A watch that gains uniformly is 5 minutes slow at 8:00 a.m. on Sunday and 5 minutes 48 seconds fast at 8:00 p.m. on the following Sunday. At what time was the watch showing the correct time?

Introduction / Context:
This question deals with a watch that gains time at a constant uniform rate. We are told how slow it is at one moment and how fast it becomes after a certain number of days and hours. From this information we must find the exact moment when the watch was correct. These problems test understanding of linear change over time and simple proportional reasoning with minutes and hours.

Given Data / Assumptions:

• The watch is 5 minutes slow at 8:00 a.m. on Sunday.
• The watch is 5 minutes 48 seconds fast at 8:00 p.m. on the following Sunday.
• The gain or change in error of the watch is uniform over time.
• We are asked to find when the watch showed the correct time between these two instants.

Concept / Approach:
The watch error changes linearly from a negative value (slow) to a positive value (fast). The total change in error divided by the time between the two observations gives the rate of change of error per hour. Knowing the rate, we can compute how long it takes for the error to go from minus 5 minutes to zero, which is exactly when the watch is correct. We then add this duration to the starting instant.

Step-by-Step Solution:
Step 1: Express all errors in minutes. At 8:00 a.m. Sunday: error = -5 minutes (watch is slow). At 8:00 p.m. next Sunday: 5 minutes 48 seconds fast. 48 seconds = 48 / 60 = 0.8 minute. So final error = +5.8 minutes. Total change in error = 5.8 - ( -5 ) = 10.8 minutes. Step 2: Find the time interval between the two observations. From Sunday 8:00 a.m. to next Sunday 8:00 a.m. = 7 days = 7 * 24 = 168 hours. From Sunday 8:00 a.m. to Sunday 8:00 p.m. of the following week is 7 days 12 hours = 7.5 days. Total hours = 7.5 * 24 = 180 hours. Step 3: Rate of change of error per hour. Rate = total change in error / total time. Rate = 10.8 minutes / 180 hours = 0.06 minute per hour. Step 4: Time needed for error to reach zero. At start the error is -5 minutes. We need an increase of 5 minutes. Time needed = 5 minutes / 0.06 minute per hour. Time needed = 5 / 0.06 hours = 83.333... hours. That is 83 hours 20 minutes. Step 5: Add this to the starting instant. From 8:00 a.m. Sunday plus 72 hours = 8:00 a.m. Wednesday (3 days later). Remaining = 11 hours 20 minutes. 8:00 a.m. Wednesday + 11 hours 20 minutes = 7:20 p.m. Wednesday. So the watch is correct at 20 minutes past 7:00 p.m. on Wednesday.

Verification / Alternative check:
If the watch is correct at 7:20 p.m. Wednesday, then by 8:00 p.m. next Sunday it has gained 5.8 minutes over 180 - 83.333... = 96.666... hours. The gain is: rate * time = 0.06 * 96.666... = 5.8 minutes, which matches the given final error, so our calculation is consistent.

Why Other Options Are Wrong:
Option a (7:00 p.m. Wednesday) and option c (15 minutes past 7:00 p.m.) correspond to smaller time intervals and would not accumulate exactly 5 minutes of correction from the initial slow reading. Option d (8:00 p.m. Wednesday) overshoots and would make the watch already fast instead of exactly correct at that moment.

Common Pitfalls:
Typical mistakes include mixing days and hours, or forgetting to convert seconds into fractional minutes. Some learners divide 10.8 by 7.5 instead of by 180 hours, which produces an incorrect rate. Another mistake is treating the change as symmetric at the midpoint in time without checking the exact rate, which only works if the start and end errors are equal in magnitude, which is not the case here.

Final Answer:
The watch was showing the correct time at 20 minutes past 7:00 p.m. on Wednesday.