let the 3 o'clock position be the zero position for both hands.
at t = 0 (5:15) the minute hand is at and is moving with speed radians per hr.
at t = 0, the hour hand is at radians, moving at radians per hr.
for t in hours, the two hands coincide when their respective positions are the same, ...
hrs = 12 min + 16.363636... sec
the hands coincide at 5:27 + 16 and sec