A bus for Shiridi leaves every 52 minutes from a bus stand. An enquiry officer tells a passenger that the last bus left 16 minutes ago and that the next bus will leave at 8:15 p.m. At what exact time did the enquiry officer give this information to the passenger?

Introduction / Context:
Time and timetable problems often involve buses or trains leaving at fixed intervals. The key skill is to relate the fixed time gap between consecutive departures to the times given in the statement. In this question, you know the interval between buses, the time of the next bus, and how long ago the last bus left when the enquiry officer spoke. You must use these pieces to reconstruct the exact time of the enquiry.

Given Data / Assumptions:
- A bus for Shiridi leaves every 52 minutes from a bus stand. - The enquiry officer says the last bus left 16 minutes ago. - He also says the next bus will leave at 8:15 p.m. - Bus departures follow a strict 52 minute schedule without gaps or extra buses.

Concept / Approach:
When buses leave at a fixed interval, consecutive departure times form an arithmetic sequence with common difference equal to the interval. If the next bus is at 8:15 p.m., the previous one must have left exactly 52 minutes earlier. That previous departure is the same “last bus” that, according to the officer, left 16 minutes before he spoke. Once we know the last departure time, we simply add 16 minutes to get the time when the officer gave the information.

Step-by-Step Solution:
Step 1: Note that buses leave every 52 minutes. Step 2: The next bus is scheduled for 8:15 p.m. Step 3: The last bus before 8:15 p.m. must have left 52 minutes earlier. Step 4: Subtract 52 minutes from 8:15 p.m.: 8:15 p.m. minus 52 minutes equals 7:23 p.m. Step 5: Therefore, the last bus departed at 7:23 p.m. Step 6: The enquiry officer says this last bus left 16 minutes ago. Step 7: Add 16 minutes to 7:23 p.m. to find the enquiry time: 7:23 p.m. + 16 minutes = 7:39 p.m. Step 8: Thus, the officer must have given the information at 7:39 p.m.

Verification / Alternative check:
We can verify the logic by working forward from the enquiry time. If the officer spoke at 7:39 p.m. and the last bus left 16 minutes earlier, that departure occurred at 7:23 p.m. Adding the fixed interval of 52 minutes to 7:23 p.m. gives 8:15 p.m., which correctly matches the announced time of the next bus. Since all times are consistent, 7:39 p.m. is validated as the enquiry time.

Why Other Options Are Wrong:
07:23 p.m: This is the time of the last bus departure, not the time of the enquiry. 06:25 p.m: From this time, adding 16 minutes would not land on a scheduled bus time that fits a 52 minute pattern ending at 8:15 p.m. 06:23 p.m: Similarly inconsistent with the 52 minute schedule and the given next departure time.

Common Pitfalls:
Students may mistakenly subtract 16 minutes from 8:15 p.m., mixing up “how long ago the bus left” with the interval between buses. Another common error is to forget that the 52 minute gap applies between consecutive buses, not between the enquiry and the next bus. Carefully separating “schedule interval” from “time since last event” prevents these confusions.

Final Answer:
The enquiry officer gave the information at 7:39 p.m.