Mahesh rows to a place 80 km away and back in a total of 20 hours. He observes that he can row 8 km downstream in the same time as 4 km upstream. What is the speed (in km/h) of the boat in still water?

Introduction / Context:
In this question, Mahesh rows to a destination and back, with a given total time for the round trip. Additionally, he notes a relationship between the time taken to cover 8 km downstream and 4 km upstream. We are asked to find the speed of the boat in still water. This is a classic boats and streams problem that combines time comparisons and round trip travel in a current.

Given Data / Assumptions:

• Distance from starting point to destination = 80 km.
• Total distance for round trip = 160 km.
• Total time for round trip = 20 hours.
• He can row 8 km downstream in the same time as 4 km upstream.
• Let b be the speed of the boat in still water (km/h).
• Let c be the speed of the current (km/h).
• Downstream speed = b + c; upstream speed = b - c.

Concept / Approach:
We use two key ideas. First, the equality of times for 8 km downstream and 4 km upstream gives a direct relationship between b and c. Second, the total time of 20 hours for rowing 80 km downstream and 80 km upstream provides another equation. By combining these, we can solve for b, the speed of the boat in still water. The algebra is straightforward once the equations are set up correctly.

Step-by-Step Solution:
Step 1: Use the equal time condition. Time for 8 km downstream = 8 / (b + c). Time for 4 km upstream = 4 / (b - c). Given that these times are equal: 8 / (b + c) = 4 / (b - c). Cross multiply: 8(b - c) = 4(b + c). 8b - 8c = 4b + 4c. Rearrange: 8b - 4b = 8c + 4c. 4b = 12c, so b = 3c. Step 2: Use the round trip time. Downstream distance = 80 km; time downstream = 80 / (b + c). Upstream distance = 80 km; time upstream = 80 / (b - c). Total time = 20 hours, so: 80 / (b + c) + 80 / (b - c) = 20. Step 3: Substitute b = 3c. Downstream speed = 3c + c = 4c. Upstream speed = 3c - c = 2c. So 80 / (4c) + 80 / (2c) = 20. Simplify: 20 / c + 40 / c = 60 / c. Thus 60 / c = 20, so c = 60 / 20 = 3 km/h. Step 4: Find the boat speed in still water. b = 3c = 3 * 3 = 9 km/h.
Verification / Alternative check:
With b = 9 and c = 3, downstream speed = 12 km/h and upstream speed = 6 km/h. Downstream time for 80 km = 80 / 12 hours and upstream time = 80 / 6 hours. 80 / 12 ≈ 6.67 hours, 80 / 6 ≈ 13.33 hours, sum ≈ 20 hours, matching the given total. Time for 8 km downstream = 8 / 12 ≈ 0.67 hours and time for 4 km upstream = 4 / 6 ≈ 0.67 hours, confirming the equality.
Why Other Options Are Wrong:
If the boat speed were 7, 5, 11 or 2 km/h, it would be impossible to choose a current speed that satisfies both the equal time condition and the total round trip time of 20 hours. Quick checks show that those speeds lead to serious mismatches in the times for the given distances.
Common Pitfalls:
A common pitfall is to misinterpret the statement about 8 km downstream and 4 km upstream, treating the distances as equal when they are not. Another frequent mistake is to plug in numerical approximations too early, which can make the algebra harder to follow.
Final Answer:
The speed of the boat in still water is 9 kmph.