SOLUTION: Shortly after 5 o'clock, when the minute hand and the hour hand of the clock made a 90 degree angle, Ian went out to walk the dog. On his return, after 5:30, the two hands again ma

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Question 1100015: Shortly after 5 o'clock, when the minute hand and the hour hand of the clock made a 90 degree angle, Ian went out to walk the dog. On his return, after 5:30, the two hands again made a 90 degree angle. For how many minutes was Ian out with his dog?
Found 2 solutions by richwmiller, Edwin McCravy:
Answer by richwmiller(17219)   (Show Source): You can put this solution on YOUR website!
24 hours?
Answer by Edwin McCravy(20054)   (Show Source): You can put this solution on YOUR website!
In 12 hours the minute hand makes 12 revolutions while the
hour hand makes only 1 revolution.  So the minute hand's
angular speed RELATIVE to the hour hands' angular speed is 
the difference between them or 12-1 = 11 revolutions per 
12 hours. 

Therefore from noon to midnight, there are 11 periods of 
time between consecutive times when the hands are exactly 
together.  That's approximately 1 hour and 5 minutes but 
EXACTLY 12/11 or 1 1/ll hours. 

So each of those periods of time is 12/11 hours (1 1/11 hours) 
in length.

At the beginning of each of those periods the hands are together.
Then, after 1/4 of one of those periods later, the hands are at 90°,
and, after 1/2 of one of those periods later, the hands are at 180°,
and, after 3/4 of one of those periods later, the hands are at 90° 
again.  Then 1/4 of one of those periods later, they are back 
together again.

So he went to walk the dog when exactly 1/4 of one of those periods
had lapsed, and returned when exactly 3/4 of that period had lapsed.

3/4 - 1/4 = 2/4 = 1/2.

So he was gone for half of one of those 12/11 hour periods or 
6/11 of an hour.  That's  32 8/11 minutes.

If you have any questions, ask me in the note form below and I'll
get back to you by email.

Edwin
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