5. A company's working hours are from 8:30 AM to 5:30 PM. During this period, the hour and minute hands of the clock overlap times.
【Analysis】Method one: At 8:30, the hour hand is between 8 and 9, and the minute hand points to 6. At 9:00, the hour hand points to 9, and the minute hand points to 12, the minute hand has surpassed the hour hand. Therefore, between 8:30 and 9:00, the minute hand and the hour hand overlap once.
Similarly, between 9:00 and 10:00, 10:00 and 11:00, the minute hand and the hour hand each overlap once. Note that, between 11:00 and 12:00, the minute hand and the hour hand overlap at exactly 12:00, and between 12:00 and 1:00, the minute hand and the hour hand also overlap at exactly 12:00, which counts as 1 time. Next, between 1:00 and 2:00, 2:00 and 3:00, 3:00 and 4:00, 4:00 and 5:00, the minute hand and the hour hand each overlap once.
At 5:00, the hour hand points to 5, and the minute hand points to 12. At 5:30, the hour hand is between 5 and 6, and the minute hand points to 6, the minute hand has surpassed the hour hand. Therefore, between 5:00 and 5:30, the minute hand and the hour hand overlap once. In summary, they overlap 9 times.
Method two: Circular track
From 8:30 to 5:30, a total of 9 hours have passed. During these 9 hours, the minute hand has run 9 laps, and the hour hand has run \(9 \div 12 = 0.75\) laps. The minute hand has run 8.25 more laps than the hour hand.
Initially, at 8:30, the hour hand is between 8 and 9, and the minute hand points to 6, the minute hand is behind the hour hand by \(75 \div 360 < 0.25\) laps. Therefore, within the 0.25 laps that the minute hand is behind, it will catch up to the hour hand once. After that, for every additional lap the minute hand runs compared to the hour hand, it catches up once. Thus, running 8.25 more laps, it catches up 9 times, meaning the minute hand and the hour hand overlap 9 times.