lets see what we know
AT 12:00 hands are together
next time will be 1:05 + x - not sure what x is yet
next time will be 2:10 + 2x
next time will be 3:15 + 3x
next time will be 4:20 + 4x
..
...
finally at 11:55 + 11x
BUT we know that 11:55 + 11x is 12:00
Therefore 11x = 5 minutes
x= 5/11 min or 0.4545454545454545 min or 27.272727272727272 sec
therefore 4:20 + 4x = 4:21:49.090909090909 PM H:M:S
Hope my addition is correct this time.
Dr. G.