A man starts walking at 3 pm. He walks at a speed of 4 km/hr on level ground,…

Concept: When a journey has several legs at different speeds, the time for each leg is distance divided by speed, and the total time is the sum of these leg times. If a stretch is covered once in each direction at two different speeds, the two times can be added and expressed as a single fraction of the round-trip time.

Application:

1. Let x km be the one-way distance on level ground and y km be the one-way distance on the hill stretch.

2. Going out: level ground takes x/4 hours (at 4 km/hr) and the uphill climb takes y/3 hours (at 3 km/hr).

3. Coming back: the downhill stretch takes y/6 hours (at 6 km/hr) and the level ground takes x/4 hours again (at 4 km/hr).

4. Total travel time from 3 pm to 9 pm is 6 hours, so x/4 + y/3 + y/6 + x/4 = 6.

5. Combine like terms: x/2 + (y/3 + y/6) = 6, and y/3 + y/6 = y/2, so the equation becomes x/2 + y/2 = 6.

6. Multiplying through by 2 gives x + y = 12, which is the one-way distance covered (level ground plus hill stretch).

Cross-check: Since only x + y matters, any split satisfying it works, e.g. x = 4 km, y = 8 km gives time = 4/4 + 8/3 + 8/6 + 4/4 = 1 + 2.667 + 1.333 + 1 = 6 hours, confirming the one-way distance of 12 km.