# Calendar Formulas

## Formulas To Solve Calendars Problem in Aptitude

A calendar is a method  of organizing the days in a year. The calendar starts on 1st January and ends on 31st December.  The year is classified into two types: ordinary years (365 days) and a leap year (366 days).In an ordinary year, there are 365 days, which means 52 x 7 + 1, or 52 weeks and one day. ### Calendar Formulas & Definitions

• A calendar is a method for systematically  organizing days for  religious, social, commercial or administrative purposes.It is done by giving names to periods of time, such as  days, weeks, months and years.
• A date is single, specific day within such a system
• The year is classified into two type:
1. Ordinary years (365 days)
2. Leap year (366 days.)z
• The calendar starts on 1st January and ends on 31st December.

### Calendar Formulas for Odd days

• Odd days are the number of days that are more than the number of days in a complete week.
• For example:         Calculate odd days for 10 and 14 days

10 days = 1 week (7 days) + 3 days. Here, 3 days are odd days

14 days = 2 weeks (14 days) + 0 day (0 odd day).

### Formulas of Calendar For Number of Odd Days

• 1 ordinary year has 1 odd day

Explanation :  In an ordinary year, there are 365 days, which means 52 x 7 + 1, or 52 weeks and one day. This additional day is called an odd day.

• 1 leap year has 2 odd days

Explanation:  A leap year has 366 days. There are 29 days in February in a leap year. There are 52 weeks and 2 odd days in a leap year.

• 100 years has 5 odd days

Expalnation :  Odd days in a leap year = (52 weeks +2) days  .In 100 years , there will be 24 leap years and 76 non-leap years. So odd days in 100 years will be (76 x 1 + 24 x 2) which is 124 odd days. This can also be written as 17 weeks + 5 days. So every 100 years will have 5 odd days.

• 200 years has 3 odd days

Expalnation: 100 years give us 5 odd days as calculated above. 200 years give us 5 x 2 = 10 .Hence , 7 days (one week) = 3 odd days.

• 300 years has 1 odd day

Expalnation: 300 years give us 5 x 3 = 15.Hence, 14 days (two weeks) = 1 odd day.

• 400 years has 0 odd day

Expalnation:  The number of odd days in 400 years will be ( 5 x 4 + 1) because 400 is itself a leap year and that is why it has one odd day extra. Thus odd days in 400 will be 0.

• Similarly, all the 4th centuries 800 years, 1200 years, 1600 years, 2000 years etc. have 0 odd day.

• Mapping of the number of odd day to the day of the week
 Days Sunday Monday Tuesday Wednesday Thursday Friday Saturday Number  of odd days 0 1 2 3 4 5 6

### Reference Chart that gives odd days for the given months

• To determine whether a year is a leap year, follow these steps
1. If the year is evenly divisible by 4, go to step 2. Otherwise, go to step 5.
2. If the year is evenly divisible by 100, go to step 3. Otherwise, go to step 4.
3. If the year is evenly divisible by 400, go to step 4. Otherwise, go to step 5.
4. The year is a leap year (it has 366 days).
5. The year is not a leap year (it has 365 days).
• For example:

1. Check for years not ending with “00”.

Year 1997 is not a leap year because it is not divisible by 4.

Year 2016 is a leap year because it is divisible by 4.

2. Now, check for years ending with “00”.

Year 2000 is a leap year because it is divisible by 4,100 and 400.

Year 1900 is not a leap year because it is divisible by 4 and 100 but not 400.

Year 1600 is a leap year because it is divisible by 4, 100, and 400.

• For this reason, the following years are not leap years:

1700, 1800, 1900, 2100, 2200, 2300, 2500, 2600

This is because they are evenly divisible by 100 but not by 400.

• The following years are leap years:

1600, 2000, 2400

This is because they are evenly divisible by both 100 and 400.

### Points to remember

• Last day of a century cannot be Tuesday or Thursday or Saturday because of the number of odd days.
• For the calendars of two different years to be the same, the following conditions must be satisfied.

a.   Both years must be of the same type. i.e., both years must be ordinary years or both years must be leap years.

b.   1stJanuary of both the years must be the same day of the week.

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