.
.
. .

.
The solution comes from
 

Subj:.....A Clock Puzzle (S486, S601)
          From the book 
           "Mathematical Puzzles of Sam Loyd" 
            Edited by Martin Gardner 
            From: Dover Publications in 1959
 

If the minute hands goes twelve times faster than the hour
hand, then they will meet eleven times during every 12-hour
period.  By taking the eleventh part of 12 hours for our
constant, we find there will be a meeting of hands every
65 and 5/11 minutes, or every 65 minutes, 27 and 3/11
seconds.  The hands will next come together, therefore, at
5 minutes, 27 and 3/11 seconds past 1 o'clock.

.
.
..