If two incorrect watches are set at 12:00 noon at correct time, when will both watches show the correct time for the first time given that the first watch gains 1 min in one hour and second watch looses 4 min in two hours.

First watch:
It shows correct time when it creates difference of 12 hours.
So, to create difference of 12 hour, time required = $$\frac{{60 \times 12}}{{24}}$$   = 30 days

Second watch:
It shows correct time when it creates difference of 12 hours.
So, to create difference of 12 hour, time required = $$\frac{{30 \times 12}}{{24}}$$   = 15 days
Now, LCM of 30 and 15 gives the time when they show correct time together. Thus, required time = 30 days, at same time.