Clocks and Calendars Notes for Reasoning

Clocks and Calendars are two classic reasoning chapters that appear in SSC CGL, CHSL, MTS, railway exams and state PCS papers, and occasionally in UPSC CSAT and bank exams. Both chapters are built on a small number of fixed facts: a clock face has 360 degrees and 60 minute divisions, and a calendar repeats its weekdays in cycles of seven days. Once you understand these facts and learn a few formulas, most questions can be solved in under a minute. These chapters are especially useful because they are short to prepare, highly predictable and rarely require long calculations.

Quick Facts at a Glance

• Minute hand speed: 6° per minute. Hour hand speed: 0.5° per minute.
• Angle between hands at H hours and M minutes = |30H − 5.5M| degrees.
• Hands coincide 22 times in 24 hours and are opposite 22 times in 24 hours.
• Odd days: Ordinary year = 1 odd day; leap year = 2 odd days.
• Leap year: Divisible by 4; a century year must be divisible by 400.

Part A: Clocks

1. Basic Facts About a Clock

• The dial is divided into 60 minute spaces; each minute space = 6°.
• Each hour mark is 30° apart (360° ÷ 12).
• The minute hand moves 360° in 60 minutes = 6° per minute.
• The hour hand moves 30° in 60 minutes = 0.5° per minute.
• So the minute hand gains 5.5° per minute over the hour hand. In minute spaces, it gains 55 minutes in 60 minutes.

2. Angle Between the Hands

Formula: Angle θ = |30H − (11/2)M|, where H is the hour and M is the minutes. If the result is more than 180°, subtract it from 360° to get the smaller angle.

Example 1: Angle at 3:40. θ = |90 − 220| = 130°.

Example 2: Angle at 10:10. θ = |300 − 55| = 245°. Smaller angle = 360 − 245 = 115°.

Example 3: Angle at 7:20. θ = |210 − 110| = 100°.

3. When Do the Hands Coincide, Become Opposite or Form a Right Angle?

Position	Times in 12 hours	Times in 24 hours
Coincide (0°)	11	22
Opposite (180°)	11	22
Right angle (90°)	22	44
In a straight line (0° or 180°)	22	44

Hands coincide every 65 5/11 minutes, not every 60 minutes. Between 11 and 1 o'clock, they coincide only once (at 12:00), which is why the count is 11 and not 12.

Time when hands coincide between H and H + 1: M = (60H)/11 minutes past H. Example: between 2 and 3, M = 120/11 = 10 10/11 minutes past 2.

Time when hands are opposite between H and H + 1: Solve |30H − 5.5M| = 180.
Example: between 4 and 5: 5.5M − 120 = 180 → M = 300/5.5 = 54 6/11 minutes past 4.

Right angle between 3 and 4: |90 − 5.5M| = 90 → M = 0 (3:00) or M = 180/5.5 = 32 8/11 minutes past 3.

4. Gaining and Losing Clocks

• If the hands of a clock coincide every less than 65 5/11 minutes, the clock gains time.
• If they coincide every more than 65 5/11 minutes, the clock loses time.
• If a clock is set right at a certain time and gains x minutes per hour, after t hours it shows x × t minutes ahead.

Example: A clock gains 5 minutes every hour. It is set right at 6 a.m. What will it show at 6 p.m. the same day? In 12 hours it gains 60 minutes, so it shows 7 p.m.

5. Mirror Images of Clocks

To find the time shown in a mirror (vertical mirror), subtract the actual time from 11:60 (that is, 12:00). If the actual time is 4:25, the mirror shows 11:60 − 4:25 = 7:35. If the time is between 11:00 and 1:00, subtract from 23:60 instead.

Part B: Calendars

6. The Concept of Odd Days

Odd days are the extra days left over after complete weeks. Since a week has 7 days, the weekday shifts by the number of odd days.

• Ordinary year (365 days) = 52 weeks + 1 day → 1 odd day.
• Leap year (366 days) = 52 weeks + 2 days → 2 odd days.
• 100 years = 76 ordinary + 24 leap years → 76 + 48 = 124 odd days → 124 mod 7 = 5 odd days.
• 200 years = 3 odd days; 300 years = 1 odd day; 400 years = 0 odd days.

7. Leap Year Rules

• A non-century year is a leap year if it is divisible by 4 (2024, 2028).
• A century year is a leap year only if divisible by 400. So 2000 was a leap year, but 1900 and 2100 are not.

8. Odd Days for Months

Month	Days	Odd days
Jan, Mar, May, Jul, Aug, Oct, Dec	31	3
Apr, Jun, Sep, Nov	30	2
February (ordinary / leap)	28 / 29	0 / 1

Day codes: 0 = Sunday, 1 = Monday, 2 = Tuesday, 3 = Wednesday, 4 = Thursday, 5 = Friday, 6 = Saturday.

