A man swims 72 km downstream and 45 km upstream, taking 9 hours for each journey. What is the speed of the current (in km/h)?

Introduction / Context:
This swimming problem is a boats and streams type question with different distances downstream and upstream but equal times. We can use the distances and times to find effective downstream and upstream speeds. Once we know those speeds, we can find the swimmer’s speed in still water and the current speed by solving simple equations. The focus is on manipulating distance time relations and using the symmetric structure of the problem.

Given Data / Assumptions:

• Downstream distance = 72 km.
• Downstream time = 9 hours.
• Upstream distance = 45 km.
• Upstream time = 9 hours.
• Speeds are assumed constant, and current is uniform.

Concept / Approach:
First we compute:

• Downstream speed = distance / time.
• Upstream speed = distance / time.

Then we let b be the swimmer’s speed in still water and c be the current speed. We use:

• b + c = downstream speed.
• b - c = upstream speed.

Solving the system gives c, the required speed of the current.

Step-by-Step Solution:
Downstream speed = 72 / 9 = 8 km/h. Upstream speed = 45 / 9 = 5 km/h. Let b be swimming speed in still water and c be stream speed. Then b + c = 8 and b - c = 5. Add the equations: 2b = 13 so b = 6.5 km/h. Subtract the equations: 2c = 3 so c = 1.5 km/h. Therefore, the speed of the current is 1.5 km/h.

Verification / Alternative check:
Using b = 6.5 km/h and c = 1.5 km/h, downstream speed is 8 km/h, giving time 72 / 8 = 9 hours for the downstream journey. Upstream speed is 5 km/h, giving time 45 / 5 = 9 hours for the upstream journey. Both match the given times, confirming that the calculation of the current speed is accurate.

Why Other Options Are Wrong:
A current speed of 1 km/h would produce different upstream and downstream speeds and would not match the given distances for 9 hours. Values such as 2 km/h or 3.2 km/h would change the speeds even more and would not yield the exact distances within 9 hours. Only 1.5 km/h leads to effective speeds of 8 km/h and 5 km/h that satisfy the problem conditions.

Common Pitfalls:
Some students average the distances or times incorrectly, while others forget that the swimmer’s own speed is different from the effective speeds. Another mistake is to divide the sum or difference of distances by total time instead of computing each speed separately. Always calculate upstream and downstream speeds first, then use the system of equations to obtain still water and current speeds.

Final Answer:
The speed of the current is 1.5 km/h.