Question 1091689: What time after 4 o'clock will the minute hand be nine minutes behind 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 after 4 o'clock when the minute hand is "9 minutes" behind the hour hand. 9 minutes is 3/20 of an hour; so you want the minute hand to be 3/20 of 360 degrees, or 54 degrees, behind the hour hand.
At x minutes after 4, the angle the hour hand makes with 12 o'clock, in degrees, is
 (120 degrees to get to 4; then 0.5 degrees for each minute after 2)
And x minutes after 4, the angle the minute hand makes with 12 o'clock, in degrees, is
You want the minute hand to be "9 minutes", or 54 degrees, behind the hour hand, so you want
The minute hand will be "9 minutes" behind the hour hand at 4:12.
Answer by Theo(13342)  (Show Source):
You can put this solution on YOUR website! i did so well on the last one that i thought i would try this one as well.
once again, the hour hand moves 1/12 of a minute mark every time the minute hand move 1 minute mark.
at 4:00 pm, the hour hand is at the 20 minute mark and the minute hand is at the 0 minute mark.
when the minute hand moves x minute marks and the hour hand moves x/12 minute marks.
since you want the minute hand to be behind the hour hand, your formula becomes:
(20 + x/12) - (0 + x) = 9 minute marks.
simplify to get 20 + x/12 - x = 9
subtract 20 from both sides to get x/12 - x = -11
multiply both sides of the equation by 12 to get x - 12x = -132
combine like terms to get -11x = -132
solve for x to get x = -132/-11 = 12.
when x = 12, the minute hand is at the 12 minute mark.
when x = 12, the hour hand is at the 10 + 12/12 = 21 minute mark.
the difference between them is 21 - 12 = 9 minute marks, with the hour hand being ahead of the minute hand by that many minute marks.
the following pictures confirms this to be true.
the minute hand will be behind the hour hand at 4:12 am or pm.
the minute hand is at the 12 minute mark and the hour hand is at the 21 minute mark.
the difference is 12 - 12 = 9 with the hour hand ahead of the minute hand.
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