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Study Materials › Arithmetic › Calendars & Clocks
Two small topics that between them gave four marks across the 2022 papers. Both run on a single idea — counting a remainder. For calendars you count days modulo 7; for clocks you track the gap between two hands moving at different constant speeds.
An odd day is the remainder when a period is divided by 7.
Ordinary year = 365 days = 52 weeks + 1 odd day.
Leap year = 366 days = 52 weeks + 2 odd days.
Leap year test: divisible by 4; but century years must be divisible by 400. 1900 was not a leap year; 2000 was.
The minute hand moves 6° per minute; the hour hand moves 0.5° per minute. The minute hand gains 5.5° per minute.
Angle between hands at H hours M minutes = |30H − 5.5M| degrees.
The hands coincide 22 times and are opposite 22 times in 24 hours — 11 of each per 12 hours, not 12, because they overlap once every 655⁄11 minutes.
If 15 August 2022 is a Monday, what is the date of the Monday immediately following 15 August 2024?
15 Aug 2022 → 15 Aug 2023 crosses February 2023 (not a leap February): 365 days, 1 odd day →
Tuesday.
15 Aug 2023 → 15 Aug 2024 crosses February 2024 (leap): 366 days, 2 odd days →
Thursday.
The next Monday after Thursday 15 August is four days later:
19 August 2024.
If 15 August 2022 is a Monday, on which day will Republic Day be celebrated in 2024?
From the previous example, 15 August 2023 is a Tuesday.
Count forward to 26 January 2024: 16 days left in August, then 30 + 31 + 30 + 31 for September to December,
then 26 in January → 164 days.
164 ÷ 7 leaves a remainder of 3, so Tuesday + 3 =
Friday.
Method: step to the nearest same-date anniversary first, then count the leftover days. Counting all
the way from the original date in one go invites arithmetic slips.
After 2022, which is the next calendar year identical to 2022?
A year repeats only if it starts on the same weekday and matches in leap status.
1 January advances by 1 day after an ordinary year and 2 after a leap year. Starting from 2022:
2023 +1, 2024 +1, 2025 +2 (2024 was leap), 2026 +1, 2027 +1, 2028 +1 → back to the same weekday in 2028.
But 2028 is a leap year and 2022 is not, so their calendars diverge from March. Continuing:
2029 +2, 2030 +1, 2031 +1, 2032 +1, 2033 +2 → same weekday again, and 2033 is an ordinary year.
Answer: 2033.
This is the whole point of the question. 2028 is offered as an option and satisfies the weekday test
alone. Matching leap status is the second condition candidates forget.
At what time between 7 and 8 o'clock is the minute hand 5 minutes behind the hour hand?
Work in minute-divisions rather than degrees. At 7:m the hour hand sits at 35 + m/12 divisions, and the minute
hand at m.
Being 5 divisions behind: m = (35 + m/12) − 5.
m − m/12 = 30 → (11/12)m = 30 → m = 360/11.
Answer: 328⁄11 minutes past 7.
Note: answers in clock questions almost always come out in elevenths, because the hands' relative
speed is 11/12 of a division per minute. An answer without an eleventh is usually a sign of an error.
If m and n are the numbers of rotations of the hour hand and the minute hand in 72 hours, find n − m.
The hour hand completes one rotation every 12 hours: 72 ÷ 12 = 6 rotations.
The minute hand completes one every hour: 72 rotations.
n − m = 72 − 6 = 66.
Note: this is a rare clock question with no elevenths in it — it asks about full rotations, not
about the hands meeting.
Also see: Time, Speed & Distance · Number System · Averages