In one hour, a boat covers 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 is a straightforward boats and streams problem where the effective downstream and upstream speeds are directly given as distances covered in one hour. From these effective speeds, we can easily compute the speed of the boat in still water by using the relation between upstream speed, downstream speed, and the boat and stream speeds. The question checks understanding of average of two opposite direction speeds in still water.

Given Data / Assumptions:

• Distance covered downstream in one hour = 12 km, so downstream speed = 12 km/h.
• Distance covered upstream in one hour = 6 km, so upstream speed = 6 km/h.
• Let b be the speed of the boat in still water.
• Let c be the speed of the stream.
• Speeds are assumed constant.

Concept / Approach:
For a boat in a stream:

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

We are given both downstream and upstream speeds numerically, so we can directly set up two linear equations in b and c and solve. The boat speed in still water is simply the average of the downstream and upstream speeds.

Step-by-Step Solution:
Downstream speed = 12 km/h, upstream speed = 6 km/h. Let b be the speed of the boat in still water and c be the speed of the stream. Then b + c = 12 and b - c = 6. Add the two equations: (b + c) + (b - c) = 12 + 6. This gives 2b = 18. So b = 18 / 2 = 9 km/h.

Verification / Alternative check:
If the boat speed in still water is 9 km/h, then the sum of upstream and downstream speeds must be 2b = 18 km/h. The given speeds 12 km/h and 6 km/h indeed add up to 18 km/h. The stream speed is then (downstream - upstream) / 2 = (12 - 6) / 2 = 3 km/h, which is a reasonable value. The given distances covered in one hour match these speeds, so the solution is consistent.

Why Other Options Are Wrong:
Options 8 km/h, 7 km/h, and 7.5 km/h would yield different sums of upstream and downstream speeds. For example, if b were 8 km/h, the sum of downstream and upstream speeds would be 16 km/h, which does not match the observed 12 + 6 = 18 km/h. Therefore only 9 km/h correctly fits the given data.

Common Pitfalls:
Students sometimes confuse the formula and think that the boat speed equals the difference between downstream and upstream speeds divided by 2, which actually gives the stream speed. It is important to remember that boat speed is (downstream + upstream) / 2 and stream speed is (downstream - upstream) / 2. Keeping the formulas straight avoids this common confusion.

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