Last reviewed 22 Sept 2026 · 5 min read
How fast the hands move
| Hand | In 60 minutes | Per minute |
|---|---|---|
| Minute hand | ||
| Hour hand | ||
| Relative (minute gains on hour) |
At hours and minutes:
If the result exceeds , the smaller angle is .
The is the hour hand's position from 12, and is how much the minute hand has gained on it.
The standard positions
- The hands coincide when , i.e. , so .
- They are opposite ( apart) when .
- They are at right angles when the expression equals — which happens twice in most hours.
Counts in 12 hours: coincide 11 times, opposite 11 times, at right angles 22 times. In 24 hours: 22, 22 and 44.
Because the hands coincide 11 times in 12 hours, successive coincidences are minutes apart — not 65 minutes, which is what a "too fast" clock question usually hinges on.
Clocks that gain or lose
A clock that shows the hands coinciding every 65 minutes of true time is gaining, because a correct clock takes minutes. The gain in a day is
More generally: if a clock gains minutes in 24 hours, then in hours of true time it shows hours.
Mirror images
The mirror image of a clock time is found by subtracting it from 11:60 (that is, from 12:00) when the time is between 1 and 11 o'clock; for times between 12:00 and 1:00, subtract from 23:60.
Example: the mirror of 4:20 is .
Worked examples
Find the angle between the hands at 3:40.
Solution. .
Find the angle at 8:10.
Solution. → the smaller angle is 175°.
At what time between 4 and 5 o'clock do the hands coincide?
Solution. → at 4:21 9/11.
At what time between 5 and 6 are the hands opposite?
Solution. — that is 6 o'clock exactly, so between 5 and 6 they are never opposite; the opposition happens at 6:00. (Between 5 and 6 the solution is negative and rejected.)
At what times between 2 and 3 are the hands at right angles?
Solution. gives or . The first gives → 2:27 3/11; the second is rejected, so there is one such time in this hour.
How many times in 24 hours are the hands at right angles?
Solution. 44 times.
A clock's hands coincide every 64 minutes of correct time. How much does it gain in a day?
Solution. A correct clock takes minutes. The clock runs minutes fast in every 64 minutes shown. In 24 hours of true time: 32 8/11 minutes gained.
A clock gains 5 minutes a day. It was set right at 8 a.m. on Monday. What will it show at 8 a.m. on Wednesday?
Solution. Two days → 10 minutes fast → it shows 8:10 a.m.
What is the mirror image of 7:25?
Solution. 4:35.
What is the angle at 5 o'clock?
Solution. 150°.