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23 times. The time interval between each 'meeting' of the minute and hour hands is 65minutes and 27.272727(recurring) seconds There are number of ways of working this out. But my method allows you to calculate this mentally. it takes 60 minutes for a minutes hand to complete 360 degrees. ie 360/60 = 6 degrees per minute. and 12 * 60 minutes (12 hours) for a hours hand to complete 360 degrees. ie 360 / (60*12) = 0.5 degrees per minute so the relative speed of minutes hand with respect to hours hand is 6-0.5 = 5.5 degrees per minute. Now time taken to complete 360 degress gives you the time interval between each meeting of both the hands and that is 360/5.5 = 65.45 minutes . TADA !!

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