Expert: Scott A Wilson - 5/19/2010

Question
1)  At what time between 2 and 3 o'clock are the hands of the clock be 15 minutes apart?

2)  At what time after between 7 and 8 o'clock are the hands of a clock be 20 minutes apart?

3) Find two different times between 11 and 12 o'clock when the hands of a clock are at right angles to each other?

If you can, you can use this:
x = the number of minute spaces moved by the minute hand
x/12 = the number of minute spaces moved by the hour hand
or if you want your own way to solve this, you can!

Thanks :)

Answer
1) The position of the hour hand at this point...
Well, there are 60 minutes in an hour, and that makes the hour hand move 5 minutes each hour.
We know that 2 o'clock puts the hour hand at 10 minutes.

Since the hour hand passes 5 minutes every hour, it goes 5/60 = 1/12 per minute.
The minute hand goes at 1 minute for each minute.

The correct time could be found by solving for x as minutes past 2 with the equation
10 + x/12 = x - 15.  This does not use x + 15 since that would put the time before 2 o'clock.

2) There are two times since at 7 o'clock, we'll look at ±20 minutes from the 7.
The base number for the hour hand is 7*5 = 35 minutes.  The equations to solve would then be
35 + x/12 = x + 20 and 35 + x/12 = x - 20.  When the 2nd equation is solved, it puts x at 60,
which means it is really 8 o'clock, so only the 1st one works.

3) When the hands are at right angles to each other, there is 15 minutes between them.
This means that 11 + x/12 = x - 15, for it it is x + 15, x is not between 0 and 60,
for then x would be negative.  The other gives 26 = 11x/12, and that solution put x
between 0 and 60.