If a regular analog clock currently reads 4:33, in how many minutes will the hour hand and the minute hand overlap?

How can I solve this? I know that the minute hand travels six degrees per minute and the hour hand travels 0.5 degrees per minutes, and I've tried setting up systems of equations, but it isn't working. Any help is greatly appreciated.

  • $\begingroup$ Find the positions as functions of time and set them equal $\endgroup$ – David Peterson Feb 22 '15 at 21:47
  • 3
    $\begingroup$ Starting from 12:00 the minute hand will overtake the hour hand eleven times in the span of 12 hrs. So they overlap after every $1\dfrac1{11}$ hours. After 4:33 the next occurrence is at $5\dfrac5{11}$ or, at about 5:27:16.4. $\endgroup$ – Jyrki Lahtonen Feb 22 '15 at 22:00

First note that in one hour the space gap b/w the hour hand and minute hand is 55 minutes on the clock, so in actual 60 minutes the minute hand gains 55 minutes over the clock hand. At 4:00 the hands are 20 minutes apart. To averlap the minute hand has to gain 20 minutes over the hour hand which take time = $\dfrac{60}{55}\times 20 = 21.81 $ minutes but this is before 4:33 so we have to find the next overlap which requires the minute hand to gain another 60 minutes over the hour hand, that is the actual time required further $\dfrac{60}{55}\times 60 = 65.45$ minutes.

So the next overlap after 4:33 will occur at $4 hours + (21.81+65.45)minutes = 5:27$ of clock.


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