8 free adults, 1 paying. The free-place count does not change, because losing 5 pupils never crosses a multiple of 10 downward past 80.
New cost:
- Coaches: £480
- Pupils: 82 × £7.25 = 574 + 20.50 = £594.50
- Paying adult: £9.50
- Total: 480 + 594.50 + 9.50 = £1,084.00
Still under £1,200, with £116 to spare.
Third coach? No. A third coach adds £240, taking the total to £1,324, which breaks the £1,200 limit. The £116 surplus is real but not nearly enough. Useful judgement: a saving of £36.25 looks like 'spare money', but spare money is only spendable against what is actually being bought.