How often do the hour and and minute hand of an analog clock point in the same direction?
The hands start pointing the same direction at midnight, then the hour hand moves clockwise (!) by 360° in 12 hours. The minute hand moves clockwise at the rate of 360° in one hour. So when does the minute hand, moving at 360°/ hour lap the hour hand moving at 30° per hour? When
360t = 30t mod 360
or equivalently
12t = t mod 12
which means t is a multiple of 12/11.
So the times are at 12n/11 hours for n = 0, 1, 2, …, 21, which means 22 times a day. The first time after midnight is 12/11 hours later, or at 5/11 of a minute past 1:05.
If you also consider the second hand, the three hands will only line up at midnight and noon.
Another way of looking at this that I personally find slightly more transparent: think about the motion of the minute hand _relative to the hour hand_. The minute hand goes around at a certain speed. The hour hand goes in the same direction at 1/12 the speed, so the difference between the two goes in the same direction at 11/12 the speed. So if the minute hand makes a full rotation every hour, the (minute hand minus hour hand) makes a full rotation every 12/11 hours.
(Obviously this is exactly equivalent to what you wrote.)