A watch which gains uniformly is 2 minutes low at noon on Monday and is 4 min.…

Concept: A watch that gains or loses time uniformly builds up error at a constant rate. Given the error at two known moments, the difference between those errors divided by the elapsed real time between them gives that constant hourly rate; the same rate can then be scaled to find exactly when the error returns to zero, i.e. when the watch shows the correct time.

Applying this here:

1. At noon on Monday the watch reads 2 minutes slow, so its error is -2 minutes relative to the correct time.

2. At 2 p.m. on the following Monday it reads 4 minutes 48 seconds fast, so its error is +4 minutes 48 seconds (+4.8 minutes).

3. The real time elapsed between these two readings is 7 full days plus 2 hours, i.e. 168 + 2 = 170 hours.

4. The error changes from -2 minutes to +4.8 minutes over these 170 hours, a total swing of 4.8 - (-2) = 6.8 minutes, so the watch gains at a constant rate of 6.8 / 170 = 0.04 minutes (2.4 seconds) every hour.

5. The watch shows the correct time exactly when its error returns to zero, that is, once it has gained back the original 2-minute deficit. At 0.04 minutes gained per hour, this takes 2 / 0.04 = 50 hours from the Monday-noon start.

6. 50 hours from Monday noon is 48 hours (2 full days, to Wednesday noon) plus 2 more hours, giving Wednesday at 2 p.m.

Cross-check: Checking this rate against the given data: after the full 170 hours, the accumulated gain is 0.04 x 170 = 6.8 minutes; starting from -2 minutes, the error becomes -2 + 6.8 = +4.8 minutes, i.e. 4 minutes 48 seconds fast -- exactly matching the second reading, which confirms the rate and the 50-hour correction time.

Result: So the watch shows the correct time at 2 p.m. on Wednesday.