Clocks and calendars
Clock and calendar questions become easier when you replace the picture or date with numbers. Learn the movement rates of clock hands and use seven-day cycles to track weekdays.
Turn the clock face into angles#
A clock face is a circle of 360°. The twelve hour marks divide it into equal sections:
360° ÷ 12 = 30° per hour mark
At exactly 3:00, the minute hand is at 12 and the hour hand is at 3. Their smaller angle is 3 × 30° = 90°.
The hands do not jump between marks. The minute hand moves:
360° ÷ 60 = 6° per minute
The hour hand completes 30° in 60 minutes, so it moves:
30° ÷ 60 = 0.5° per minute
At 2:20, the minute hand is at 20 × 6° = 120°. The hour hand has moved 2 × 30° plus 20 × 0.5° = 70°. The smaller angle is 120° - 70° = 50°.
Use the smaller angle unless told otherwise#
Two hands create two angles that total 360°. If their positions differ by 280°, the smaller angle is 360° - 280° = 80°. Read whether the question asks for the smaller angle, larger angle, or clockwise separation.
Treat weekdays as a cycle#
Weekdays repeat every seven days. To find the weekday after a number of days, divide by seven and use the remainder.
If today is Tuesday, 17 days later is:
17 = 14 + 3
Fourteen days complete two full weeks. Move three more days: Wednesday, Thursday, Friday.
For a date interval, be clear whether the starting day is counted. “Ten days after Monday” moves forward ten times; it does not count Monday as day one.
Check month lengths and leap years#
Months do not all have the same length. April, June, September, and November have 30 days; February has 28 in a common year and 29 in a leap year; the others have 31.
In the Gregorian calendar, a year divisible by 4 is usually a leap year. A century year such as 1900 must also be divisible by 400, so 1900 was not a leap year but 2000 was.
Draw a short seven-column weekday cycle or write the hand angles. A small conversion is more reliable than trying to rotate the entire problem mentally.
Key points
- A full clock face is 360 degrees, with 30 degrees between consecutive hour marks.
- The minute hand moves 6 degrees per minute and the hour hand moves 0.5 degrees per minute.
- Weekdays repeat in a seven-day cycle, so only the remainder after division by seven changes the weekday.
- Gregorian leap years are divisible by 4, except century years must also be divisible by 400.