The watch's gain is consistent, adding the same amount of time each hour. We are given:
The elapsed time between 10:00 AM on Monday and 1:00 PM on Tuesday is 1 day and 3 hours, which equals 27 hours. During this period, the watch gains 4.5 minutes (from 2 minutes slow to 2.5 minutes fast).
The gain rate is calculated by dividing total gain by total time:
Rate of gain \(= \frac{4.5 \text{ minutes}}{27 \text{ hours}} = \frac{1}{6} \text{ minutes per hour}.\)
To find when the watch showed the accurate time, consider that it was 2 minutes slow at 10:00 AM on Monday. We need to determine when it gained those 2 minutes. The time to gain 2 minutes is:
Time taken to gain 2 minutes \(= \frac{2 \text{ minutes}}{\frac{1}{6} \text{ minutes per hour}} = 12 \text{ hours}.\)
Starting at 10:00 AM on Monday, adding 12 hours results in 10:00 PM on Monday, when the watch displayed the correct time.