Minibus Firm Comparison
A school sports department is comparing two minibus firms for away fixtures. Firm A charges a £45 booking fee plus £0.55 per mile. Firm B charges a £20 booking fee plus £0.80 per mile. Each quote is for a round trip, and the department expects most round trips to be between 30 and 70 miles. The school has set a strict budget of £75 per trip, and the business manager will not approve a firm that goes over that limit for any planned trip. Twelve fixtures are planned: eight are 40-mile round trips and four are 70-mile round trips. The head of sport wants to use the cheaper firm for every trip but also wants to know whether the 70-mile trips break the per-trip budget. A colleague suggests negotiating a flat £5 discount per booking with Firm B rather than changing firms. The department needs a clear algebraic comparison before booking.
1. Form expressions for the cost of a round trip of m miles with each firm. Solve to find the mileage at which the two costs are equal, and state which firm is cheaper for the planned 30–70 mile range.
Read the worked answer
Let m be the round-trip mileage. Firm A: cost = 45 + 0.55m. Firm B: cost = 20 + 0.80m. Set equal: 45 + 0.55m = 20 + 0.80m → 25 = 0.25m → m = 100. For m < 100, Firm B is cheaper; for m > 100, Firm A is cheaper. Check m = 40: A = 67, B = 52, so B is cheaper. Therefore throughout the planned 30–70 mile range, Firm B is cheaper.
2. Using Firm B, determine the greatest round-trip distance that stays within the £75 per-trip budget. Check the 70-mile trips and state the financial consequence.
Read the worked answer
Using Firm B, solve 20 + 0.80m ≤ 75 → 0.80m ≤ 55 → m ≤ 68.75 miles. A 70-mile round trip costs 20 + 0.80(70) = 20 + 56 = £76, which is £1 over the per-trip budget. For the planned fixtures, eight 40-mile trips cost 8 × 52 = £416 and four 70-mile trips cost 4 × 76 = £304, total £720. The 70-mile trips breach the per-trip limit by £1 each, total £4, even though the overall total is not capped.
3. Evaluate the colleague's £5 discount proposal with Firm B. Does it bring every planned trip within the per-trip budget, and what is the new total cost?
Read the worked answer
With a £5 discount per booking, Firm B becomes 15 + 0.80m. At 70 miles: 15 + 56 = £71, within £75. At 40 miles: 15 + 32 = £47. Total cost = 8 × 47 + 4 × 71 = 376 + 284 = £660, a saving of £60 compared with £720. The discount makes every planned trip comply with the per-trip budget, so it is the better operational choice if the discount is confirmed in writing.