Clock angles, hands together and opposite, gaining time, odd days, leap years, and day-of-week calculation, with worked examples and practice
Two small but reliable topics. Both reward a memorised formula and a calm head rather than long working. Clocks are a geometry-of-angles problem on a 360 degree dial; calendars are a remainder problem in modulo 7. They build on the divisibility and remainder ideas in number system and simplification.
A clock face is 360°, divided into 12 hours, so each hour mark is 30° apart. There are 60 minutes, so each minute mark is 6° apart.
| Hand | Rate |
|---|---|
| Minute hand | 360° in 60 minutes = 6° per minute |
| Hour hand | 360° in 720 minutes (12 hours) = 0.5° per minute |
| Relative gain (minute over hour) | 6 - 0.5 = 5.5° per minute |
The minute hand gains 5.5° on the hour hand every minute. This single number drives almost every clock question.
| Concept | Rule |
|---|---|
| Angle between hands | abs( 30 times H minus 5.5 times M ), where H is the hour and M the minutes |
| Hands coincide (overlap) | 11 times every 12 hours, so once every 720/11 = 65 and 5/11 minutes |
| Hands at right angle (90°) | 22 times every 12 hours |
| Hands in a straight line (180°, opposite) | 11 times every 12 hours |
| Faulty clock gaining or losing | compare against the true 24 hour day |
Note: in 12 hours the hands overlap only 11 times (not 12), because between 11 and 1 o'clock they overlap just once, at 12 o'clock. The same logic gives 11 straight-line positions and 22 right angles per 12 hours.
The day of the week repeats every 7 days, so only the remainder after dividing by 7 matters. That remainder is called the number of odd days.
| Period | Odd days |
|---|---|
| Ordinary year (365 days) | 365 mod 7 = 1 |
| Leap year (366 days) | 366 mod 7 = 2 |
| 100 years | 5 |
| 200 years | 3 |
| 300 years | 1 |
| 400 years | 0 |
A leap year is divisible by 4, except century years, which must be divisible by 400. So 2000 was a leap year, but 1900 was not.
| Months | Odd days |
|---|---|
| 31-day months (Jan, Mar, May, Jul, Aug, Oct, Dec) | 3 |
| 30-day months (Apr, Jun, Sep, Nov) | 2 |
| February (ordinary) | 0 |
| February (leap) | 1 |
Map the odd-day count to a day, counting from a known anchor. A standard convention takes 0 odd days as Sunday: 0 Sunday, 1 Monday, 2 Tuesday, 3 Wednesday, 4 Thursday, 5 Friday, 6 Saturday. Whichever anchor you use, apply it consistently within one problem.
Find the angle between the hands at 3:40.
Angle = abs( 30 times 3 minus 5.5 times 40 ) = abs( 90 minus 220 ) = abs( minus 130 ) = 130°.
At 4:00 the hour hand is at 120°, the minute hand at 0. The minute hand must close a 120 degree gap, gaining 5.5° per minute.
Time = 120 / 5.5 = 21 and 9/11 minutes. So the hands coincide at about 4:21 and 9/11.
At 3:00 the gap is 90° (hour hand at 90, minute hand at 0). For the next right angle the minute hand must reach 90° ahead minus or plus 90.
To be 90° apart the first time, the minute hand needs to gain 90 minus 90 = 0 (it is already 90 behind, but they are 90 apart at exactly 3:00). The next 90 degree position needs a relative change of 180°: 180 / 5.5 = 32 and 8/11 minutes, so at about 3:32 and 8/11.
A clock gains 5 minutes every hour. If it is set correctly at 8:00 a.m., what is the true time when it shows 2:00 p.m.?
The faulty clock shows 6 hours elapsed (8 a.m. to 2 p.m.). It runs fast: in 65 of its own minutes only 60 true minutes pass. So 6 hours = 360 shown minutes correspond to 360 times 60/65 = 332.3 true minutes, which is about 5 hours 32 minutes. True time is roughly 1:32 p.m.
If 1 January 2020 was a Wednesday, what day was 1 January 2021?
2020 is a leap year (divisible by 4, not a century), so it carries 2 odd days. Wednesday plus 2 days = Friday.
What day of the week was 15 August 1947?
Count odd days up to that date from the 400-year and year-by-year rules. The accepted result is Friday. India became independent on a Friday. For any unfamiliar date, build the total odd days from completed centuries, the years since, the leap years among them, and the days in the current year, then map the remainder to a day.