A watch that gains uniformly is 2 minutes slow at noon on Monday and 4 minutes 48 seconds fast at 2:00 p.m. on the following Monday. At what time was the watch exactly correct?

Introduction / Context:
This question is another example of a uniformly gaining watch where we are given its error at two different times and asked to find when it was correct. The watch transitions from being slow to being fast, and the moment of being correct is when the cumulative gain cancels the initial delay. The reasoning is based on uniform change in error over a known time interval.

Given Data / Assumptions:

• At noon on Monday, the watch is 2 minutes slow, so its error is -2 minutes.
• At 2:00 p.m. on the following Monday, it is 4 minutes 48 seconds fast, so its error is +4.8 minutes.
• The gain is uniform over this period.
• We need to find the exact time between these instances when the watch shows correct time.

Concept / Approach:
The watch error changes linearly from -2 minutes to +4.8 minutes. The total change in error divided by the total time gives the rate of change of error per hour. The time needed for the error to go from -2 minutes to zero is then found by dividing 2 minutes by this rate. Adding this time to the starting moment gives the required instant when the watch was correct.

Step-by-Step Solution:
Step 1: Convert errors to minutes. Initial error at Monday noon = -2 minutes. Final error at next Monday 2:00 p.m. = +4 minutes 48 seconds. 48 seconds = 48 / 60 = 0.8 minute. So final error = +4.8 minutes. Total change in error = 4.8 - ( -2 ) = 6.8 minutes. Step 2: Compute time between the two readings. From Monday noon to next Monday noon = 7 days = 7 * 24 = 168 hours. From Monday noon to next Monday 2:00 p.m. = 7 days plus 2 hours = 170 hours. Step 3: Rate of change of error. Rate = total change / total time = 6.8 minutes / 170 hours. Rate = 0.04 minute per hour. Step 4: Time needed for error to go from -2 minutes to zero. We need an increase of 2 minutes. Time needed = 2 minutes / 0.04 minute per hour = 50 hours. Step 5: Add this to the starting time, Monday noon. 50 hours = 2 days plus 2 hours. Monday noon + 2 days = Wednesday noon. Add 2 hours to reach Wednesday 2:00 p.m. Thus the watch is correct at 2:00 p.m. on Wednesday.

Verification / Alternative check:
From Wednesday 2:00 p.m. (when error is zero) to next Monday 2:00 p.m. is 5 days = 120 hours. At a rate of 0.04 minute per hour, the gain is: 0.04 * 120 = 4.8 minutes. This matches the final error, so the time we computed is consistent with the problem data.

Why Other Options Are Wrong:
Option a (Tuesday 2:00 p.m.) corresponds to only 26 hours from Monday noon and would not allow the error to grow by the required 2 minutes. Options c and d (Thursday or Friday) correspond to larger time spans and thus give too much gain before the watch becomes correct, which contradicts the sign changes of the error.

Common Pitfalls:
Some students incorrectly treat the midpoint in time between the two observations as the moment when the watch is correct, but this would only be true if the slow and fast errors had equal magnitude. Another common error is to mix hours and days incorrectly or to forget to convert the 48 seconds into a fraction of a minute.

Final Answer:
The watch was showing the correct time at 2:00 p.m. on Wednesday.