A man rows his boat 60 km downstream and 30 km upstream, taking 3 hours for each trip. What is the speed of the stream?

Introduction / Context:
This boats and streams problem gives different distances downstream and upstream but the same time for each journey. That information allows you to calculate downstream and upstream speeds directly, and then deduce the speed of the stream and the speed of the boat in still water. The question specifically asks for the speed of the stream, making it a simple but instructive exercise in relative speed.

Given Data / Assumptions:

• Downstream distance = 60 km; time taken = 3 hours.
• Upstream distance = 30 km; time taken = 3 hours.
• Let speed of stream be s km/h.
• Let speed of boat in still water be b km/h.
• Downstream speed = b + s.
• Upstream speed = b − s.

Concept / Approach:
We first compute the effective speeds from the given distance and time for each direction:

• Downstream speed = distance / time = 60 / 3 = 20 km/h.
• Upstream speed = distance / time = 30 / 3 = 10 km/h.

Then we use the standard relations b + s = 20 and b − s = 10 to solve for b and s. Finally, we report s, the speed of the stream.

Step-by-Step Solution:
Step 1: Downstream speed = 60 / 3 = 20 km/h.Step 2: Upstream speed = 30 / 3 = 10 km/h.Step 3: Let b be speed in still water and s be speed of stream. Then b + s = 20 and b − s = 10.Step 4: Add the two equations: (b + s) + (b − s) = 20 + 10 ⇒ 2b = 30.Step 5: So b = 30 / 2 = 15 km/h.Step 6: Substitute back into b + s = 20: 15 + s = 20 ⇒ s = 5 km/h.Step 7: Thus, the speed of the stream is 5 km/h.

Verification / Alternative check:
With b = 15 and s = 5, downstream speed = 15 + 5 = 20 km/h, so time for 60 km downstream is 60 / 20 = 3 hours. Upstream speed = 15 − 5 = 10 km/h, so time for 30 km upstream is 30 / 10 = 3 hours. Both match the conditions given in the question, confirming that s = 5 km/h is correct.

Why Other Options Are Wrong:
10 km/h or 15 km/h as stream speeds would imply impossible or negative still-water speeds when solving b + s = 20 and b − s = 10. A stream speed of 45 km/h is clearly impossible because it would exceed the downstream speed itself. 7.5 km/h does not satisfy both equations simultaneously. Only 5 km/h yields consistent downstream and upstream speeds with the given travel times.

Common Pitfalls:

• Assuming upstream distance is also 60 km instead of 30 km.
• Mixing up which speed corresponds to downstream or upstream motion.
• Arithmetic mistakes when adding or subtracting the equations b + s = 20 and b − s = 10.
• Guessing the answer without verifying that both given times are satisfied.

Final Answer:
The speed of the stream is 5 km/h.