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 (in km/h)?

Introduction / Context:
This question tests your understanding of downstream and upstream speeds and how to extract the speed of the current from given distances and times. Here, the man takes the same time for two trips but covers different distances downstream and upstream, giving you enough information to find the stream speed.

Given Data / Assumptions:

Downstream distance = 60 km, time taken = 3 hours.
Upstream distance = 30 km, time taken = 3 hours.
Let the speed of the boat in still water be b km/h.
Let the speed of the stream be c km/h.
Downstream speed = (b + c) km/h.
Upstream speed = (b - c) km/h.

Concept / Approach:
From distance and time, we can compute effective downstream and upstream speeds. Once we have these two effective speeds, we can find b and c using the relationships:
b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 Our main interest is c, the speed of the stream.

Step-by-Step Solution:
Step 1: Find downstream speed. Downstream speed v_d = distance / time = 60 / 3 = 20 km/h. Step 2: Find upstream speed. Upstream speed v_u = 30 / 3 = 10 km/h. Step 3: Compute the speed of the stream. c = (v_d - v_u) / 2 = (20 - 10) / 2 = 10 / 2 = 5 km/h.

Verification / Alternative check:
Now compute the still-water speed: b = (v_d + v_u) / 2 = (20 + 10) / 2 = 30 / 2 = 15 km/h. Using b = 15 and c = 5, downstream speed = 20 km/h and upstream speed = 10 km/h, which reproduce the original times of 60/20 = 3 h downstream and 30/10 = 3 h upstream. This consistency confirms our answer.

Why Other Options Are Wrong:
10 km/h would imply a still-water speed of 15 km/h and give upstream speed = 5 km/h, leading to an upstream time of 30 / 5 = 6 hours, not 3 hours.
15 km/h and 45 km/h as current speeds are unrealistic here; they lead to negative or inconsistent upstream speeds and do not match the given trip times.

Common Pitfalls:
Some students mistakenly average distances or times instead of calculating speeds separately. Others mix up the formulas for b and c or forget to divide the difference of speeds by 2. Always compute effective downstream and upstream speeds first, and then use the standard formulas to find the still-water and current speeds.

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