The minute and the hour hand of a watch meet every 65 minutes.
How much time does the watch lose or gain ?
In order to find out how much time the clock gains or loses, we need to find out how frequently the hands of a properly running clock meet.
Assume that the hands are together at a given time. An hour later, the minute hand will be in the same position, and the hour hand will be 30 degrees ahead. Assume x minutes pass from that time, and define the functions m(x) and h(x) as the position of the minute and hour hand x minutes later, given as degrees from where the minute hand is at x=0:
m(x) = 6x
h(x) = 30 + (x/60)*30 = 30 + x/2
When the hands meet, m(x) = h(x); thus, 6x = 30 + x/2, and x=60/11 (i.e. 5 minutes and 27.27272727... seconds). Thus, on a properly running clock the hands meet about every 65 minutes and 27 seconds; the clock in the question is therefore gaining about 27 seconds every 65 minutes, or about 25 seconds per hour.
Solution
