An absent-minded watch repairman connected the hour hand to the minute hand pinion and the minute hand to the hour hand pinion and set the clock at 6AM which was the correct time then. How soon after 6AM (at what time) will this clock give the correct time again?
[Math] Algebra clock problem
algebra-precalculus
Best Answer
Hint: Try writing out formulae for the positions of the hands on a correct clock and one the wrong clock in terms of time.
Correct clock:
the long hand is at $(\text{time in mins})\times (6^{\circ}), \mod 360^{\circ}$, since it starts straight up and moves by 360 degrees each hour;
the short hand is at $180^{\circ} +(\text{time in mins})\times \frac{1}{2}^{\circ}, \mod 360^{\circ}$, since it starts straight down and moves by 360 degrees every 12 hours.
Now write similar equations for the wrong clock, and look for values of time from which the long-hand equations give the same angle and so do the short hand equations.