A boat travelling upstream covers a distance of 29 km at a speed of 6 km/h, and while travelling downstream it covers the same distance at 12 km/h. What is the speed of the boat in still water (in km/h)?

Introduction / Context:
You are given the upstream and downstream speeds for the same distance and asked for the boat speed in still water. This is one of the most fundamental types of boats and streams questions, reinforcing the formula that the still water speed is the average of upstream and downstream speeds.

Given Data / Assumptions:
- Upstream speed = 6 km/h. - Downstream speed = 12 km/h. - Distance covered each way = 29 km (though it is not directly needed for the still water speed). - Let b be the speed in still water and c be the current speed. - Upstream speed = b - c, downstream speed = b + c.

Concept / Approach:
Since upstream and downstream speeds are known, we can add them and divide by 2 to get the speed in still water. This is because (b - c + b + c) / 2 = (2b) / 2 = b. It is not necessary to use the distance or times in the calculation of b for this kind of question.

Step-by-Step Solution:
Step 1: Let upstream speed u = 6 km/h and downstream speed d = 12 km/h. Step 2: Speed in still water b = (u + d) / 2. Step 3: b = (6 + 12) / 2 = 18 / 2 = 9 km/h.

Verification / Alternative check:
Using b = 9 km/h, we can find the current speed c as (d - u) / 2 = (12 - 6) / 2 = 3 km/h. Then upstream speed should be b - c = 9 - 3 = 6 km/h and downstream speed should be 9 + 3 = 12 km/h. These match the given speeds, confirming that 9 km/h is correct for the still water speed.

Why Other Options Are Wrong:
- 6 km/h would correspond to a zero current, contradicting the different upstream and downstream speeds. - 8 km/h, 10 km/h, and 11 km/h are not equal to the average of 6 and 12 and would not reproduce the same upstream and downstream speeds when combined with a current.

Common Pitfalls:
Some learners incorrectly average distances or times, or think that the still water speed equals the higher of the two speeds. Others forget that current affects both directions equally and that the still water speed must lie exactly midway between upstream and downstream speeds. Remember the simple rule: boat in still water = (downstream + upstream) / 2.

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