If the minute hand and hour hand coincide after every 65 minutes, how much…

Concept: On a correctly-running analog clock, the minute and hour hands coincide once every 65 and 5/11 minutes (this is a fixed fact of the hands' relative speeds, independent of how well the clock keeps time). If a clock or watch is observed to coincide at some OTHER interval, the watch itself is running fast or slow: a SHORTER observed interval means the hands sweep past each other sooner than they should, so the watch gains time; a LONGER interval means it loses time. The size of the gain or loss scales up over all the cycles that occur in a day.

Application:

1. This watch's hands coincide every 65 minutes, which is 5/11 minutes shorter than the standard 65(5/11)-minute interval, so the watch gains 5/11 minutes every time it completes one such 65-minute cycle.

2. A full day has 24 x 60 = 1440 minutes, giving 1440/65 cycles of 65 minutes each in a day.

3. Total gain in a day = (number of cycles) x (gain per cycle) = (1440/65) x (5/11) = (1440 x 5)/(65 x 11) = 7200/715 = 1440/143 minutes.

4. Dividing 1440 by 143 gives a quotient of 10 with a remainder of 10, so the watch gains 10(10/143) minutes in a day.

Cross-check: 1440/143 is approximately 10.07, matching a whole-number part of 10 plus a small fraction. Independently, roughly 1440/65 = 22 cycles occur in a day, and 22 x 5/11 minutes is close to 10 minutes, confirming the size of the gain by a second route.

Hence, the watch gains 10(10/143) minutes in a day.