At what time between 4 and 5 o'clock will the hands of a watch point in opposite directions?

The hands of a clock are in exactly opposite directions 11/2 minutes past every hour. So, looking at the time between 4 and 5 o'clock, the hands will be in opposite directions at 22 minutes past 4.

Here's the step by step calculation:

1. We know that in 60 minutes, the minute hand gains 55 minutes on the hour hand (because in 60 minutes, the minute hand makes one full circle, 360 degrees, and the hour hand moves 30 degrees, which is 1/12 of the full circle, so the difference is 360 - 30 = 330 degrees, which is 11/12 of the full circle, and 11/12 of 60 minutes is 55 minutes).

2. To be in opposite directions, they must be 30 minutes apart.

3. At 4 o'clock, they are 20 minutes apart (because the hour hand is at 4 and the minute hand is at 12, which is 20 minutes ahead of 4).

4. Therefore, the minute hand must gain another 10 minutes on the hour hand, which will take 10/55 of an hour, or approximately 10.91 minutes.

5. So, the hands will be in opposite directions at approximately 22 minutes past 4.