Amith can row a boat a distance d km upstream and the same distance d km downstream in a total time of 5 hours 15 minutes. He can also row the boat 2d km upstream in 7 hours. How long will it take Amith to row the distance 2d km downstream?

Introduction / Context:
This is a multi step boats and streams problem involving an unknown distance and two different journey scenarios. It requires careful algebraic modelling of upstream and downstream times and is a good test of problem solving with averages and relative speeds on water.

Given Data / Assumptions:
- Amith rows a distance d km upstream and d km downstream in 5 hours 15 minutes. - 5 hours 15 minutes = 5.25 hours. - He rows 2d km upstream in 7 hours. - Let b be the speed of the boat in still water (km/h). - Let c be the speed of the current (km/h). - Upstream speed = b - c, downstream speed = b + c.

Concept / Approach:
Time equals distance divided by speed. The given information can be translated into two equations involving d, b, and c. However, we can work smartly by focusing on times per unit distance instead of finding b and c explicitly. By letting t up denote time to travel d km upstream and t down denote time to travel d km downstream, we can first compute t up from the 2d upstream journey, then use the total 5.25 hours for d up and d down to find t down. From that, the time for 2d downstream is simply 2 * t down.

Step-by-Step Solution:
Step 1: For upstream, speed is b - c, so time for distance d upstream is d / (b - c). Step 2: For downstream, speed is b + c, so time for distance d downstream is d / (b + c). Step 3: From the first scenario, total time for d upstream and d downstream is 5.25 hours. Step 4: So d / (b - c) + d / (b + c) = 5.25. Call this Equation (1). Step 5: From the second scenario, 2d km upstream takes 7 hours, so 2d / (b - c) = 7. Step 6: Divide both sides by 2 to get d / (b - c) = 3.5 hours. Step 7: Substitute d / (b - c) = 3.5 into Equation (1): 3.5 + d / (b + c) = 5.25. Step 8: Hence d / (b + c) = 5.25 - 3.5 = 1.75 hours. Step 9: Time to row 2d km downstream = 2 * [d / (b + c)] = 2 * 1.75 = 3.5 hours. Step 10: 3.5 hours = 3 hours 30 minutes.

Verification / Alternative check:
Notice that upstream time per d is 3.5 hours and downstream time per d is 1.75 hours. For the 5.25 hour combined trip, 3.5 + 1.75 = 5.25, which matches the given total. For the 2d upstream trip, 2 * 3.5 = 7 hours, which matches the given 7 hours. Therefore the derived values are consistent, and 2d downstream must indeed take 2 * 1.75 hours = 3.5 hours.

Why Other Options Are Wrong:
- 4 hours 10 minutes and 4 hours 1 minute are larger than 3.5 hours and would not align with the ratio between upstream and downstream times. - 3 hours 15 minutes and 3 hours 45 minutes give downstream times per d that do not combine correctly to produce the 5.25 and 7 hour totals given in the question.

Common Pitfalls:
Learners often try to find b and c directly and get lost in algebra, or they misuse the distances d and 2d. Another common mistake is to convert 5 hours 15 minutes incorrectly, or to mix hours and minutes during calculations. Working in decimal hours and focusing on time per d is a clean, efficient approach here.

Final Answer:
Amith will take 3 hours 30 minutes to row 2d km downstream.