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Calendar Reasoning Notes

Calendar is a way to display relationships between day, month and year.

Every year that divides by four is leap year and all other are ordinary years. In the ordinary year there are 365 days and 366 days in Leap year i.e. in a simple year, total 52 weeks and 1 day, and in one leap year there are 52 weeks and 2 extra days.


Normal and Solar Year

The main and smallest unit of time measurement is the day. The timeline of one day is equal to the time spent in a whole round on the Earth’s own axis and when the earth takes a full circle of the Sun, then the time taken is equal to one solar year.

One solar year = 365 days, 5 hours, 48 minutes and 47½ seconds is equal to approximately 365.2422 days. It was revised to 365 days as the year which was called General Year .

Leap years

A normal year is having 365 days and a solar year having 365.2422 days. In this way, every normal year is less than 0.2422 days from the solar year. If the year is calculated on the basis of 365 days, then every year, the average year will be less than 0.2422 days from the solar year. Thus the importance of calendar will be reduced. If this sequence continues for 4 years, the normal year in this period will be 0.2422 × 4 days from the solar year i.e. ‘0.9688 days. This period is equivalent to about 1 day. Thus, after every 4 years, it is added in the form of an amendment, the 1 day is added in the month of February, which is February 29 and the year is of 366 days. Thus we can say that the year which is completely divided from 4 or the centenary year which is completely divided from 400, is called leap year; Eg -1996, 2000, 2004, 2016 etc.

In the leap year, both the first day of the year and the last day are uneven, i.e., the last day increases one day compared to the first day of the year; For example, if the first day of leap year i.e. January 1 is wednesday, then the last day of the same year i.e. December 31, will be on Thursday.

Centenary Leap Year

Leap year comes in every 4 years. Hence leap year 4 is contained in the multiplier. Thus, if the number of the year divided from 4, it will be leap years, but this rule does not apply in the century. Century leap year comes after 400 years. These are multiples of 400 years. Thus, if it is a part of 400 in the century year, then it will be a leap year.

cycle of days

The seventh part of any week is called day. There are 7 days in a week. Monday, Tuesday, Wednesday, Thursday, Friday, Saturday and Sunday. A cycle of weeks is completed in seven days. After this, the days start getting recursive again. .

The 28th, 30th, or 31st part of any month, or 365th part of the year, is called Date . This is determined by numbers.

Context of Odd Days

Extra days left after this full cycle of days are called odd days – if starting from Monday, then 1 full cycle and 1 extra day or 1 asymmetrical day will be available in 8 days time period. Each days comes in one full circle one after another. After this, the days start getting recursive again.

Any day remained after the full cycle of seven days is called an odd day. Therefore, to find the odd days, divide the number of days by 7. The quotient (or bar) that is received in this division indicates towards the full circle of the days and the remainder which indicates the number of odd days.

Number of odd days in 14 days = 14 = 2 times 0 remaining

= 0 days

Number of odd days in 20 days = 20/7 = 2 times 6 remaining

= 6 days

Number of odd days in 30 days = 30/7 = 4 times 2 remaining

= 2 days

Number of uneven days in 365 days = 365/7 = 52 times 1 remaining

⇒ Notes – The number of odd days can not be greater than 6.

Use of Odd Days

To know which day will be on a certain date, it is necessary to use an odd day. The beginning of the century is Monday, that is the day, on 1st January 1 is Monday. Therefore, in the 7-day cycle in relation with century, the first day is Monday and the last day is Sunday.

Calendar Reasoning Notes 

Calendar related important facts

(i) One Normal Year = 365 days = 52 weeks + 1 day = 1 odd day

(ii) one leap year = 366 days = 52 weeks + 2 days = 2 odd days

(iii) 100 years = 76 ordinary years + 24 leap years

= (76 + 24 × 2) Heterogeneous day

= 124 Asynchronous day

= 17 weeks + 5 days

= 5 Odd Days

(iv) In 200 years the number of odd days 2 × 5/7 = 1 bar 3 remaining = 3 days

(v) In the 300 years the number of odd days – 2 bar 1 remaining = 1 day

(vi) The number of odd days in 400 years = 4 × 5 = 20 days

Because the fourth century is leap years. Therefore, the number of odd days is to add 1 more: Number of odd days in 400 years

20 + 1/7 = 3 times remaining 0

= 0 days

So there is no odd day in 400 years.

