In a watch, the minute hand crosses the hour hand for the third time exactly after every 3 hr 18 min and 15s of normal time. What is the time gained or lost by this watch in one day?

In a normal watch, the minute hand crosses the hour's hand after every 1 hour 5 minutes and 27 seconds.
So, the third time the hour's hand crosses the minute's hand is after 3 hours 16 minutes and 21 seconds.

In this watch, the time taken for this to happen is 3 hours 18 minutes and 15 seconds.
Hence, the watch loses 1 minute and 54 seconds after every 3 hours 18 minutes and 15 seconds.
18 minutes and 15 seconds = 1095 seconds = 1095/3600 $$\approx$$ .304 hours.
=> 3 hours 18 minutes and 15 seconds = 3.304 hours
So, time lost in a day is $$1\frac{54}{60}*\frac{24}{3.304} = \frac{114}{60}*\frac{24}{3.304} \approx 13.8$$
So, the time lost by the watch in a day is approximately equal to 13 minutes and 48 seconds.