In city X, buses are always punctual. How long, at the most, will Mr Roy have to wait for a bus? Statement I: Mr Roy arrives at the bus stand at 9:00 a.m. Statement II: There is a bus at 10:00 a.m. and possibly another bus even earlier.

Introduction / Context:
This question examines understanding of the phrase at the most and the idea of punctuality in public transport. Mr Roy arrives at a bus stand in city X, where buses are always punctual. We have information about his arrival time and about at least one scheduled bus. The goal is to determine the maximum possible waiting time for Mr Roy, given that a bus is guaranteed at a particular time and may possibly arrive earlier.

Given Data / Assumptions:

• Buses in city X are always punctual, so they arrive exactly at their scheduled times.
• Statement I: Mr Roy arrives at the bus stand at 9:00 a.m.
• Statement II: There is a bus at 10:00 a.m., and there may possibly be another bus even earlier than 10:00 a.m.
• We are asked for the maximum time Mr Roy might have to wait.
• The question is about the worst case waiting time, not the minimum.

Concept / Approach:
The key concept is to distinguish between minimum and maximum waiting time. The phrase at the most refers to the longest duration Mr Roy might need to wait, given the information. Since a bus is definitely scheduled at 10:00 a.m. and buses are punctual, Mr Roy is guaranteed to catch that bus at the latest. If there is an earlier bus, his actual waiting time will be less, but the maximum possible waiting time is determined by the latest guaranteed bus that he might need to take.

Step-by-Step Solution:
Step 1: From statement I, we know that Mr Roy is present at the bus stand at 9:00 a.m. Step 2: Statement I alone does not tell us when the next bus is due, so we cannot determine his waiting time from this statement alone. Step 3: From statement II, we know that there is a bus at 10:00 a.m., and there might also be another bus before 10:00 a.m., but the time of any earlier bus is not specified. Step 4: Statement II alone does not mention when Mr Roy arrives, so it cannot give his waiting time. Thus statement II alone is not sufficient. Step 5: Now combine both statements. Mr Roy arrives at 9:00 a.m., and buses are punctual. We are told there is definitely a bus at 10:00 a.m. Step 6: Because of punctuality, the 10:00 a.m. bus will arrive exactly at 10:00 a.m. So in the worst case, if there is no earlier bus that he can catch, he will wait from 9:00 a.m. to 10:00 a.m. Step 7: The waiting time from 9:00 a.m. to 10:00 a.m. is 1 hour. Step 8: If there is an earlier bus, Mr Roy will catch it and wait less than 1 hour. But the question asks for the maximum possible waiting time, which is based on the latest guaranteed bus time compatible with his arrival. Step 9: Therefore, the maximum time Mr Roy might need to wait is 1 hour.

Verification / Alternative check:
Assume there is an earlier bus at 9:20 a.m. In that case, his waiting time is 20 minutes. Assume another scenario where there is no earlier bus, and the only bus after 9:00 a.m. is at 10:00 a.m. Then his waiting time is 60 minutes. Since the question asks at the most, we consider the scenario in which he has to wait the longest while remaining consistent with the given information. The latest guaranteed bus is at 10:00 a.m., so 1 hour is the maximum waiting time.

Why Other Options Are Wrong:
Option a is wrong because, without any bus schedule, statement I alone cannot give a waiting time. Option b is wrong because, without knowing when Mr Roy arrives, statement II alone cannot produce a waiting duration. Option c is wrong because neither statement on its own allows computation of waiting time; we need both arrival time and bus time. Option e is wrong because, with both statements combined, we clearly get a maximum waiting time of 1 hour.

Common Pitfalls:
Some students confuse the phrase at the most with at least. At least would refer to the minimum guaranteed wait, while at the most is about the worst case. Others ignore the possibility of an earlier bus and think that punctuality implies only the 10:00 a.m. bus. However, the phrase possibly another bus even earlier clearly allows earlier departures. The safe approach is to base the maximum waiting time on the latest guaranteed bus after the arrival time, ignoring optional earlier buses for the maximum calculation.

Final Answer:
Using both statements together, the maximum time that Mr Roy may have to wait for a bus is 1 hour. Correct option: The data in both statements I and II together are sufficient to answer the question.