Choosing a Minibus: Two Linear Quotes and a Fixed Budget
Mpho's class is organising a trip to a nature reserve 180 km from school. There are 42 students and 4 teachers, and the school will pay for the teachers. Two hire companies have quoted for a minibus and driver on a return journey of d kilometres:
- Savanna Tours: fixed charge of P600 plus P5.50 per kilometre.
- Delta Coaches: fixed charge of P300 plus P8.00 per kilometre.
Both quotes are for one vehicle that seats all 46 passengers. The class has P1,400 from a sponsored walk, and each student has agreed to contribute P25. The organiser must confirm the booking on Friday. The route is not yet fixed: the direct route is 360 km return, while travelling out by one gate and back by another scenic gate would be 400 km return. Nobody wants to ask parents for more money, but cancelling would lose the deposit already promised to the reserve.
1. Form an equation to find the return distance at which the two quotes cost the same, solve it, and explain what the answer means for a trip of 360 km.
Read the worked answer
Set up. Let d be the return distance in km. Savanna: C = 600 + 5.50d. Delta: C = 300 + 8.00d.
Solve. 600 + 5.50d = 300 + 8.00d → 600 − 300 = 8.00d − 5.50d → 300 = 2.50d → d = 120 km.
Check. At d = 120: Savanna = 600 + 660 = P1,260; Delta = 300 + 960 = P1,260. ✓
Interpretation. At exactly 120 km return the quotes are equal. Delta's higher rate per kilometre (P2.50 more) is offset by its P300 lower fixed charge only while the distance is short: for d < 120 km Delta is cheaper, and for d > 120 km Savanna is cheaper. The fixed charge dominates small trips; the rate dominates long ones.
For 360 km. The distance is well past the break-even point, so Savanna should be cheaper - confirmed by substitution: 600 + 1,980 = P2,580 against 300 + 2,880 = P3,180. Answering "Savanna" without the break-even calculation would be a guess; the break-even distance is what makes the choice defensible if the route changes.
2. Using the 360 km direct route, find the cheaper quote and test it against the class budget. If there is a shortfall, calculate the extra amount each student would have to pay.
Read the worked answer
Quotes at d = 360.
- Savanna: 600 + 5.50 × 360 = 600 + 1,980 = P2,580.
- Delta: 300 + 8.00 × 360 = 300 + 2,880 = P3,180.
Savanna is cheaper by P600, which matches 2.50 × 360 − 300 = 600.
Budget. Sponsored walk P1,400 + students' contributions 42 × P25 = P1,050. Total = P2,450. The teachers' fares are paid by the school, so they add nothing to this fund.
Shortfall. 2,580 − 2,450 = P130.
Extra per student. 130 ÷ 42 = P3.095…, so P3.10 each, which raises 42 × 3.10 = P130.20 - enough, with 20c spare. Rounding down to P3 would raise only P126 and leave the class P4 short, so money must always be rounded up when collecting.
Judgement. The gap is small (about 5% of the total), so the trip is viable; the class should book Savanna and collect P3.10, or run one more small fundraiser. Booking Delta would need P730, i.e. P17.38 per student, for the same journey.
3. The scenic route is 400 km return. Which company is cheaper for that route, and by how much? The class keeps the P2,450 budget: how much further than 360 km could they travel on the direct route before the shortfall doubles?
Read the worked answer
Scenic route, d = 400.
- Savanna: 600 + 5.50 × 400 = 600 + 2,200 = P2,800.
- Delta: 300 + 8.00 × 400 = 300 + 3,200 = P3,500.
Savanna is cheaper by P700. The gap widens with distance because Savanna saves P2.50 on every kilometre once past 120 km: 2.50 × 400 − 300 = 700. ✓
Shortfall at 360 km. 2,580 − 2,450 = P130.
Rate of change. With Savanna, every extra kilometre adds P5.50 to the bill while the budget stays at P2,450, so the shortfall grows by P5.50 per km.
Doubling the shortfall. An extra P130 is needed: 130 ÷ 5.50 = 23.6 km. So the shortfall reaches about P260 at 360 + 23.6 ≈ 384 km return (check: 600 + 5.50 × 384 = P2,712, and 2,712 − 2,450 = P262 ≈ 2 × P130).
Judgement. The scenic route at 400 km would leave a shortfall of 2,800 − 2,450 = P350, i.e. P8.34 per student. The class has roughly 24 km of slack before costs double the current gap, so the scenic option is only affordable with a second fundraiser.