SOLUTION: starting at 8am determine the number of times in the next 24 hours for which the hour and minute hands on a clock form a 180% angle

The answer is 22 times in 24 hours.  Here's how to get it:

The hour hand makes 1 complete revolution in 12 hours.  That's
 or 30 degrees/hour.  The minute hand make 1 complete
revolution in 1 hour, so that's 360 degrees/hour.

The angle between the hands is therefore increasing at the rate of
(360-30) degrees/hour, or 330 degrees/hour.  Therefore in 24 hours
the angle between the hands has increased by 24x330° or 7920°.
Each revolution is 360° so we divide and get 22 
revolutions of the angle between the hands exactly.  Since the hands form 
a 180° angle exactly once each revolution, that's 22 times in 24 hours.

That's the answer, but for fun let's find those 22 times as a check.

At 8:00 the hands form a 120° angle.  Therefore the angle between the hands
must increase by 60° in order to be 180° apart for the first time.  The angle
between the hands increases at 330 degrees/hour, so it will take it 60/330 or
2/11 ths of an hour for the hands to form a 180° angle.  2/11ths of an
hour is  or  minutes after 8:00, or 

8:10 10/11, the first time after 8:00 the hands will form a 180° angle.

 or  hours later the hands
will form a 180° angle again.   of an hour is  or
 minutes, so that will be 1 hour and  minutes later which will be 
at 9:15 15/11 or 9:16 4/11.  

So here are the 22 exact times when the hands will form a 180° angle.

 1.  8:10 10/11
 2.  9:16 4/11
 3.  10:21 9/11
 4.  11:27 3/11
 5.  12:32 8/11
 6.  1:38 2/11
 7.  2:43 7/11
 8.  3:49 1/11
 9.  4:54 6/11
10.  6:00
11.  7:05 5/11
12.  8:10 10/11
13.  9:16 4/11
14.  10:21 9/11
15.  11:27 3/11
16.  12:32 8/11
17.  1:38 2/11
18.  2:43 7/11
19.  3:49 1/11
20.  4:54 6/11
21.  6:00
22.  7:05 5/11


Edwin