The clock advances by 10 minutes every 24 hours. Over a period of 24 hours, the clock will be 10 minutes ahead of the actual time. The observation period spans 1 day and 5 hours (from 10 am to 3 pm the following day), so the total elapsed time is:
\ \[\ 24 \, \text{hours} + 5 \, \text{hours} = 29 \, \text{hours}.\ \]\
Given the clock gains 10 minutes every 24 hours, over 29 hours it will gain:
\ \[\ \frac{10 \times 29}{24} = 12.08 \, \text{minutes}.\ \]\
Therefore, when the watch indicates 3 pm, the actual time is:
\ \[\ 3 : 00 - 12.08 \, \text{minutes} = 2 : 47 : 52 \, \text{pm}.\ \]\
In conclusion, the true time is roughly 48 minutes past 2 pm.