Solution
The Hasty Wristwatch
Show the problem again
A perfect clock has its hour and minute hands coincide every 720/11 minutes of its own time. A faulty wristwatch, whose two hands still move in the correct 12:1 ratio, is observed to have its hands coincide every 65 minutes of true time. How many minutes does the watch gain over 24 hours of true time? Give the exact answer as a reduced improper fraction, or as a decimal rounded to 2 or 3 places.
Worked solution
The watch's own time between successive overlaps is always 720/11 = 65 5/11 minutes, because that follows from the 12:1 ratio of the hand speeds. Since it takes only 65 true minutes, the watch runs fast by a factor (720/11)/65 = 144/143. In 1440 true minutes it therefore shows 1440 × 144/143 minutes, a gain of 1440/143 minutes, which is **1440/143** (about 10.07 minutes).
Source: Classic clock-hand riddle (the watch whose hands meet at the wrong interval) in wide circulation; no traceable original source. Statement written for AxiomIQ.