A motorboat travels 10 miles downstream on a river in 20 minutes and then takes 30 minutes to travel the same 10 miles upstream. What is the speed of the river current in mph?

Introduction / Context:
This boats and streams question tests understanding of how a boat speed in still water combines with the river current to give different speeds downstream and upstream. By using distance, time, and speed relationships, we can separate the speed of the current from the speed of the motorboat itself.

Given Data / Assumptions:
- Distance each way on the river = 10 miles. - Time taken downstream = 20 minutes. - Time taken upstream = 30 minutes. - Let the speed of the boat in still water be b mph. - Let the speed of the current be c mph. - Downstream speed = b + c, upstream speed = b - c.

Concept / Approach:
Speed is equal to distance divided by time. We first convert the times from minutes to hours to stay in miles per hour. Using the given times and common distance, we find the effective downstream and upstream speeds. These give us two equations in b and c. Adding and subtracting the equations allows us to solve for the speed of the current c.

Step-by-Step Solution:
Step 1: Convert 20 minutes to hours: 20 / 60 = 1/3 hour. Step 2: Downstream speed = distance / time = 10 / (1/3) = 30 mph. Step 3: Convert 30 minutes to hours: 30 / 60 = 1/2 hour. Step 4: Upstream speed = distance / time = 10 / (1/2) = 20 mph. Step 5: Therefore, b + c = 30 and b - c = 20. Step 6: Add the two equations: (b + c) + (b - c) = 30 + 20 gives 2b = 50. Step 7: So b = 50 / 2 = 25 mph. Step 8: Substitute b into b + c = 30 to get 25 + c = 30, so c = 5 mph.

Verification / Alternative check:
Using b = 25 mph and c = 5 mph, downstream speed = 25 + 5 = 30 mph. Time downstream = 10 / 30 = 1/3 hour, which is 20 minutes, as given. Upstream speed = 25 - 5 = 20 mph. Time upstream = 10 / 20 = 1/2 hour, which is 30 minutes, also matching the data. So the calculated current speed is fully consistent with the problem conditions.

Why Other Options Are Wrong:
- 8 mph or 7 mph would give upstream or downstream times different from 20 and 30 minutes. - 6 mph or 4 mph also fail when checked back into the equations b + c = 30 and b - c = 20. - Only 5 mph satisfies both the downstream and upstream conditions exactly.

Common Pitfalls:
Many learners forget to convert minutes into hours, which leads to wrong speeds. Others mistakenly average the two times rather than working with speeds. A further mistake is to assume the current speed is half of one of the given speeds without using equations. Always use distance / time carefully and set up equations for downstream and upstream speeds.

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