Calendar and clock questions look like they need a calendar or a watch in front of you. They do not. A handful of rules, all built on the numbers 7, 30 and 5.5, answer almost every question at this level, and the arithmetic is simple enough to do in your head once the rules are clear.
This guide covers the calendar rules first (odd days, leap years, and how the weekday shifts from one year to the next), then the clock rules (the angle formula and mirror images), with worked examples and a practice set.
Part 1: calendars
Odd days
A weekday repeats every 7 days. So what matters is not the number of days, but the remainder after dividing by 7. That remainder is called the number of odd days, and it tells you how far the weekday moves.
| Period | Days | Odd days (remainder after ÷ 7) |
|---|---|---|
| One week | 7 | 0 |
| February in a normal year | 28 | 0 |
| February in a leap year | 29 | 1 |
| A 30-day month | 30 | 2 |
| A 31-day month | 31 | 3 |
| A normal year | 365 | 1 |
| A leap year | 366 | 2 |
| 100 years | 5 | |
| 400 years | 0 |
The last two rows are useful for questions about dates centuries apart. 100 years contain 24 leap years and 76 normal years, giving 76 + 48 = 124 odd days, and 124 ÷ 7 leaves 5.
Leap years
- A year divisible by 4 is a leap year: 2024 and 2028.
- A century year (ending in 00) is a leap year only if it is divisible by 400.
- So 2000 was a leap year, but 1900 was not, and 2100 will not be.
Worked examples
Example 1 (days after): Today is Wednesday. What day will it be after 50 days?
- 50 ÷ 7 = 7 weeks and remainder 1.
- Wednesday + 1 = Thursday.
Example 2 (month to month): 1 March is a Tuesday. What day is 1 April in the same year?
- March has 31 days, which is 3 odd days.
- Tuesday + 3 = Friday.
Example 3 (year to year, no leap day): 15 August 2025 was a Friday. What day was 15 August 2026?
- The year in between contains no 29 February (2026 is not a leap year), so there is 1 odd day.
- Friday + 1 = Saturday.
Example 4 (year to year, with a leap day): 1 January 2024 was a Monday. What day was 1 January 2025?
- 2024 is a leap year, and 29 February 2024 falls between the two dates. So there are 2 odd days.
- Monday + 2 = Wednesday.
Part 2: clocks
How the hands move
- The minute hand moves 360° in 60 minutes: 6° per minute.
- The hour hand moves 30° in 60 minutes: 0.5° per minute.
- So the minute hand gains 5.5° per minute on the hour hand.
From these comes the angle formula. At H hours and M minutes:
- Angle between the hands = |30H − 5.5M|
The bars mean "ignore the minus sign". If the answer is more than 180°, subtract it from 360° to get the smaller angle, which is what questions usually want.
Mirror images
In a mirror, a clock's left and right swap. To find the time shown in the mirror, subtract the real time from 11:60 (which is just 12:00 written so you can subtract minutes easily).
- 11:60 − 8:25 = 3:35.
- For times between 11:00 and 12:59, subtract from 23:60 instead. For example, 23:60 − 12:20 = 11:40.
Facts worth knowing
| In 24 hours, the hands… | Number of times |
|---|---|
| Overlap | 22 |
| Are at right angles | 44 |
| Point in opposite directions (a straight line, 180°) | 22 |
Worked examples
Example 5 (angles): Find the angle between the hands at 5:30, and at 9:00.
- 5:30: |30 × 5 − 5.5 × 30| = |150 − 165| = 15°. The hour hand has moved halfway past 5, so the hands are close together.
- 9:00: |270 − 0| = 270°. That is more than 180°, so the smaller angle is 360 − 270 = 90°.
Example 6 (mirror): A clock shows 8:25. What time does its mirror image show?
- 11:60 − 8:25 = 3:35.
- Check: in the real clock the minute hand points at 5 (25 minutes). In the mirror it points at 7, which is 35 minutes.
Common mistakes
- Treating every century year as a leap year. 1900 was not.
- Adding 2 odd days for a leap year when 29 February does not fall between the two dates.
- Counting today as one of the days in "after N days" questions.
- Forgetting that the hour hand moves too. At 5:30 it is not on 5, it is halfway to 6.
- Forgetting to give the smaller angle when the formula gives more than 180°.
Practice set
- Find the angle between the hands at 3:40.
- Today is Monday. What day will it be after 30 days?
- Which of these is a leap year: 1900, 2000 or 2100?
- A clock shows 2:50. What does its mirror image show?
- 5 June is a Sunday. What day is 5 July in the same year?
- 1 January 2026 is a Thursday. What day is 1 January 2027?
- Find the angle between the hands at 6:15.
- Today is Friday. What day was it 40 days ago?
Answers:
- 130°. |90 − 5.5 × 40| = |90 − 220| = 130.
- Wednesday. 30 ÷ 7 leaves 2, and Monday + 2 = Wednesday.
- 2000 only. 1900 and 2100 are century years not divisible by 400.
- 9:10. 11:60 − 2:50 = 9:10.
- Tuesday. June has 30 days, which is 2 odd days. Sunday + 2 = Tuesday.
- Friday. 2026 is not a leap year, so there is 1 odd day.
- 97.5°. |180 − 5.5 × 15| = |180 − 82.5| = 97.5.
- Sunday. 40 ÷ 7 leaves 5. Going back 5 days from Friday: Thursday, Wednesday, Tuesday, Monday, Sunday.
What to do next
- Learn the odd-days table and the leap-year rule for century years.
- Learn the angle formula and practise it on ten random times.
- Solve 10 calendar and clock questions a day for five days.
- See the reasoning plan for where this topic fits, and revise number series for the same remainder-style thinking.
A note on dates and numbers. Exam patterns, vacancies and schedules change from year to year. Always confirm the current details in the latest notification on the Railway Recruitment Boards website .
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