9. Finding the Day of a Given Date

Example: What day was 15 August 1947?

1. 1600 years → 0 odd days. 300 years (1601–1900) → 1 odd day.
2. 46 years (1901–1946): 11 leap years and 35 ordinary → 22 + 35 = 57 → 57 mod 7 = 1 odd day.
3. 1 January to 15 August 1947 (non-leap): Jan 3, Feb 0, Mar 3, Apr 2, May 3, Jun 2, Jul 3, Aug 15 → total 31 → 31 mod 7 = 3 odd days.
4. Total = 0 + 1 + 1 + 3 = 5 → Friday.

This matches the historical fact that India became independent on a Friday.

10. Shortcut: Day Shift From a Known Date

If you know the day of one date, count odd days to the target date. Example: If 1 January 2025 was a Wednesday, 1 January 2026 is Wednesday + 1 = Thursday (2025 is not a leap year).

11. Repeating Calendars

• A calendar repeats when the total odd days in between is a multiple of 7 and the leap-year status matches.
• For an ordinary year, the calendar often repeats after 6 or 11 years, depending on its position in the leap cycle.
• For a leap year, the calendar repeats after 28 years (within the same century rules).

Example: The calendar of 2024 (a leap year) will repeat in 2052.

Common Mistakes to Avoid

1. Using 30H − 6M instead of 30H − 5.5M in the angle formula.
2. Saying hands coincide 24 times a day instead of 22.
3. Treating 1900 or 2100 as leap years.
4. Counting the starting date when calculating odd days to a later date.
5. Forgetting to take the smaller angle when the result exceeds 180°.

Practice Questions with Answers

1. Angle between the hands at 5:30? Answer: |150 − 165| = 15°.
2. At what time between 6 and 7 do the hands coincide? Answer: 360/11 = 32 8/11 minutes past 6.
3. If 26 January 2026 is a Monday, what day is 26 January 2027? Answer: Tuesday (2026 has 1 odd day).
4. Is 2100 a leap year? Answer: No, it is not divisible by 400.

More Solved Examples

Clock example: A clock shows 8:00 a.m. on Monday and loses 10 minutes every 24 hours. What time will it show at 8:00 a.m. on Thursday (actual time)? From Monday 8 a.m. to Thursday 8 a.m. is 3 days, so the clock loses 30 minutes. It will show 7:30 a.m.

Clock example: How many degrees does the hour hand move from 2:15 to 5:45? Time elapsed = 3 hours 30 minutes = 210 minutes. Hour hand moves 0.5° per minute, so 210 × 0.5 = 105°.

Calendar example: If the 5th of a month is a Tuesday, what day is the 27th? The difference is 22 days = 3 weeks + 1 day. So the 27th is Wednesday.

Calendar example: How many Sundays can a non-leap year have at most? A non-leap year has 52 weeks + 1 day, so it can have 52 or 53 Sundays. It has 53 Sundays only if it starts on a Sunday. A leap year has 53 Sundays if it starts on a Saturday or Sunday.

Exam Tip: Use Known Anchor Dates

Remembering one or two anchor dates speeds up calendar questions. For example, 1 January 2001 was a Monday, and 1 January 2026 was a Thursday. From an anchor, you only need to count odd days forward or backward instead of starting from year 1. In the exam, this can cut the time for a calendar question from two minutes to thirty seconds.

Related Notes

• Number and Letter Series Notes
• Direction Sense and Seating Arrangement Notes
• Time and Work Notes
• All Reasoning Notes

Frequently Asked Questions

What is the formula for the angle between clock hands?

Angle = |30H − 5.5M| degrees. If the result exceeds 180°, subtract it from 360° for the smaller angle.

How many times do clock hands coincide in a day?

22 times in 24 hours, or 11 times in 12 hours.

How many times are the hands at right angles in a day?

44 times in 24 hours.

How fast does the minute hand gain over the hour hand?

It gains 5.5 degrees per minute, or 55 minute spaces in 60 minutes.

What are odd days?

Odd days are the extra days left after counting complete weeks in a period.

How many odd days are in a leap year?

A leap year has 2 odd days; an ordinary year has 1.

Is 1900 a leap year?

No. A century year is a leap year only if it is divisible by 400.

How many odd days are there in 400 years?

Zero. That is why the calendar pattern repeats every 400 years.

What day was 15 August 1947?

It was a Friday.

After how many years does a leap year calendar repeat?

Generally after 28 years, provided no non-leap century year falls in between.

How do you find the mirror image time of a clock?

Subtract the actual time from 11:60. For example, 3:15 appears as 8:45.

Are clocks and calendars asked in UPSC CSAT?

They appear occasionally in CSAT and more regularly in SSC and state exams.