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

Introduction / Context:
This question combines knowledge of speed, distance, and time with boats and streams concepts. You are given the sum of upstream and downstream speeds and one actual upstream journey. From this, you must work out the downstream speed and then determine the time taken to travel a different downstream distance.

Given Data / Assumptions:
- Sum of upstream and downstream speeds = 82 km/h. - Upstream distance covered = 105 km. - Time taken upstream = 3 hours. - Required: time to cover 126 km downstream. - Let upstream speed be u km/h and downstream speed be d km/h.

Concept / Approach:
First, find the upstream speed using speed = distance / time. Then use the given sum u + d = 82 to find downstream speed d. Finally, use the standard formula time = distance / speed to find the time for the downstream trip of 126 km. An approximate decimal answer is acceptable when the division is not exact.

Step-by-Step Solution:
Step 1: Upstream speed u = distance / time = 105 / 3 = 35 km/h. Step 2: We know u + d = 82, so 35 + d = 82. Step 3: Solve for d: d = 82 - 35 = 47 km/h. Step 4: Downstream distance required = 126 km. Step 5: Time taken downstream = distance / downstream speed. Step 6: Time = 126 / 47 hours. Step 7: 47 * 2 = 94 and 126 - 94 = 32, so 126 / 47 is a little more than 2.6. Step 8: Actual value 126 / 47 is approximately 2.68 hours, which rounds to about 2.7 hours.

Verification / Alternative check:
If we take 2.7 hours as an approximate time, the implied distance at 47 km/h is 47 * 2.7 = 126.9 km, which is very close to 126 km, allowing for rounding. Using 2.6 hours would give only 122.2 km, which is too low. Thus 2.7 hours is the best approximation among the options.

Why Other Options Are Wrong:
- 2.5 hours and 2.6 hours imply speeds that do not match 47 km/h when combined with 126 km. - 2.8 hours would suggest a distance greater than 126 km at 47 km/h. - 3 hours is clearly too long and would correspond to 141 km at 47 km/h.

Common Pitfalls:
A common error is to average 82 and 35 instead of using their relationship, or to miscalculate the upstream speed. Another frequent mistake is to round too early and then get a significantly different answer. Always find the exact division first, then round to the nearest option provided.

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