Common as a warm-up or as a filler question on a timed screen. Fast to answer once you have the two rates.
The rates
- Minute hand: 360 degrees in 60 minutes = 6 degrees per minute.
- Hour hand: 360 degrees in 12 hours = 0.5 degrees per minute.
- Relative speed: 5.5 degrees per minute.
That third number does most of the work.
The angle formula
At H hours and M minutes:
- Hour hand position: 30H + 0.5M degrees from 12.
- Minute hand position: 6M degrees.
Angle = |30H - 5.5M|, and if that exceeds 180, subtract from 360.
Example: 3:15. |90 - 82.5| = 7.5 degrees. Not zero - the hour hand has crept a quarter of the way toward 4, which is exactly the trap in this question.
How often do the hands overlap?
They start together at 12:00. The minute hand gains 5.5 degrees per minute, so it takes 360/5.5 = 65 5/11 minutes to lap the hour hand.
In 12 hours (720 minutes) that gives 720 / (720/11) = 11 overlaps, and 22 in 24 hours.
The intuitive answer is 24, and it is wrong: there is no overlap around 12:00 and another near 1:05 in the same way there is at every other hour, because the 11:00 hour has no overlap of its own - the 12:00 one serves it.
Right angles and opposites
Same relative-speed logic:
- Hands at 180 degrees: also 11 times in 12 hours, 22 per day.
- Hands at 90 degrees: twice per lap, so 22 times in 12 hours and 44 per day.
The fast method under time pressure
For "angle at H:M", compute 30H, then subtract 5.5M, take the absolute value, fold above 180. Two multiplications and a subtraction - well under the few seconds a timed screen allows.
Practise arithmetic speed with the mental math drill.