Question 1091690: At what time between 2 and 3 o'clock will the minute hand be 12 minutes spaces ahead of the hour hand?
Found 2 solutions by greenestamps, Theo:
Answer by greenestamps(13200)   (Show Source):
You can put this solution on YOUR website!
I find it easiest to work problems involving the angle between the hands of a clock by measuring all angles relative to 12 o'clock.
The arithmetic required to solve this kind of problem is based on the fact that the hour hand moves 1/12 of a revolution, or 30 degrees, in 1 hour (60 minutes), so its rate of rotation is 0.5 degrees per minute; and the minute hand moves one full rotation (360 degrees) in an hour (60 minutes), so its rate of rotation is 6 degrees per minute.
You want to find the time between 2 and 3 o'clock when the minute hand is "12 minutes" ahead of the hour hand. 12 minutes is 1/5 of an hour; so you want the minute hand to be 1/5 of 360 degrees, or 72 degrees, ahead of the hour hand.
At x minutes after 2, the angle the hour hand makes with 12 o'clock, in degrees, is
 (60 degrees to get to 2; then 0.5 degrees for each minute after 2)
And x minutes after 2, the angle the minute hand makes with 12 o'clock, in degrees, is
You want the minute hand to be "12 minutes", or 72 degrees, ahead of the hour hand, so you want
The minute hand will be "12 minutes" ahead of the hour hand at 2:24.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! it's scary, but i think i have it.
2:00 pm is at the 10 minute mark.
3:00 pm is at the 15 minute mark.
1 revoution of the minute hand is 60 minutes.
in those same 60 minutes, the hour hand has movee 5 minute marks (between 2:00 pm and 3:00 pm).
the hour hand is therefore moving at a rate that is 1/12 the rate of the minute hand.
so, every minute, the minute hand is moving 1 minute mark and, every minute, the hour hand is moving 1/12 of a minute mark.
this makes sense, because in 1 hour the minute hand moves 60 minute marks while the hour hand moves 1/12 * 60 = 5 minute marks.
at 2:00 pm, the hour hand is at the 10 minute mark and the minute hand is at the 0 minute mark.
if you let x represent the number of minute marks that the minute hand moves in x minute, then x/12 is equal to the number of minute marks that the hour hand moves in x minutes.
the hour hand starts at the 10 minute mark and moves x/12 minute marks in x minutes.
the minute hand starts at the 0 minute mark and move x minute marks in x minutes.
you want the minute hand to be 12 minute marks ahead of the hour hand.
your formula becomes (0 + x) - (10 + x/12) = 12 minute marks.
simplify this to get x - 10 - x/12 = 12
add 10 to both sides to get x - x/12 = 22
combine like terms to get 11x/12 = 22
multiply both sides by 12 to get 11x = 264
solve for x to get x = 264/11 = 24
in 24 minutes, the minute hand moves from the 0 minutes mark + 24 minutes to the 24 minute mark.
in 24 minutes, the hour hand moves from the 10 minute mark + 24/12 = 2 minutes to the 12 minute mark.
24 minute mark minus 12 minute mark = 12 minute marks.
the minute hand is 12 minute marks ahead of the hour hand.
the following picture of a clock simulator on the web shows this to be true.
you can see that at 2:24 am or pm, the minute hand is at the 24 minutes mark and the hour hand is at the 12 minute mark, therefore the minute hand is 12 minutes ahead of the hour hand.
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