The sum of a boat’s upstream and downstream speeds is 82 km/h. The boat travels 105 km upstream in 3 hours. How much time (in hours) will it take to cover 126 km downstream?

Introduction / Context:
This question combines upstream and downstream speeds with information about actual travel time. We are given the sum of upstream and downstream speeds and the time taken to travel a known upstream distance. From that we find the individual upstream speed and obtain the downstream speed. Finally we use the downstream speed to compute how long it will take to travel another known distance with the current. This kind of question tests algebraic manipulation and application of the basic speed relations.

Given Data / Assumptions:

• Upstream distance covered = 105 km.
• Time taken upstream = 3 hours.
• Sum of upstream and downstream speeds = 82 km/h.
• Downstream distance to be covered = 126 km.
• Speeds are constant and current is uniform.

Concept / Approach:
We first compute upstream speed from distance and time. Then we use the given sum of upstream and downstream speeds to find downstream speed. Finally, time downstream is simply distance divided by downstream speed. Rounding to one decimal place is enough to match the options given.

Step-by-Step Solution:
Upstream speed = distance / time = 105 / 3 = 35 km/h. Let u be upstream speed and d be downstream speed. Given u + d = 82 km/h. We have u = 35, so 35 + d = 82. Thus d = 82 - 35 = 47 km/h. Downstream speed = 47 km/h. Downstream distance to be covered = 126 km. Time downstream = distance / speed = 126 / 47 hours. Compute 126 / 47 which is approximately 2.68 hours. Rounding to one decimal place gives about 2.7 hours.

Verification / Alternative check:
We can estimate 47 * 2.7 = 47 * (2 + 0.7) = 94 + 32.9 = 126.9 km, which is very close to 126 km. Using the exact fraction 126 / 47 is about 2.68 hours, so option 2.7 hours is the best match among the choices. The upstream speed of 35 km/h is consistent with 105 km in 3 hours and the sum of speeds being 82 km/h, confirming the calculations.

Why Other Options Are Wrong:
If the time were 2.5 hours, the distance covered at 47 km/h would be only 117.5 km, which is too short. If the time were 2.6 hours, the distance would be 122.2 km, still less than 126 km. If the time were 2.8 hours, the distance would be 131.6 km, which is too large. Only around 2.7 hours gives a distance close to 126 km, matching the required value.

Common Pitfalls:
A common mistake is to misinterpret the sum of speeds 82 km/h as twice the still water speed without checking the upstream data. Another error is rounding the intermediate speeds too early, which can produce a noticeably different final time. It is good practice to keep fractions exact until the last step and then round only when comparing with the given options.

Final Answer:
The boat will take approximately 2.7 hours to travel 126 km downstream.