SOLUTION: It is between 2 and 3pm. A person looking his watch mistakes the hour hand for the minute hand and thinks that the time of the day is 55 minutes earlier than it really is. What is

To ask "what is the true time ?" in this situation is the same as to ask:
  

     At what time between 2:00 pm  and 3:00 pm the angle between the minute and the hour hands is exactly 30 degrees ?  

         ( 30 degrees  correspond to 5 minutes, 1/12 of the full revolution of the minute hand )


And the answer to it is the solution of this equation:

 = 30.        (! 6t is the angular position of the minute hand measured in degrees from the "vertical up" position;
                            ( (60 + (6/12)*t) is the position of the hour hand measured in degrees from the "vertical up" position).


Or, which is the same,

 = 30.

Simplify and solve:

60 - 0.5t - 6t = 30  ====>  5.5t = 60 - 30  ====>  5.5t = 30  ====>

t =  = 5.(45) = 5 minutes and 27.(27) seconds = 5 minutes and 27.(27) seconds (approximately).


Here the record 5.(45)  means 5 followed periodical decimal fraction with the period 45;
                27.(27) means 27 followed periodical decimal fraction with the period 27.