Clock hand speeds
The minute hand moves 6° per minute (360° in 60 minutes).
The hour hand moves 0.5° per minute (30° per hour).
So the minute hand gains 5.5° per minute on the hour hand.
The angle formula
At H hours and M minutes, the angle between the hands is
θ = |30H − 5.5M|
If the answer is more than 180°, subtract it from 360° to get the smaller angle.
Time | Calculation | Angle |
|---|---|---|
3:00 | 30 × 3 − 0 | 90° |
3:30 | 90 − 165 | 75° |
7:20 | 210 − 110 | 100° |
When the hands coincide
The hands overlap 11 times in 12 hours (22 times in a day), not 12, because the overlap between 11 and 1 happens only once, at 12:00.
Between H and H + 1 o'clock, the hands coincide at 60H/11 minutes past H. Between 4 and 5: 240/11 = 21 9/11 minutes past 4.
Fast and slow clocks
A clock that gains g minutes per hour shows g × (hours elapsed) minutes too much.
Example. A clock set right at 8 a.m. gains 5 minutes an hour. At 2 p.m. (6 hours later) it shows 2:30 p.m.
Calendars: odd days
Days beyond complete weeks are odd days.
Period | Odd days |
|---|---|
ordinary year (365 days) | 1 |
leap year (366 days) | 2 |
100 years | 5 |
400 years | 0 |
Moving forward: each ordinary year moves the same date forward by one weekday; a leap year moves dates after 29 February forward by two.
Example. 1 January 2023 was a Sunday. 2023 is not a leap year, so 1 January 2024 was a Monday.
Counting days: 100 days after a Wednesday is 100 = 14 × 7 + 2 days later: a Friday.
Leap years
A year is a leap year if it is divisible by 4, except century years, which must be divisible by 400. So 2000 was a leap year but 1900 was not.
Repeating calendars
A year has the same calendar as an earlier year when the total odd days between them is a multiple of 7 and both are of the same type (leap or ordinary). 2021 and 2027 share a calendar.
Shortcuts and tricks
The standard way places each hand separately: the minute hand moves 6° a minute and the hour hand 0.5° a minute. The shortcuts below combine these into one formula.
The angle formula
Example: the angle at 8:20.
Standard way: the minute hand is at 4, so 120°. The hour hand is at 8 × 30 + 20 × 0.5 = 250°. The angle is 250 − 120 = 130°.
Shortcut: θ = |30H − 5.5M| = |240 − 110| = 130°.
When the hands meet
Example: when do the hands coincide between 2 and 3?
Standard way: the minute hand must gain 60° on the hour hand at 5.5° a minute: 60 ÷ 5.5 = 10 10/11 minutes past 2.
Shortcut: the hands meet at H × 60/11 = H × 5 5/11 minutes past H: 2 × 5 5/11 = 10 10/11 minutes past 2.
Same date next year
Example: 15 March 2025 was a Saturday. What day is 15 March 2026?
Standard way: 365 days = 52 weeks + 1 day, so it moves one day forward: Sunday.
Shortcut: move one day forward each year, and two if the year in between contains a 29 February.
Repeating calendars
Example: which year has the same calendar as 2019?
Standard way: add odd days year by year until the total is a multiple of 7 and the year type matches.
Shortcut: for an ordinary year, check where it falls after the last leap year. One year after a leap year repeats in 6 years; two or three years after, in 11 years. 2019 is three years after 2016, so it repeats in 2030.
Exam tips
Always reduce clock angles to the smaller angle unless asked for the reflex angle.
In calendar questions, count odd days, not total days.
A century year is a leap year only if divisible by 400.