A boat travels 24 km upstream in 6 hours and 20 km downstream in 4 hours. What are the speeds (in km/h) of the boat in still water and of the current, respectively?

Introduction / Context:
This question provides separate upstream and downstream journeys of a boat over different distances and times. From this information we can compute the effective speeds in each direction and then determine the speed of the boat in still water and the speed of the current. This is a direct application of the relationships between upstream speed, downstream speed, boat speed and current speed.

Given Data / Assumptions:

• Upstream distance = 24 km, time = 6 hours.
• Downstream distance = 20 km, time = 4 hours.
• Let boat speed in still water be b km/h.
• Let current speed be c km/h.
• Upstream speed = b - c.
• Downstream speed = b + c.
• We must find b and c.

Concept / Approach:
First we compute the effective upstream and downstream speeds by dividing distance by time. Once upstream and downstream speeds are known, we use: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 These formulas are obtained by solving the linear system b - c = upstream speed and b + c = downstream speed. The calculations here are simple and rely only on accurate arithmetic.

Step-by-Step Solution:
Step 1: Compute upstream speed. Upstream speed = distance / time = 24 / 6 = 4 km/h. Step 2: Compute downstream speed. Downstream speed = 20 / 4 = 5 km/h. Step 3: Use the formulas for boat and current speeds. b = (downstream speed + upstream speed) / 2 = (5 + 4) / 2 = 9 / 2 = 4.5 km/h. c = (downstream speed - upstream speed) / 2 = (5 - 4) / 2 = 1 / 2 = 0.5 km/h.
Verification / Alternative check:
Check upstream with these values: b - c = 4.5 - 0.5 = 4 km/h, which matches the calculated upstream speed. Check downstream: b + c = 4.5 + 0.5 = 5 km/h, matching the calculated downstream speed. Therefore the pair 4.5 km/h and 0.5 km/h satisfies all conditions.
Why Other Options Are Wrong:
Pairs such as 4.5 km/h and 3 km/h or 5 km/h and 2 km/h imply upstream and downstream speeds that are far from 4 and 5 km/h respectively. Similarly, 4 km/h and 2 km/h or 6 km/h and 1 km/h do not yield upstream and downstream speeds equal to 4 km/h and 5 km/h when plugged into b - c and b + c.
Common Pitfalls:
A common error is miscomputing the initial upstream or downstream speeds from distance and time, for example mixing up the distances or times. Another pitfall is to average the two effective speeds but forget to halve the difference to find the current speed, which leads to reporting only b and not c.
Final Answer:
The speed of the boat in still water and the speed of the current are 4.5 km/h and 0.5 km/h respectively.