A man travels 28 km downstream in a motor boat and immediately returns the same 28 km upstream. The upstream trip takes twice as long as the downstream trip. If the speed of the river flow were doubled, the total time for the downstream and upstream trip together would be 672 minutes. Find the speed of the boat in still water and the speed of the river flow.

Introduction / Context:
This is a more challenging boats and streams problem that involves two scenarios. In the first scenario, you know that the upstream journey takes twice as long as the downstream journey. In the second scenario, the speed of the stream is doubled, and the combined time for going downstream and upstream is given. From this information, you must determine both the speed of the boat in still water and the speed of the stream. This type of problem tests multi-step reasoning and algebraic skill.

Given Data / Assumptions:

• Distance downstream from start to destination = 28 km.
• Distance upstream back to starting point = 28 km.
• In the first situation, time upstream is twice time downstream.
• Let speed of boat in still water be b km/h.
• Let speed of river flow be s km/h.
• Downstream speed = b + s; upstream speed = b − s.
• In the second situation, the speed of the stream becomes 2s.
• Total time downstream and upstream together in the second situation = 672 minutes = 11.2 hours.

Concept / Approach:
The problem has two stages:

• First, use the time ratio condition (upstream time is twice downstream time) to relate b and s.
• Second, use the changed stream speed and the total time in the second scenario to solve for the numerical values of b and s.

We will derive equations from each scenario and solve them step by step.

Step-by-Step Solution:
Step 1: In the first scenario, downstream time t_d = 28 / (b + s) and upstream time t_u = 28 / (b − s).Step 2: Condition: t_u = 2 t_d ⇒ 28 / (b − s) = 2 * 28 / (b + s).Step 3: Cancel 28: 1 / (b − s) = 2 / (b + s).Step 4: Cross multiply: b + s = 2(b − s) ⇒ b + s = 2b − 2s ⇒ −b + 3s = 0 ⇒ b = 3s.Step 5: In the second scenario, stream speed is doubled to 2s. Then downstream speed = b + 2s and upstream speed = b − 2s.Step 6: Using b = 3s, downstream speed = 3s + 2s = 5s and upstream speed = 3s − 2s = s.Step 7: Total time for 28 km downstream and 28 km upstream is 11.2 hours: 28/(5s) + 28/s = 11.2.Step 8: Factor 28: 28(1/(5s) + 1/s) = 11.2.Step 9: Inside bracket, 1/(5s) + 1/s = 1/(5s) + 5/(5s) = 6/(5s).Step 10: So total time = 28 * 6 / (5s) = 168 / (5s) = 11.2.Step 11: Solve 168 / (5s) = 11.2 ⇒ 168 = 11.2 * 5s = 56s ⇒ s = 168 / 56 = 3 km/h.Step 12: From b = 3s, b = 3 * 3 = 9 km/h.

Verification / Alternative check:
Check the first scenario: with b = 9 and s = 3, downstream speed = 12 km/h and upstream speed = 6 km/h. Downstream time = 28/12 hours; upstream time = 28/6 hours, which is exactly twice as long. For the second scenario, new downstream speed = 9 + 2*3 = 15 km/h and new upstream speed = 9 − 2*3 = 3 km/h. Total time = 28/15 + 28/3 = 28/15 + 140/15 = 168/15 = 11.2 hours or 672 minutes, exactly as given. This confirms the correctness of the values.

Why Other Options Are Wrong:
Pairs like 12 km/h & 3 km/h or 8 km/h & 2 km/h do not satisfy both the time ratio and the second-scenario timing when checked carefully. The pair 9 km/h & 6 km/h gives impossible upstream or downstream speeds in the doubled-stream scenario. Only 9 km/h, 3 km/h fulfills all the given conditions.

Common Pitfalls:

• Forgetting to convert 672 minutes into hours (11.2 hours).
• Not correctly using the fact that the upstream time is twice the downstream time.
• Making algebraic mistakes when simplifying 28/(5s) + 28/s.
• Confusing the first scenario's speeds with the second scenario's doubled-stream speeds.

Final Answer:
The speed of the boat in still water is 9 km/h and the speed of the river flow is 3 km/h.