My watch, which gains time uniformly, was 2 minutes slow at noon on Sunday,…

Concept: When a watch gains or loses time at a uniform (constant) rate, its error changes linearly with elapsed real time. If two signed error readings are known at two different times (slow taken as negative, fast as positive), the total error swing over the interval between them fixes the watch's rate of gain per hour. The watch shows the correct time exactly when its error crosses zero -- by proportionality, this happens after a time (measured from the first reading) equal to: (time between the two readings) x (size of the initial error) / (total error swing).

1. Find the time between the two readings: Sunday noon to the following Sunday 2 PM is 7 days plus 2 hours, i.e. 170 hours.

2. Write the two error readings with sign, in seconds: first reading = -2 minutes = -120 seconds (slow); second reading = +4 minutes 48 seconds = +288 seconds (fast).

3. Total error swing over the 170 hours = 288 - (-120) = 408 seconds.

4. The watch shows the correct time when its error becomes 0, i.e. after it closes the initial 120-second deficit. By proportionality, time taken = 170 hours x 120 seconds / 408 seconds = 50 hours.

5. 50 hours after Sunday noon is 2 days and 2 hours later, which is Tuesday at 2 PM.

Cross-check: rate of gain = 408 seconds / 170 hours = 2.4 seconds per hour. In 50 hours the watch gains 50 x 2.4 = 120 seconds = 2 minutes, exactly closing the initial 2-minute deficit -- confirming Tuesday 2 PM.