Excursion Bus Quotes: Break-Even and Budget Reality
Two bus companies quote for a school excursion. Company P charges a fixed ₹2,500 plus ₹18 per kilometre. Company Q charges a fixed ₹1,800 plus ₹22 per kilometre. The quotes are for a single round trip, and the distance is the total kilometres travelled. The organising teacher has a budget cap of ₹5,000 for transport and is considering destinations 130 km and 190 km away (round-trip distances). A student draws both cost rules on a graph and concludes that 'P is always cheaper for long trips because its line is less steep,' but the graph's vertical axis starts at ₹1,500, which makes the intersection look different from its true position. The teacher also has a handwritten note saying that at 150 km both companies charge exactly the same.
1. Find the distance at which the two companies charge the same amount, and explain which company is cheaper on either side of that distance.
Read the worked answer
Let d be the round-trip distance in km. Set the costs equal:
2,500 + 18d = 1,800 + 22d 700 = 4d d = 175 km.
At 175 km both charge 2,500 + 18(175) = 2,500 + 3,150 = ₹5,650, and also 1,800 + 22(175) = 1,800 + 3,850 = ₹5,650.
For d < 175, Q is cheaper because its fixed charge is lower and the per-km difference has not yet outweighed that. For d > 175, P is cheaper because its lower rate per km gradually wins. The break-even is 175 km.
2. Check the 130 km and 190 km trips against the ₹5,000 cap. State clearly whether either destination is feasible under the cap.
Read the worked answer
At 130 km:
- P: 2,500 + 18 × 130 = 2,500 + 2,340 = ₹4,840
- Q: 1,800 + 22 × 130 = 1,800 + 2,860 = ₹4,660
Both are under ₹5,000, and Q is cheaper by ₹180.
At 190 km:
- P: 2,500 + 18 × 190 = 2,500 + 3,420 = ₹5,920
- Q: 1,800 + 22 × 190 = 1,800 + 4,180 = ₹5,980
Both exceed ₹5,000. The 190 km trip is not feasible under the cap with either quote; the teacher must raise the budget, shorten the distance, or negotiate a new quote.
3. Critique the student's graph conclusion and the handwritten note that both companies charge the same at 150 km.
Read the worked answer
The student is right that P's line is less steep, but wrong to say P is 'always' cheaper for long trips. A lower slope only means P eventually becomes cheaper; before the intersection Q is cheaper. Truncating the vertical axis at ₹1,500 distorts where the lines appear to meet visually, so the graph alone is unreliable without solving the equations.
Check 150 km:
- P: 2,500 + 18 × 150 = 2,500 + 2,700 = ₹5,200
- Q: 1,800 + 22 × 150 = 1,800 + 3,300 = ₹5,100
The note is false: Q is cheaper by ₹100 at 150 km. Equality occurs at 175 km, not 150 km.