At exactly 12 o'clock noon the hour hand of a clock begins to move at six times its normal speed, and the minute hand begins to move backwards at five-sixths its normal speed. When the two hands next coincide, what will be the correct time?
When you say "coincide", do you mean when they are "on top of each other?".
If that is what you mean, then:
Every minute on a regular Clock, the hour-hand moves forward 1/2 minute, while the minute-hand moves backwords 5/6 minute.....So, since they are moving towards each other then:
1/2m + 5/6m =60 minutes.
Solve for m and m = 45 minutes(on a regular Clock).
So, the hour-hand will be: 1/2 x 45 =22.5 minutes on your faulty Clock. And:
5/6 x 45 =37.5 minutes (backwards) for the minute-hand. Or: 60 - 37.50 =22.50 when both hands will be on top of each other.
So 45 minutes will have passed on a regular Clock, or the time will be:12:45 pm.
