How many times do the hands overlap in a working day?

Question

An office clock runs from 9:00 am to 5:00 pm. How many times during those eight hours do the hour and minute hands point in exactly the same direction? List the times to the nearest minute, and explain why the answer is not simply eight.

Accepted Answer

Seven times: at about 9:49, 10:55, 12:00, 1:05, 2:11, 3:16 and 4:22. The overlap times. The hands coincide at 12:00 and then every 12/11 hours, which is 65 and 5/11 minutes. Counting from noon backwards and forwards, the overlaps in a 12-hour cycle fall at 12:00, 1:05:27, 2:10:55, 3:16:22, 4:21:49, 5:27:16, 6:32:44, 7:38:11, 8:43:38, 9:49:05 and 10:54:33. Between 9:00 and 5:00. From the list: 9:49, 10:55, 12:00, 1:05, 2:11, 3:16, 4:22. The next overlap, 5:27, is after the working day ends, and the previous one, 8:44, is before it starts. That is seven overlaps. Why not eight. Eight hours cont…

How many times do the hands overlap in a working day?

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