The current of a stream flows at 1 km/h. A motor boat goes 35 km upstream from a point and then returns to the starting point, taking a total of 12 hours for the round trip. What is the speed of the motor boat in still water?

Introduction / Context:
Here we are given the speed of the stream, the distance travelled upstream and downstream, and the total time taken for the round trip. The task is to determine the boat's speed in still water. This is a classic application of relative speed concepts in boats and streams.

Given Data / Assumptions:

Speed of the stream c = 1 km/h.
Let the speed of the boat in still water be b km/h.
Distance upstream from start to turning point = 35 km.
Distance downstream back to start = 35 km.
Total time for upstream and downstream = 12 hours.
Upstream speed = (b - c) km/h.
Downstream speed = (b + c) km/h.

Concept / Approach:
Time is equal to distance divided by speed. For the round trip, we sum the time taken upstream and downstream and set this equal to 12 hours. Since the stream speed is known, the only unknown is b. Solving the resulting equation gives the boat's still-water speed.

Step-by-Step Solution:
Step 1: Write expressions for times upstream and downstream. Upstream time t₁ = 35 / (b - 1). Downstream time t₂ = 35 / (b + 1). Total time t₁ + t₂ = 12 hours. Step 2: Set up the equation. 35 / (b - 1) + 35 / (b + 1) = 12. Divide both sides by 35: 1 / (b - 1) + 1 / (b + 1) = 12 / 35. Step 3: Combine the fractions on the left. (b + 1 + b - 1) / (b^2 - 1) = 12 / 35. 2b / (b^2 - 1) = 12 / 35. Step 4: Cross-multiply and solve for b. 2b * 35 = 12(b^2 - 1). 70b = 12b^2 - 12. Rearrange: 12b^2 - 70b - 12 = 0. Divide by 2: 6b^2 - 35b - 6 = 0. Solve the quadratic: discriminant Δ = 35^2 + 4 * 6 * 6 = 1225 + 144 = 1369 = 37^2. Thus b = [35 ± 37] / 12. Positive solution: (35 + 37)/12 = 72/12 = 6. So b = 6 km/h.

Verification / Alternative check:
With b = 6 km/h and c = 1 km/h: upstream speed = 5 km/h, downstream speed = 7 km/h. Upstream time = 35 / 5 = 7 h. Downstream time = 35 / 7 = 5 h. Total time = 7 + 5 = 12 h, which matches the given total time exactly.

Why Other Options Are Wrong:
8 km/h, 7.5 km/h or 5.5 km/h for b would yield total times that are not equal to 12 hours when you recompute upstream and downstream times using c = 1 km/h and distance 35 km each way.

Common Pitfalls:
Mistakes often arise from incorrect simplification of the fraction equation or from arithmetic slips when solving the quadratic. Another pitfall is choosing the negative root of the quadratic, which is not physically meaningful for speed. Always check that your final speed values give positive upstream and downstream speeds and reproduce the given total time.

Final Answer:
The speed of the motor boat in still water is 6 km/h.