Calculating the number of times the hour and minute hands are at 180 degrees on a clock:

There are a few key points to consider when determining the number of times the hour and minute hands overlap at a 180-degree angle on a clock:

- The minute hand moves 360 degrees in 60 minutes, which means it moves 6 degrees every minute.
- The hour hand moves 360 degrees in 12 hours, which means it moves 30 degrees every hour.

Determining the positions of the hands:

1. At 12:00:
- The hour hand is at 12 while the minute hand is at 12. They are not at a 180-degree angle.

2. Between 12:00 and 1:00:
- The hands move closer to each other but do not overlap at a 180-degree angle.

3. At 1:00:
- The hour hand is closer to 1 while the minute hand is at 12. They are not at a 180-degree angle.

4. Between 1:00 and 2:00:
- The hands move closer to each other but do not overlap at a 180-degree angle.

5. At 2:00:
- The hour hand is at 2 while the minute hand is at 12. They are not at a 180-degree angle.

Calculating the exact times:

The hour and minute hands overlap at a 180-degree angle approximately 22 times in a 12-hour period. This is because the minute hand gains on the hour hand at a rate of 5.5 degrees per minute (30 degrees per hour - 6 degrees per hour), which means they will be at a 180-degree angle every 65 and 5/11 minutes.

Therefore, the hour and minute hands are at a 180-degree angle on the clock approximately 22 times in a 12-hour period.

Within 24 hrs...i think its happen 23 times....