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

Introduction / Context:
This is a standard boats and streams question where you are given upstream and downstream journeys over known distances and times. From these, you can find the upstream and downstream speeds and then deduce the speeds of the boat in still water and of the current.

Given Data / Assumptions:

• Upstream distance = 24 km, time taken = 6 hours.
• Downstream distance = 20 km, time taken = 4 hours.
• Upstream speed = speed of boat in still water minus speed of current.
• Downstream speed = speed of boat in still water plus speed of current.
• All speeds are in km/h.

Concept / Approach:
Upstream speed is found by dividing upstream distance by upstream time. Similarly, downstream speed is found from downstream data. Once these two effective speeds are known, we use the formulas: boat speed in still water = (upstream speed + downstream speed) / 2 current speed = (downstream speed - upstream speed) / 2 These come from solving the two basic equations for upstream and downstream speeds.

Step-by-Step Solution:
Step 1: Upstream speed = 24 km / 6 h = 4 km/h. Step 2: Downstream speed = 20 km / 4 h = 5 km/h. Step 3: Let b be the speed of the boat in still water and s be the speed of the current. Step 4: Then b - s = 4 and b + s = 5 from the upstream and downstream speeds. Step 5: Add the two equations: (b - s) + (b + s) = 4 + 5 ⇒ 2b = 9 ⇒ b = 4.5 km/h. Step 6: Subtract the first equation from the second: (b + s) - (b - s) = 5 - 4 ⇒ 2s = 1 ⇒ s = 0.5 km/h.

Verification / Alternative check:
Check the effective speeds. Upstream effective speed = b - s = 4.5 - 0.5 = 4 km/h, which matches 24 km in 6 hours. Downstream effective speed = b + s = 4.5 + 0.5 = 5 km/h, matching 20 km in 4 hours. This confirms that the calculated speeds are correct.

Why Other Options Are Wrong:
Pairs like 4.5 km/h and 3 km/h or 5 km/h and 2 km/h would give very different upstream and downstream speeds that do not fit the given times and distances. For example, with 5 km/h and 2 km/h, the upstream speed would be 3 km/h, not the required 4 km/h. Only the pair 4.5 km/h and 0.5 km/h satisfies both journeys.

Common Pitfalls:
Learners sometimes forget to compute separate upstream and downstream speeds before applying the formulas, or they accidentally swap them. Another common mistake is mixing up the addition and subtraction when finding the still water and current speeds. Always write the equations clearly and solve them step by step.

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.