In one hour, a boat travels 12 km along the stream and 6 km against the stream. What is the speed of the boat in still water (in km/h)?

Introduction / Context:
This question uses basic boats and streams concepts to find the speed of a boat in still water. When a boat travels with the stream and against the stream, the current either helps or opposes the motion, creating two different effective speeds. The middle value of these speeds gives the speed in still water.

Given Data / Assumptions:
- Speed along the stream (downstream speed) = 12 km/h. - Speed against the stream (upstream speed) = 6 km/h. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Downstream speed = b + c, upstream speed = b - c.

Concept / Approach:
Downstream the current adds to the boat speed, and upstream it subtracts. Therefore we can write two simple equations and then find the average of the downstream and upstream speeds to get the still water speed: b = (downstream + upstream) / 2. This is a standard shortcut result in boats and streams problems.

Step-by-Step Solution:
Step 1: b + c = 12 (downstream speed). Step 2: b - c = 6 (upstream speed). Step 3: Add the equations: (b + c) + (b - c) = 12 + 6. Step 4: This gives 2b = 18. Step 5: Divide by 2 to find b: b = 18 / 2 = 9 km/h. Step 6: So the speed of the boat in still water is 9 km/h.

Verification / Alternative check:
Use the average formula directly: speed in still water = (downstream speed + upstream speed) / 2 = (12 + 6) / 2 = 18 / 2 = 9 km/h. This quick method matches the algebraic solution, confirming the result.

Why Other Options Are Wrong:
- 8 km/h and 7 km/h do not satisfy both equations b + c = 12 and b - c = 6 together. - 7.5 km/h is simply the plain average of 9 and 6, not of 12 and 6, and does not work when checked in the equations. - 6 km/h would imply zero current, which contradicts the problem information.

Common Pitfalls:
Learners sometimes incorrectly average 12 and 6 and then subtract something for the current, or they confuse which speed is upstream and which is downstream. Remember that the boat speed in still water is always the simple average of upstream and downstream speeds, while the current is half of their difference.

Final Answer:
The speed of the boat in still water is 9 km/h.