It is now 8:00 p.m. When can Hemant get the next bus for Ramnagar from Dhanpur? Consider the following statements: I. Buses for Ramnagar leave the bus stand after every 30 minutes until 10:00 p.m. II. Fifteen minutes ago, one bus had left the bus stand for Ramnagar.

Introduction / Context:
This data sufficiency problem involves bus departure times from Dhanpur to Ramnagar. The current time is 8:00 p.m., and Hemant wants to know when he can catch the next bus. Two statements give the departure frequency and the time of a recent departure. The challenge is to determine whether any one statement or both together allow us to find the exact time of the next bus.

Given Data / Assumptions:

• The current time for Hemant is exactly 8:00 p.m.
• Statement I: Buses leave every 30 minutes until 10:00 p.m., but the starting time of the schedule is not specified.
• Statement II: A bus left for Ramnagar fifteen minutes ago.
• All times mentioned refer to the same day and bus route, and the every 30 minutes pattern is regular and continuous during service hours.

Concept / Approach:
The key is to recognise that knowing only the frequency of bus departures is not enough without an anchor time. Similarly, knowing only one recent departure time does not guarantee knowledge of the repeating pattern unless frequency is known. When both frequency and one departure time are known, we can generate the full schedule and identify the next bus after a given current time.

Step-by-Step Solution:
Step 1: Consider statement II alone. Fifteen minutes ago, at 7:45 p.m., a bus left Dhanpur for Ramnagar. Without any information about the frequency of departures, we cannot infer whether the next bus is in 15 minutes, 30 minutes, one hour, or some other interval. Statement II alone is therefore not sufficient. Step 2: Consider statement I alone. Buses leave every 30 minutes until 10:00 p.m., but the statement does not specify from which exact time this pattern starts. For example, the bus times might be 7:30, 8:00, 8:30, and so on, or they might be 7:45, 8:15, 8:45, and so on. Without the starting point, we cannot determine whether a bus is due at 8:00 p.m., 8:15 p.m., or some other time, so statement I alone is not sufficient. Step 3: Combine statements I and II. From statement II, we know a bus left at 7:45 p.m. From statement I, we know that buses leave every 30 minutes. Therefore, the departure times are 7:45 p.m., 8:15 p.m., 8:45 p.m., and so on, continuing until service stops at 10:00 p.m. Step 4: Determine the next bus after 8:00 p.m. Given that the departures are at 7:45 p.m. and then 8:15 p.m., the next departure after 8:00 p.m. is at 8:15 p.m. Step 5: Since the combined information from both statements uniquely fixes the schedule and gives Hemant the next bus time as 8:15 p.m., we conclude that both statements are necessary and sufficient.

Verification / Alternative check:
To verify, imagine trying different starting times for the 30 minute schedule without statement II. Many different patterns satisfy statement I, so the next bus after 8:00 p.m. cannot be uniquely determined. Conversely, using only statement II, we can construct many different repeating patterns that include a departure at 7:45 p.m. but differ in interval length. Only when we enforce both the 30 minute interval and the specific departure at 7:45 p.m. do we get a fixed schedule that answers the question.

Why Other Options Are Wrong:
Option A is wrong because statement I alone lacks an initial time. Option B is wrong because statement II alone does not specify the timetable interval. Option C incorrectly claims that either statement alone is sufficient. Option E, which states that even both statements together are not sufficient, is contradicted by the clear derivation of the next bus at 8:15 p.m. only option D correctly states that both statements I and II together are necessary and sufficient.

Common Pitfalls:
Learners may assume that a 30 minute schedule must start on the hour or the half hour, which is not given. Another common error is to assume that the last bus departure is always at the closing time of service, rather than up to that time. Carefully reading the exact wording and combining the anchor departure time with the frequency avoids such mistakes.

Final Answer:
The data in both statements I and II together are necessary to answer the question, so the correct option is D.