(vii) Number of days in the months having 31 Days = 

= 4 bar 3 remaining

= 3 days

(viii) The last day of a century can not be Tuesday, Thursday or Saturday but it can be Wednesday, Friday and Sunday.

(ix) The first day of a century can be on Monday, Tuesday, Thursday or Saturday.

(x) If there is a day on any date of any month in a normal year, then the next year on the same date of the same month the day will be shifted by one. Like – If on January 1, 2001 there is Monday, Then on , January 1 2002 there will be Tuesday.

In each leap year, the day of any particular day or date will increase by 2. i.e. on January 1, 1996 there will be Saturday, then on January 1, 1997 there will be Monday.

(xii) The last day of the normal year is the same as its first day i.e. if it is Thursday on January 1, then on 31st December of that year will also Thursday and on 1st January of next year Will be Friday. In a Leap year if there is Thursday on 1 January of the year, then on, 31 December of that year will be Saturday.

(xiii) 7 Number of days to go back = number of days to move forward.

(xiv) The role of ‘after’, ‘before’, ‘to’ and ‘from’ are very important related to the questions to this chapter. So this information is necessary.

If the word ‘after’ has been used to find a date or day in any question, then calculate the date by increasing the day by one from the given day; For example, what will be the date after 4 days of March 15, so here is the date after the 4 days of March 15. 15 + 4 + 1 = 20 March Similarly, on the third day after Thursday, i.e. three days after tomorrow, i.e. the fourth day will be Monday. Apart from this, if there is a question based on ‘before’ or ‘after’, then the time is decreaded or increased based on the condition given in the question, For Eg. three days before Thursday was Monday. Similarly, the date 5 days after August 12 will be, 12 + 5 i.e. 17th August.

(xv) The total number of days in each week is 7. Therefore, on adding or subtracting 7 days to any day, the result will be the same day.

The questions asked in this chapter are generally divided into four parts.

(a) based on different dates: On a given Date, the Number of Days asked will be after the given date or before the given date.

After Calculation – If the Number of Days will be completely Divisible by 7.Then the day will be same as asked in the question.

(b) Based on the year intervals :First of all, in the questions, Calculate the difference between the two years given. Add the Number of Leap years between these years. Then divide it by 7. If the remainder comes to zero, then it will be the same day as asked in the question, if the remainder is obtained in the digits, then calculate the answer by increasing number of days if asked about later days or calculate the answer by decreasing number of days if asked about previous days.

(c) Questions on the Mixed Problem: In these type of Questions first of all calculate the difference of number of Years. Then Calculate the difference in Number of Days by Date based on the rules given above.

(d) Based on the basic calendar: Before solving these type of questions, it is necessary to keep in mind the below Facts: 

If 31 December 1600 -> Sunday

31 December 1700 -> Friday

31 December 1800 ->Wednesday

31 December 1900 -> Monday

31 December 2000) -> Sundays

Type 1

Example: If the 11th day of the month is Saturday, which of the following days will occur five times in that month?

(a) Tuesday (b) Sunday (c) Friday (d) Saturday

Solution: (c) According to the question, if the 11th day of the month is Saturday then (11-7) = 4 date will also be Saturday. In this way, All the Days before Date 4 will arrive five times in the month. i.e. Friday, Thursday and Wednesday.

Type 2

Example: Jatin remembers correctly that her mother’s birthday is after 12 March and before March 17, while her sister correctly remembers that her mother’s birthday is after 10 March but before March 14. On what date of March was his mother’s birthday?

(A) 15 (b) 16 (c) 14 (d) can not be determined (e) None of the above

Solution: (e) Mother’s birthday, according to Jatin, -> 13, 14, 15, 16 March According to Jatin’s sister -> 11, 12, 13 March,

Hence Jatin’s Mother’s birthday was on 13 March.