City A to City B is a downstream journey on a stream which flows at a speed of…

Concept

For two bodies that start moving toward each other from opposite ends of a fixed distance, the time to their first meeting is (total distance) ÷ (sum of their speeds). On water, a boat's effective speed changes with direction: effective speed = still-water speed + stream speed when moving downstream, and still-water speed − stream speed when moving upstream. If a boat reaches its destination before the other and reverses direction, its effective speed changes accordingly, and any later meeting must be re-worked from that turnaround point using the new relative speed.

Application

1. Downstream and upstream speeds: Boat P travels from City A towards City B, which is the downstream direction, so its effective speed = 25 + 5 = 30 km/hr. Boat Q travels from City B towards City A, against the stream, so its effective speed = 15 − 5 = 10 km/hr.

2. First meeting: the boats approach each other, so their closing speed = 30 + 10 = 40 km/hr. First meeting time = 300 ÷ 40 = 7.5 hours.

3. Time for P to reach City B: P covers the full 300 km at 30 km/hr, so it arrives at City B after 300 ÷ 30 = 10 hours.

4. Q's position at that moment: in those same 10 hours, Q (still moving toward City A at 10 km/hr) has covered 10 × 10 = 100 km from City B, so Q is 100 km from City B (200 km from City A) and has not yet reached City A.

5. Direction change: on reaching City B, P turns around and now heads back towards City A — this leg is upstream for P, so its effective speed becomes 25 − 5 = 20 km/hr. Q continues towards City A unchanged, at 10 km/hr, since it still needs 300 ÷ 10 = 30 hours in total to reach City A.

6. Closing the gap: from the 10-hour mark, both boats move in the same direction (towards City A), with P behind Q by 100 km but faster, so the gap closes at 20 − 10 = 10 km/hr.

7. Second meeting time: closing 100 km at 10 km/hr takes 100 ÷ 10 = 10 more hours, so the second meeting happens at 10 + 10 = 20 hours from the start.

8. Meeting point: from the 10-hour mark, P (now moving upstream from City B) covers 20 km/hr × 10 hours = 200 km, so the second meeting is 200 km from City B (100 km from City A).

Cross-check

Checking with Q's position at 20 hours: Q has been moving towards City A at a steady 10 km/hr the whole time (it only reaches City A at the 30-hour mark), so at 20 hours it has covered 10 × 20 = 200 km from City B — the same 200-km-from-City-B point found for P above, confirming the second meeting point and time.

Result

So the two boats meet for the first time after 7.5 hours, and meet again after 20 hours, 200 km from City B (100 km from City A).