Coach hire: seats, staff ratios and cost comparison
A secondary school is hiring coaches for a trip to an outdoor activity centre. There are 178 pupils and 12 staff going. Each coach seats 48 people. Regulations require at least one member of staff on each coach, and the organiser's own plan is to have at least 2 staff on every coach and no more than 3 empty seats in total across the convoy. Company A charges £420 per coach plus £3.50 per person for insurance. Company B charges £480 per coach with insurance included, and gives 5% off the total bill when four or more coaches are booked. The parent-teacher association has pledged £2,000 towards transport, and the school does not want to use its main budget. Eight more pupils ask to join after the first booking is planned. The organiser needs to decide the number of coaches, the company, and what to charge pupils.
1. Work out the minimum number of coaches needed and check the seating and staffing conditions.
Read the worked answer
Total people = 178 + 12 = 190.
Coaches needed = 190 ÷ 48 = 3.958..., so round up to 4 coaches.
Seats = 4 × 48 = 192, so 192 − 190 = 2 empty seats. This meets the 'no more than 3 empty seats' aim.
Staffing: 12 staff across 4 coaches gives 3 per coach, which is above both the legal minimum of 1 and the organiser's target of 2 per coach. So 4 coaches works on every count.
Judgement point: rounding 3.958 down to 3 coaches would leave 190 − 144 = 46 people without a seat, so the answer must be rounded up even though the decimal is close to 4.
2. Calculate the cost with each company, recommend one, and work out a fair charge per pupil given the £2,000 pledge.
Read the worked answer
Company A: 4 × £420 = £1,680 for coaches. Insurance = 190 × £3.50 = £665. Total = £1,680 + £665 = £2,345.
Company B: 4 × £480 = £1,920. The 5% discount applies because 4 coaches are booked: £1,920 × 0.95 = £1,824. Insurance is included. Total = £1,824.
Difference: £2,345 − £1,824 = £521, so Company B is cheaper.
Shortfall against the £2,000 pledge = £1,824 − £2,000 → actually the pledge exceeds the Company B cost, so the trip is covered with £176 to spare. If the pledge is treated as a fixed contribution and the balance is charged to pupils, the amount needed is £1,824 − £2,000 = −£176, i.e. nothing is needed, and the surplus could go towards the activity centre.
However, if the PTA money is capped at £2,000 and other costs are added, a contingency charge of £1 per pupil (raising £178) would cover a shortfall of up to £176. State clearly which assumption is being used; this is the judgement the question is testing.
3. Eight more pupils join. Recalculate the number of coaches and compare the two companies again, including what this means for pupil charges.
Read the worked answer
New total = 178 + 8 + 12 = 198.
Coaches = 198 ÷ 48 = 4.125, so round up to 5 coaches. That gives 5 × 48 = 240 seats and 42 empty seats, so the 'no more than 3 empty seats' aim cannot be met with 48-seat coaches; the organiser must accept the wastage or look at a mixed coach-and-minibus arrangement.
Staffing: 12 staff across 5 coaches is not possible at 2 each (that needs 10, so it works: some coaches get 2, others 3). The legal minimum of 1 per coach is satisfied.
Company A: 5 × £420 = £2,100; insurance 198 × £3.50 = £693; total £2,793.
Company B: 5 × £480 = £2,400; 5% discount applies, so £2,400 × 0.95 = £2,280.
Company B is still cheaper, by £513.
Shortfall = £2,280 − £2,000 = £280 across 186 pupils, which is £280 ÷ 186 = £1.51 per pupil to the nearest penny, or £2 per pupil if a whole-pound charge is preferred (raising £372, leaving about £92 as contingency).
The step change from 4 to 5 coaches costs £456 with Company B, which is why the organiser should confirm numbers before booking.