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

Introduction / Context:
This round trip boats and streams problem asks you to determine the one way distance to a point on a river, given the boat speed in still water, the speed of the current, and the total time for the trip. The key here is to correctly compute upstream and downstream speeds and then use the total time for going and returning.

Given Data / Assumptions:
- Speed of boat in still water = 5 km/h. - Speed of river current = 1 km/h. - Total time for going to the place and coming back = 75 minutes. - 75 minutes = 75 / 60 = 1.25 hours. - Let one way distance be d km. - Downstream speed = 5 + 1 = 6 km/h. - Upstream speed = 5 - 1 = 4 km/h.

Concept / Approach:
Time downstream equals distance divided by downstream speed, and time upstream equals distance divided by upstream speed. The total round trip time is the sum of these two times. Setting this equal to 1.25 hours gives an equation in d that can be solved to find the distance to the place. This method is standard for such symmetric round trip problems.

Step-by-Step Solution:
Step 1: Time taken downstream = d / 6 hours. Step 2: Time taken upstream = d / 4 hours. Step 3: Total time = d / 6 + d / 4. Step 4: Given total time = 1.25 hours. Step 5: So d / 6 + d / 4 = 1.25. Step 6: Use common denominator 12: (2d / 12) + (3d / 12) = 1.25. Step 7: This simplifies to 5d / 12 = 1.25. Step 8: Multiply both sides by 12: 5d = 1.25 * 12. Step 9: 1.25 * 12 = 15, so 5d = 15. Step 10: Therefore d = 15 / 5 = 3 km.

Verification / Alternative check:
If the place is 3 km away, time downstream = 3 / 6 = 0.5 hour (30 minutes). Time upstream = 3 / 4 = 0.75 hour (45 minutes). Total time = 0.5 + 0.75 = 1.25 hours = 75 minutes, which matches the problem statement. Thus, the distance is confirmed as 3 km.

Why Other Options Are Wrong:
- 4 km or 5 km would give total times greater than 1.25 hours. - 2 km would give a total time less than 1.25 hours. - 6 km is far too large and would imply much longer travel times than given.

Common Pitfalls:
Learners may forget to convert minutes to hours, or incorrectly average the speeds instead of calculating separate upstream and downstream times. Another error is to assume 75 minutes is one way time instead of total round trip time. Carefully translating the language into a time equation is essential here.

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