Clock and calendar questions look fiddly but collapse to a couple of formulas and a short list of facts worth memorising outright.
Clock hand speeds
Minute hand = 6 deg/min; hour hand = 0.5 deg/min. Angle between hands = |30H − 5.5M| degrees.
Calendars
A normal year has 1 odd day; a leap year has 2. The day of the week repeats every 7 odd days.
How to Approach It
- Use the angle formula for clocks — the angle is |30H − 5.5M| degrees. Plug in hour and minute, then take the smaller angle if asked.
- Recall the coincidence counts — in twelve hours the hands coincide 11 times and form a right angle 22 times. A whole family of questions falls to these.
- Count odd days for calendars — a normal year leaves 1 odd day, a leap year 2. Sum the odd days, take the remainder mod 7, and shift the weekday by that much.
- Apply the leap-year rule — divisible by 4, except a century year must be divisible by 400. So 2000 was a leap year while 1900 was not.
Techniques & Methods
- Angle formula — angle between hands = |30H − 5.5M| degrees. e.g. 3:00 → |90 − 0| = 90°.
- Coincidence counts — hands coincide 11 times and form right angles 22 times per 12 hours.
- Odd-days method — normal year leaves 1 odd day, leap year 2; map the remainder to a weekday.
- Leap-year rule — divisible by 4, but a century year must be divisible by 400. e.g. 2000 was a leap year; 1900 was not.
The Edge
Memorise the counts: in 12 hours the hands coincide 11 times (not 12) and are at a right angle 22 times. A whole family of clock questions is answered the moment you recall these. Leap year rule: divisible by 4, except century years which must be divisible by 400. So 2000 was a leap year; 1900 was not.Worked example
What is the angle between the hands at 3:00?
- Use the hand-angle formula: angle = |30H − 5.5M| degrees, where H is the hour and M the minutes.
- At 3:00, H = 3 and M = 0.
- Substitute: |30 × 3 − 5.5 × 0| = |90 − 0| = 90.
- So the hands are 90 degrees apart — which makes sense, since 3 o'clock is exactly a quarter turn.
Answer: 90 degrees
Worked example
How many times do the clock hands coincide in a full day (24 hours)?
- Recall the key fact: in 12 hours the hands coincide 11 times, not 12, because the hour hand is also moving forward.
- A full day is two such 12-hour cycles.
- So multiply: 11 × 2 = 22.
- The hands coincide 22 times in 24 hours.
Answer: 22 times
Worked Drills
Worked example
Between 3 and 4 o'clock, the hands of a clock coincide at:
- The hands coincide at (60H)/11 minutes past the hour, here H = 3.
- = 180/11 = 16 4/11 minutes past 3.
Answer: b) 16 4/11 min past 3
Worked example
In a full day (24 hours), the hands of a clock are at right angles:
- Right angles occur 22 times per 12 hours.
- Over 24 hours: 22 × 2 = 44 times.
Answer: b) 44 times
Worked example
15 August 1947 fell on which day of the week?
- Counting odd days up to that date gives the weekday offset.
- It was a Friday.
Answer: b) Friday
Worked example
If today is Wednesday, the day after 61 days will be:
- Find the number of odd days: 61 mod 7 = 5.
- Wednesday + 5 days = Monday.
Answer: b) Monday
Worked example
The angle between the hands of a clock at 7:20 is:
- Apply |30H − 5.5M| with H = 7, M = 20.
- = |30 × 7 − 5.5 × 20| = |210 − 110| = 100 degrees.
Answer: b) 100 degrees
Worked example
After the leap year 2024, the next leap year that is not a century year is:
- Leap years occur every 4 years (non-century years are unaffected by the 400 rule).
- 2024 + 4 = 2028.
Answer: b) 2028
Worked example
Angle between the hands at 6:00:
- Apply |30H − 5.5M| with H = 6, M = 0.
- = |180 − 0| = 180 degrees.
Answer: d) 180
Worked example
Number of days in a leap year:
- A leap year has one extra day in February.
- 365 + 1 = 366 days.
Answer: c) 366
⚠ Watch out
- The hands coincide 11 times per 12 hours, not 12.
- A century year is a leap year only if divisible by 400.
- Watch 12-hour versus 24-hour framing in the question.
Takeaways
- The angle between hands is |30H − 5.5M| — take the smaller angle if asked.
- Per 12 hours: hands coincide 11 times and form right angles 22 times (double for a full day).
- Calendars: 1 odd day per normal year, 2 per leap year; reduce mod 7 to shift the weekday.
- Leap year = divisible by 4, but century years only if divisible by 400.