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

Introduction / Context:
This boats and streams question combines information about total journey time with a key relationship between upstream and downstream travel times over different distances. Using this relationship, we find the ratio between upstream and downstream speeds, then combine it with the total time for a round trip to determine the speed of the boat in still water.

Given Data / Assumptions:

• Distance from starting point to destination = 80 km.
• Total distance for the round trip = 80 km upstream + 80 km downstream = 160 km.
• Total time for the round trip = 20 hours.
• 8 km downstream takes the same time as 4 km upstream.
• Let b be the speed of the boat in still water (km/h).
• Let s be the speed of the stream (km/h).
• Upstream speed = b - s, downstream speed = b + s.

Concept / Approach:
First, use the time equality 8 km downstream = 4 km upstream to relate upstream and downstream speeds. This gives a ratio that leads to a simple connection between b and s. Next, express the total round trip time in terms of upstream and downstream speeds over 80 km each. Solving the resulting equation yields the value of the stream speed and then the boat speed in still water.

Step-by-Step Solution:
Step 1: From the time equality, 8 / (b + s) = 4 / (b - s). Step 2: Cross multiply: 8(b - s) = 4(b + s). Step 3: Expand: 8b - 8s = 4b + 4s. Step 4: Rearrange: 8b - 4b = 8s + 4s ⇒ 4b = 12s ⇒ b = 3s. Step 5: Upstream speed = b - s = 3s - s = 2s; downstream speed = b + s = 3s + s = 4s. Step 6: Time to go 80 km upstream = 80 / (2s) = 40 / s. Step 7: Time to come 80 km downstream = 80 / (4s) = 20 / s. Step 8: Total time for the round trip = 40 / s + 20 / s = 60 / s hours. Step 9: Given that total time = 20 hours, so 60 / s = 20 ⇒ s = 60 / 20 = 3 km/h. Step 10: Therefore b = 3s = 3 * 3 = 9 km/h.

Verification / Alternative check:
With b = 9 km/h and s = 3 km/h, upstream speed = 6 km/h, downstream speed = 12 km/h. Time for 80 km upstream = 80 / 6 ≈ 13.33 hours. Time for 80 km downstream = 80 / 12 ≈ 6.67 hours. Total time ≈ 20 hours, which matches the given total. Also, 8 km downstream takes 8 / 12 hours, and 4 km upstream takes 4 / 6 hours; both are 2/3 hour, confirming the condition.

Why Other Options Are Wrong:
If the boat speed were 7, 5, 2 or 11 km/h, the resulting upstream and downstream speeds could not satisfy both the total time of 20 hours and the specific timing relationship between 8 km downstream and 4 km upstream. Only 9 km/h accurately fits both conditions.

Common Pitfalls:
A common error is to treat 8 and 4 as directly giving a speed ratio without considering that times are equal, not distances. Another mistake is to assume a simple average speed for the round trip, which does not hold when speeds differ. Always derive equations from time relationships and solve them carefully.

Final Answer:
Mahesh's boat speed in still water is 9 km/h.