calendar math

Now as we know the basics of Calendar - Its usage, how to read a calendar, how year, month, week and days are represented in a calendar, let us learn calculations based on the Gregorian calendar and how it can be used to find a day or date of an event in future or present. Let's recap a few already learnt concepts:

• A calendar represents a year.
• A Year is broken down into 12 months. There are two kinds of years, ordinary and leap year. An ordinary year has 365 days and a leap year has 366 days. The reason is leap year has 29 days in February and the ordinary year has 28 days. 
• No fixed number of days or weeks make a month, if nothing else is specified, we take:

1. A month has 4 weeks.
2. A month has 30 days.
3. A year has 52 weeks.
4. A year has 365 days.

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How to determine which year is a leap year?
(1) If a year is not a century then the year which is divisible by 4 is a leap year. Example, 1952, 2008, 2020.

(2) Every 4th century is a leap year. Century year to be leap year should be divisible by 400. For Example, 1200, 800.

Concept of ‘odd-days’
The number of days more than the complete weeks are called odd days in a given period. Divide the total number of days in the month of January by 7, this gives a remainder 3. Thus generally any month that has 31 days has 3 odd days and a month which has 30 days has 2 odd days. February month with 28 days has 0 odd days. Leap year has 52 weeks and 2 odd days. Ordinary years have 52 weeks and 1 odd day.

The number of odd days in a century: There are 24 leap years and 76 non–leap years. Each leap year has 2 odd days and each non-leap year has 1 odd day. Therefore, 24 × 2 + 76 × 1 = 124 total odd days. So net odd days = 124 ∕ 7. The remainder is 5 – this is the number of odd days in a century.

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Questions

1. What day is December 25, 2082?

Solution: Finding the day of the week for any date using "The Key Value Method"

This method uses codes for different months and years to find the day of the week. Let's see

• Take the last 2 digits of the year. In our example, this is 82.
   
• Divide it by 4, and drop the remainder.Therefore, 82 ∕ 4 = 20, remainder is 2, for us only "20" is important.
• Add the day of the month. In our example, 20 + 25 = 45.
   
• Add the month's key value, from the following table.

  January	1
  February	4
  March	4
  April	0
  May	2
  June	5
  July	0
  August	3
  September	6
  October	1
  November	4
  December	6

  The month is December, with a key value of 6.
  45 + 6 = 51.
   
• If your date is in January or February of a leap year, subtract 1. In our case it is December, so need not have to do anything here.
   
• Add the century code from the following table.

  1700s	1800s	1900s	2000s
  4	2	0	6

  Our year is 2082, so we look at the table for the 2000s, and get the code 6.
  Now we add this to our running total: 51 + 6 = 57. As in the Gregorian calendar repeats every four hundred years, all we have to do is add or subtract 400 until we have a date that is in the table. Example, 2482 - 400 = 2082.

• Add the last two digits of the year. 57 + 82 = 139.
   
• Divide by 7 and take the remainder. This time, 1 means Sunday, 2 means Monday, and so on. A remainder of 0 means Saturday. 139 ∕ 7 gives the remainder 6, so December 25, 2082 will be on Friday.

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2. Todays is Monday, what day will it be after 60 days?
Solution: It will be Friday. Each day is repeated after 7 days so 63^rd day will be Monday, therefore 60^th will be Friday.

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3. If 18 January 2020 is Saturday, what will be 18 January 2022?
Solution: Calculate the number of days between the two dates:

Number of days between 18 January 2020 to 18 January 2021: 366 days (2020 is a leap year)

Number of days between 18 January 2021 to 18 January 2022: 365 days 

Total number of days: 366 + 365 = 731

731 ÷ 7 gives the remainder 3, therefore, it will be 3 days after Saturday, which is Tuesday

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4. Calculate the duration between 12 October 2008 and 14 April 2020.
Solution: Write dates in a year, month, day format
 

	Year	Month	Day
	2020 2019	04 16	14
−	2008	10	12
	11	6	2

Answer - 11 years 6 months 2 days.

How? Subtract the bigger date from the smaller date, but take care of the borrowing rule. Minuend − Subtrahend = Difference

• If the minuend month number is smaller than the subtrahend month number: Borrow 12 months from the year and subtract year by one. Like in the example above.
• If the minuend days is smaller than the subtrahend: Borrow one complete month from the month and subtract the month number by one. Like in the below example.

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5.  Calculate the duration between 12 October 2008 and 4 November 2020.
Solution:
As 4 is less than 12, borrow 1 month, October i.e. 31 days is added to 4 and total days is now 35. Now you can easily subtract 12 from 35.

	Year	Month	Day
	2020	11 10	4 35
−	2008	10	12
	12	0	23

Answers: Total duration is 12 years 23 days.