A boat can move at 5 km/h in still water. If the river is flowing at 1 km/h, it takes the boat 75 minutes to go to a place and return. How far (in km) is the place from the starting point?

Introduction / Context:
This problem involves a round trip in a river with a known boat speed in still water and known stream speed. We are given the total time for the trip to a certain place and back. From these data, we must determine the one way distance. By computing upstream and downstream speeds, expressing the round trip time in terms of distance, and equating it to the given total time, we can solve for the distance.

Given Data / Assumptions:

• Boat speed in still water = 5 km/h.
• Stream speed = 1 km/h.
• Total time for the journey to the place and back = 75 minutes.
• Let the one way distance be x km.
• Speeds are constant and water current is uniform.

Concept / Approach:
The downstream speed is boat speed plus stream speed, and the upstream speed is boat speed minus stream speed. Time taken downstream is distance divided by downstream speed, and time taken upstream is distance divided by upstream speed. The sum of these two times equals the given total time converted into hours. Solving this equation yields x, the required distance.

Step-by-Step Solution:
Let x be the one way distance in km. Downstream speed = 5 + 1 = 6 km/h. Upstream speed = 5 - 1 = 4 km/h. Total time = 75 minutes = 75 / 60 hours = 1.25 hours. Time downstream = x / 6 hours. Time upstream = x / 4 hours. Total time equation: x / 6 + x / 4 = 1.25. Find common denominator 12: (2x + 3x) / 12 = 1.25. So 5x / 12 = 1.25. Multiply both sides by 12: 5x = 1.25 * 12 = 15. Thus x = 15 / 5 = 3 km.

Verification / Alternative check:
With x = 3 km, downstream time is 3 / 6 = 0.5 hours (30 minutes). Upstream time is 3 / 4 = 0.75 hours (45 minutes). Total time is 0.5 + 0.75 = 1.25 hours, which equals 75 minutes as given. This confirms that the distance 3 km is consistent with all data in the problem.

Why Other Options Are Wrong:
If x were 4 km, downstream and upstream times would be 4 / 6 and 4 / 4, summing to more than 75 minutes. A distance of 5 km or 6 km would further increase times and contradict the given total time. Therefore, only 3 km produces the exact 75 minute round trip duration.

Common Pitfalls:
Many learners forget to convert minutes into hours, or they mistakenly add or subtract the stream speed incorrectly when computing upstream and downstream speeds. Others may think that total time should be divided equally between upstream and downstream legs, which is not true due to different speeds. Writing out the full equation x / 6 + x / 4 = 1.25 and solving carefully avoids these issues.

Final Answer:
The place is 3 km away from the starting point.