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.