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At what time between 9 and 10 o'clock will the hands of a clock be together?

Question

At what time between 9 and 10 o'clock will the hands of a clock be together?

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Solution

The minute hand moves 12 times as fast as the hour hand. At 9 o'clock, the minute hand is at 45 minutes and the hour hand is at 9 hours.

The minute hand needs to catch up to the hour hand. The minute hand is 15 minutes (or 1/4 of a circle) behind the hour hand.

Since the minute hand moves 12 times as fast as the hour hand, it will take 1/12 of an hour (or 5 minutes) for the minute hand to catch up 1/4 of a circle to the hour hand.

Therefore, the hands of the clock will be together at 9:27.27, or approximately 9:27 and 16 seconds.

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