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

Introduction / Context: This is a slightly more algebraic boats and streams problem that uses symbolic distance d rather than concrete numbers. We are told the total time for an upstream and downstream trip of distance d each, and the time for an upstream trip of distance 2d. Using these conditions, we can determine the upstream and downstream speeds and then compute the time needed for a downstream trip of length 2d. The problem tests comfort with symbolic manipulation and relative speed concepts.

Given Data / Assumptions:

• Time to row d km upstream and d km downstream = 5 hours 15 minutes = 5.25 hours.
• Time to row 2d km upstream = 7 hours.
• Let upstream speed be u km/h and downstream speed be v km/h.
• Speeds are constant and d is a positive distance.

Concept / Approach: We can express times in terms of distance and speed:

• Time upstream for distance d = d / u.
• Time downstream for distance d = d / v.
• Time upstream for distance 2d = 2d / u.

From the second condition, 2d / u = 7, which gives u in terms of d. We substitute that into the first condition d / u + d / v = 5.25 and solve for v. Once v is known, time for 2d downstream is 2d / v.

Step-by-Step Solution: Let u be upstream speed and v be downstream speed. Given 2d / u = 7, so u = 2d / 7. Total time for one upstream d and one downstream d is d / u + d / v = 5.25 hours. Substitute u: d / (2d / 7) + d / v = 5.25. Simplify d / (2d / 7) = 7 / 2 = 3.5 hours. So 3.5 + d / v = 5.25. Then d / v = 5.25 - 3.5 = 1.75 hours. Thus v = d / 1.75. Time for 2d downstream = 2d / v = 2d / (d / 1.75) = 2 * 1.75 = 3.5 hours. 3.5 hours = 3 hours 30 minutes.

Verification / Alternative check: We can verify by choosing a convenient value for d, for example d = 7 km. Then u = 2d / 7 = 2 km/h. Time for 2d upstream is 14 / 2 = 7 hours, which matches the given condition. From d / v = 1.75, with d = 7 we get v = 7 / 1.75 = 4 km/h. Then time for one upstream 7 km is 7 / 2 = 3.5 hours, and one downstream 7 km is 7 / 4 = 1.75 hours. Total is 3.5 + 1.75 = 5.25 hours, as required. Time for 2d downstream is 14 / 4 = 3.5 hours or 3 hours 30 minutes, confirming the result.

Why Other Options Are Wrong: Options such as 4 hours 10 minutes or 4 hours 1 minute would correspond to different downstream speeds that conflict with the earlier conditions about 5.25 hours and 7 hours. Similarly, 3 hours 15 minutes is less than the calculated required time. Only 3 hours 30 minutes is consistent with all given time relations.

Common Pitfalls: Students sometimes attempt to guess d rather than treating it symbolically, and may choose inconsistent values. Another common mistake is to confuse the upstream and downstream times or forget to convert 5 hours 15 minutes into 5.25 hours. Working carefully with algebra and converting mixed time formats into pure hours helps avoid such misunderstandings.

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