A motorboat can travel 10 miles downstream on a river in 20 minutes and takes 30 minutes to travel the same 10 miles upstream. What is the speed of the river current (in miles per hour)?

Introduction / Context:
Here we have a motorboat moving in a river. We know the time it takes to travel a fixed distance downstream and the time to travel the same distance upstream. From these times we can derive the downstream and upstream speeds, and then find the speed of the current by using the basic boats and streams relations. The problem uses miles and minutes, so careful unit conversion is important.

Given Data / Assumptions:

• Distance downstream = 10 miles.
• Distance upstream = 10 miles.
• Time downstream = 20 minutes.
• Time upstream = 30 minutes.
• Speeds of boat and current are constant.
• We need the speed of the current in miles per hour (mph).

Concept / Approach:
Let b be the boat speed in still water and c be the current speed. Then:

• Downstream speed = b + c.
• Upstream speed = b - c.

Using distance = 10 miles and the times in hours, we can find numerical values for b + c and b - c. Solving these two linear equations gives both b and c. The answer required is c, the speed of the current.

Step-by-Step Solution:
Convert times to hours: 20 minutes = 20 / 60 = 1/3 hour, 30 minutes = 30 / 60 = 1/2 hour. Downstream speed = distance / time = 10 / (1/3) = 30 mph. Upstream speed = distance / time = 10 / (1/2) = 20 mph. Let b be boat speed in still water and c be current speed. Then b + c = 30 and b - c = 20. Add the equations: 2b = 50 so b = 25 mph. Subtract the equations: 2c = 10 so c = 5 mph.

Verification / Alternative check:
With b = 25 mph and c = 5 mph, downstream speed is 25 + 5 = 30 mph, giving time 10 / 30 hours, which is 1/3 hour or 20 minutes. Upstream speed is 25 - 5 = 20 mph, giving time 10 / 20 = 1/2 hour or 30 minutes. These match the given times exactly, confirming that 5 mph is the correct current speed.

Why Other Options Are Wrong:
If the current were 6 mph or 7 mph, the implied upstream and downstream speeds would not produce the given travel times for 10 miles. A current of 8 mph would be even more inconsistent. Only 5 mph leads to the correct pair of effective speeds, 30 mph and 20 mph, that match both the downstream and upstream journeys.

Common Pitfalls:
Learners sometimes forget to convert minutes into hours and end up with speeds in mixed units. Another common mistake is to confuse downstream and upstream formulas and write b - c for downstream speed. A third error is to average the two speeds incorrectly rather than solving the simple system of equations. Keeping track of units and using the linear equations b + c and b - c is the safest method.

Final Answer:
The speed of the river current is 5 mph.