The distance between two cities A and B is 330 km. A train leaves A at 8 a.m. towards B at 60 km/h, and another train leaves B at 9 a.m. towards A at 75 km/h. At what time do the two trains meet?

Introduction / Context:
This is a classic two trains problem where the trains start from opposite ends of a straight track at different times and speeds. The key is to carefully account for the fact that the first train begins its journey one hour earlier than the second train, and then use relative speed to determine the time at which they meet.

Given Data / Assumptions:

• Distance between cities A and B = 330 km.
• Train 1 leaves A at 8 a.m. with speed = 60 km/h.
• Train 2 leaves B at 9 a.m. with speed = 75 km/h.
• Both trains move along the same straight route towards each other.

Concept / Approach:
First, we calculate the distance covered by Train 1 during the hour before Train 2 starts. Then we compute the remaining distance between the trains at 9 a.m. After that, both trains are moving towards each other, so we can use their combined (relative) speed to find how long it takes them to meet after 9 a.m. Finally, we add this time to 9 a.m. to get the meeting time.

Step-by-Step Solution:
Step 1: From 8 a.m. to 9 a.m., only Train 1 moves. Distance covered by Train 1 = 60 km/h * 1 hour = 60 km. Step 2: Remaining distance between the trains at 9 a.m. = 330 - 60 = 270 km. Step 3: After 9 a.m., both trains move towards each other. Combined speed = 60 + 75 = 135 km/h. Step 4: Time taken after 9 a.m. to meet = Remaining distance / Combined speed = 270 / 135 hours. Step 5: 270 / 135 = 2 hours. Step 6: Therefore, trains meet at 9 a.m. + 2 hours = 11 a.m.
Verification / Alternative check:
We can confirm distances: By 11 a.m., Train 1 has travelled from 8 to 11 a.m. at 60 km/h, so distance = 3 * 60 = 180 km. Train 2 has travelled from 9 to 11 a.m. at 75 km/h, so distance = 2 * 75 = 150 km. Total distance = 180 + 150 = 330 km, exactly the distance between the two cities, confirming that 11 a.m. is the correct meeting time.

Why Other Options Are Wrong:

• 10 a.m.: At this time Train 2 has only travelled 1 hour and total distance covered is less than 330 km.
• 10:30 a.m.: Also too early; combined distance is still less than 330 km.
• 11:30 a.m.: Too late; by then the trains would have already crossed each other.

Common Pitfalls:

• Ignoring the one hour head start of Train 1.
• Using relative speed from 8 a.m. without accounting for Train 2 starting later.
• Performing unit conversion mistakes or miscalculating 270 / 135.

Final Answer:
The two trains meet at 11 a.m.