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 a speed of 12 km/h. What is the speed (in km/h) of the boat in still water?

Introduction / Context:
This question gives the effective speeds of a boat while travelling upstream and downstream over the same distance. The objective is to find the speed of the boat in still water. Because the distance is identical in both directions, we can ignore it when applying the standard relationships between upstream speed, downstream speed, boat speed and current speed.

Given Data / Assumptions:

• Upstream distance = 29 km.
• Upstream speed = 6 km/h.
• Downstream distance = 29 km.
• Downstream speed = 12 km/h.
• Let boat speed in still water be b km/h.
• Let current speed be c km/h.
• Upstream speed = b - c and downstream speed = b + c.
• We need to find b.

Concept / Approach:
When we know the upstream and downstream speeds, the boat speed and current speed can be found using the formulas: b = (downstream speed + upstream speed) / 2 c = (downstream speed - upstream speed) / 2 These come from solving the two linear equations b + c = downstream speed and b - c = upstream speed. Here, the distances are not needed to compute these speeds because the given speeds already incorporate the effect of the current.

Step-by-Step Solution:
Step 1: Write equations from the given speeds. b - c = 6. b + c = 12. Step 2: Add these equations to eliminate c. (b - c) + (b + c) = 6 + 12. 2b = 18. b = 18 / 2 = 9 km/h. Step 3: If required, we can also compute current speed. Substitute b = 9 into b + c = 12 to get 9 + c = 12, so c = 3 km/h.
Verification / Alternative check:
Check upstream speed with b = 9 and c = 3. Upstream speed = b - c = 9 - 3 = 6 km/h, which matches the given upstream speed. Downstream speed = b + c = 9 + 3 = 12 km/h, matching the given downstream speed. Thus the computed boat speed in still water is consistent.
Why Other Options Are Wrong:
If b were 6, 8, 10 or 11 km/h, there would be no non negative current speed c that makes both upstream and downstream speeds equal to 6 km/h and 12 km/h. Each such choice would lead to contradictions in at least one of the equations b - c = 6 or b + c = 12.
Common Pitfalls:
A frequent error is to try to calculate average speed from 6 km/h and 12 km/h as if they are simple averages, instead of using the correct formulas for b and c. Another mistake is to mix up b and c and mistakenly report the current speed instead of the boat speed in still water.
Final Answer:
The speed of the boat in still water is 9 kmph.