Calendars and clocks
Calendar questions use odd days, the days left over after full weeks. Clock questions use the fact that the minute hand gains 5½° per minute on the hour hand.
Key facts
| Item | Fact |
|---|---|
| Odd days | Days left over after complete weeks: 8 days = 1 week + 1 odd day |
| Ordinary year | 365 days = 52 weeks + 1 odd day |
| Leap year | 366 days = 52 weeks + 2 odd days; 2000 was a leap year, 1900 was not |
| 100, 200, 300, 400 years | 5, 3, 1 and 0 odd days |
| Months | 31-day month: 3 odd days; 30-day month: 2; February: 0 (1 in a leap year) |
| Odd-day number to day | 0 Sunday, 1 Monday, 2 Tuesday, 3 Wednesday, 4 Thursday, 5 Friday, 6 Saturday |
| Clock hands | Minute hand 6° a minute; hour hand ½° a minute; the minute hand gains 5½° a minute |
| Hands together | 11 times in 12 hours (22 times a day), once every 65 5/11 minutes |
| Hands at right angles / opposite | 22 / 11 times in 12 hours |
Worked example 1: day of the week for a date
What day of the week was 15 August 1947?
- Standard method (odd days): years completed before the date = 1,946 = 1,600 + 300 + 46.
- 1,600 years → 0 odd days; 300 years → 1; 46 years have 11 leap years (1904 to 1944) and 35 ordinary years → 11 × 2 + 35 = 57, which leaves 1.
- January to July 1947 → 3 + 0 + 3 + 2 + 3 + 2 + 3 = 16, which leaves 2. The 15 days of August leave 1.
- Total = 0 + 1 + 1 + 2 + 1 = 5 odd days → Friday.
- Shortcut (key-value method): last two digits + (those digits ÷ 4, whole part) + date + month key + century key = 47 + 11 + 15 + 3 + 0 = 76. 76 ÷ 7 leaves 6, and in this method 1 = Sunday … 6 = Friday, 0 = Saturday.
- Keys: months Jan 1, Feb 4, Mar 4, Apr 0, May 2, Jun 5, Jul 0, Aug 3, Sep 6, Oct 1, Nov 4, Dec 6 (subtract 1 for January and February of a leap year); centuries 1900s 0, 2000s 6.
- Check: both methods give Friday.
Friday.
Worked example 2: same date, two years later
10 March 2024 was a Sunday. What day of the week is 10 March 2026?
- Standard method: 10 March 2024 to 10 March 2025 is 365 days, because 29 February 2024 has already passed → 1 odd day. 10 March 2025 to 10 March 2026 is also 365 days → 1 odd day.
- Total 2 odd days: Sunday + 2 = Tuesday.
- Shortcut: the same date moves forward 1 day a year, or 2 days if a 29 February falls in between. None does here, so +1 + 1 → Tuesday.
- Check: 730 days = 104 weeks + 2 days.
Tuesday.
Worked example 3: angle between the hands
Find the angle between the hands of a clock at 4:20.
- Standard method: the hour hand moves 30° an hour and ½° a minute, so at 4:20 it is at 4 × 30 + 20 × ½ = 130° from 12. The minute hand is at 20 × 6 = 120°.
- Angle = 130° − 120° = 10°.
- Shortcut: angle = |30h − 5.5m| = |30 × 4 − 5.5 × 20| = |120 − 110| = 10°.
- Check: both methods give 10°, which is below 180°, so no adjustment is needed.
10°.
Worked example 4: when the hands meet
At what time between 3 and 4 o'clock are the hands of a clock together?
- Standard method: at 3:00 the minute hand is 90° behind the hour hand. It gains 6 − ½ = 5½° a minute, so it catches up in 90 ÷ 5½ = 180/11 = 16 4/11 minutes.
- Shortcut: the hands meet (60/11) × h minutes past h o'clock: (60/11) × 3 = 180/11 = 16 4/11 minutes past 3.
- Check: at 16 4/11 minutes past 3, the minute hand is at 6 × 180/11 ≈ 98.2° and the hour hand at 90 + ½ × 180/11 ≈ 98.2°.
16 4/11 minutes past 3 (about 3:16).
SSC, railway and state PSC papers ask 'what day was a given date', leap-year traps and clock angles; bank papers ask them less often. Placement tests like 'when are the hands together or at right angles', which the 5½° a minute idea solves.
2 marks: “Is 1900 a leap year? Give a reason.” No. A century year is a leap year only if it is divisible by 400, and 1900 is not.
5 marks: “Find the angle between the hands of a clock at 4:20, and the time between 3 and 4 o'clock when the hands are together.” |30 × 4 − 5.5 × 20| = 10°; the hands meet at (60/11) × 3 = 16 4/11 minutes past 3.
10 marks: “What are odd days? Explain how to find the day of the week for a given date, and find the day on 15 August 1947.” Define odd days; table for years and centuries; leap-year rule; steps; full working for 15 August 1947 giving Friday.
Odd days: year 1, leap year 2, 100 years 5, 400 years 0. A century is a leap year only if ÷ 400. Clock angle = |30h − 5.5m|; hands meet every 65 5/11 minutes.