Choosing a Bus Operator
A school is comparing three bus operators for an excursion. Operator P charges a fixed booking fee of ₹2,400 plus ₹18 per student. Operator Q charges ₹1,800 fixed plus ₹22 per student. Operator R charges ₹2,700 fixed plus ₹16 per student. The excursion committee has 120 students confirmed and a maximum transport budget of ₹4,500. A parent volunteer claims that because Q has the highest per-student rate, it must be the most expensive for any group size. The committee also wants to know whether the choice would change if 60 more students join, making 180. All charges are in rupees, no other fees apply, and the committee will not split the group between operators. The number of students is the only variable; teachers travel free and are not included in the per-student charge.
1. For what number of students do operators P and Q charge the same amount? Solve the equations and interpret the result.
Read the worked answer
Let x be the number of students. Cost P = 2400 + 18x and cost Q = 1800 + 22x. Set them equal: 2400 + 18x = 1800 + 22x, so 600 = 4x and x = 150. At x = 150, each charges 2400 + 18(150) = 5100, which is also 1800 + 22(150). For fewer than 150 students, Q is cheaper because its fixed fee is lower; for more than 150 students, P becomes cheaper because its per-student rate is lower. The break-even point is 150 students, not a universal 'Q is most expensive' rule.
2. With 120 students and a budget of ₹4,500, which operators are feasible and which is cheapest? Test the parent volunteer's claim.
Read the worked answer
For x = 120: P = 2400 + 18(120) = 2400 + 2160 = ₹4,560; Q = 1800 + 22(120) = 1800 + 2640 = ₹4,440; R = 2700 + 16(120) = 2700 + 1920 = ₹4,620. Only Q is within the ₹4,500 budget. Q is also the cheapest, saving ₹120 compared with P and ₹180 compared with R. The parent's claim is false for this group: Q has the highest per-student rate, yet it is cheapest because its fixed booking fee is much lower. The fixed fee matters for a small group.
3. If 60 more students join, making 180, which operator is cheapest and by how much? Does the committee's choice change, and what minimum budget is needed?
Read the worked answer
For x = 180: P = 2400 + 18(180) = 2400 + 3240 = ₹5,640; Q = 1800 + 22(180) = 1800 + 3960 = ₹5,760; R = 2700 + 16(180) = 2700 + 2880 = ₹5,580. R is now cheapest. It saves ₹60 compared with P and ₹180 compared with Q. The committee's choice changes from Q to R, so the minimum transport budget needed is ₹5,580. This shows why comparing only the per-student rate is unreliable; the fixed fee and the group size together decide the cheapest option.