# 20. Calendars

Introduction:-
Solar Year: A solar Year consists of 365 days 5 hours 48 minutes 48 sec.

Leap and Ordinary year:
A Non-leap year (or) Ordinary year has 365 days, where as a leap year has one extra day because of 29 days in the month of February. Every year which is divisible by 4 is called a leap year. A leap year consists of 366 days (52 weeks + 2 days). There extra 2days are the odd days, so a leap year has 2 odd days. An ordinary year has 1 odd day (52weeks+1day)

Leap year: A year which is divisible by 4 is a leap year, except century year which are not multiple of 400.
Note: Every century, which is multiple of 400, is a leap year
Ex: 400, 800, 1200, 1600, 2000…

Finding the no. of odd days:
In a given period, the number of days more than the Complete weeks are called odd days.
1. One ordinary year = 365 days = (52 weeks + 1 day)
So an ordinary year has 1 odd day
2. One leap year = 366 days (52 weeks + 2 days)
So a leap year has 2 odd days
3. 100 years = 76 ordinary years + 24 leap year
= (76 × 1 + 24 × 2)
= 124 odd days
= (17 weeks + 5 days) = 5 odd days
No. of odd days in 100 years = 5
No. of odd days in 200 years = 5 + 5 = 10 – 7 = 3 odd days
No. of odd days in 300 years = 5 + 5 + 5 = 15-14 = 1 odd day
No. of odd days in 400 years = 5+ 5 + 5+ 6 = 21-21 = 0 odd days
Similarly 400, 800, 1200, 1600…. Years has 0 odd day

Codes for the weeks:
 No. of days 1 2 3 4 5 6 7 Day Monday Tuesday Wednesday Thursday Friday Saturday Sunday

Codes for the century years:
 100 years – 5 500 -5 900-5 1300-5 1700-5 2100-5 200 years – 3 600-3 1000-3 1400-3 1800-3 2200-3 300 years – 1 700-1 1100-1 1500-1 1900-1 2300-1 400 years – 0 800-0 1200-0 1600-0 2000-1 2400-0

No. of Odd days in month wise:
 MONTH No. of dates Odd days January 31 3 February 28/29 0/1 March 31 3 April 30 2 May 31 3 June 30 2 July 31 3 August 31 3 September 30 2 October 31 3 November 30 2 December 31 3

Solved Examples
Ex. 1 What was the day of the week on 16th July, 1776?

Sol: 16th July, 1776 = (1775 years + period from 1.1.1776 to 16.7.1776)
Counting of odd days:
Number of odd days in 1600 years = 0
Number of odd days in 100 years = 5
75 years = 18 leap years + 57 ordinary years
= (18× 2 + 57 × 1) odd days = 93 odd days
= (13 weeks + 2 days) = 2 odd days
1775 years have = (0+5+2) odd days = 7 days = 0 odd day.
 Jan. Feb. March. April. May. June. July (31 +29 + 31 +30 + 31 + 30 + 16) = 198 days

198 days = (28 weeks +2 days) = 2 odd days.
Total number of odd days = (0+2) = 2
Hence, the required day is Tuesday

Ex. 2 What was the day of the week on 15th August, 1947?
Sol: 15th August, 1947 = (1946 years + period from 1.1.1947 to 15.8.1947)
Odd days in 1600 years = 0
Odd days in 300 years = (5×3) = 15 = 1
46 years = (11 leap years + 35 ordinary years)
= (11× 2 + 35 × 1) odd days = 57 odd days
= (8 weeks + 1 days) = 1 odd day.
Odd days in 1946 years = (0+1+1) =2.
 Jan. Feb. March April May June. July Aug. (31 +28 +31 +30 +31 +30 +31 + 15) = 227 days

227 days = (32 weeks + 3 days) = 3 odd days
Total number of odd days = (2+3) = 5
Hence, the required day is Friday.

Ex. 3 What was the day of the week on 4th June, 2002?
Sol:
4th June, 2002 = (2001 years + Period from 1.1.2002 to 4.6.2002)
Odd days in 1600 years = 0
Odd days in 400 years = 0
Odd days in 1 Ordinary year = 1
Odd days in 2001 years = (0 + 0 +1) = 1
 Jan. Feb. March April May June (31 + 28 + 31 + 30 +31 4) = 155 days

= 22 weeks + 1 day = 1 odd day
Total number of odd days = (1+1) = 2
Required day is Tuesday

Ex: 4 on what dates of March 2005 did Friday fall?
Sol:
First we find the day on 1.3.2005
1.3.2005 = (2004 years + period from 1.1.2005 to 1.3.2005)
Odd days in 1600 years = 0
Odd days in 400 years = 0
4 years = (1 leap year + 3 ordinary years)
= (1× 2 + 3 ×1) odd days = 5 odd days
 Jan. Feb. March (31 +28 +1) = 60 days = (8 weeks + 4 days) = 4 odd days.

Total number of odd days = (0+0+5+4) = 9 = 2 odd days
.: 1.3.2005 was Tuesday. So, Friday lies on 4.3.2005
Hence, Friday lies on 4th, 11th, 18th and 25th of March, 2005.

Ex: 5 Prove that the calendar for the year 2003 will serve for the year 2014.
Sol.
We must have same day on 1.1.2003 and 1.1.2014
So, number of odd days between 31.12.2002 and 31.12.2013 must be 0.
This period has 3 leap years and 8 ordinary years.
Number of odd days = (3 × 2 + 8 ×1) = 14 = 0 odd day
:. Calendar for the year 2003 will serve for the year 2014.

EXERCISE
1. What was the day of the week on 15th Aug, 1947?
(a) Friday
(b) Sunday
(c) Thursday
(d) Tuesday
(e) Saturday
2. How many odd days in 256 days?
(a) 5
(b) 6
(c) 3
(d) 4
(e) 2
3. How many odd days in 15 years?
(a) 5
(b) 4
(c) 3
(d) 6
(e) 1
4. How many odd days in 189 days?
(a) 0
(b) 2
(c) 3
(d) 4
(e) 5
5. What was the day of the week on 21st June 2014?
(a) Sunday
(b) Friday
(c) Monday
(d) Thursday
(e) Saturday
6. If January 1, 2007 was Monday then what day of the week lies on January 1, 2008?
(a) Tuesday
(b) Sunday
(c) Thursday
(d) Friday
(e) Monday
7. If January 1, 2008 was Friday then what day of the week lies on Jan 1, 2010?
(a) Sunday
(b) Monday
(C) Thursday
(d) Saturday
(e) Friday
8. On 6th march, 2005 Monday falls. What was the day of the week on 6th march, 2004?
(a) Sunday
(b) Saturday
(c) Monday
(d) Tuesday
(e) Friday
9. On 05th Feb 2006 Friday falls. What was the day of the week on 5th feb, 2004?
(a) Thursday
(b) Sunday
(c) Tuesday
(d) Friday
(e) Saturday
10. Which of the following is a leap year?
(a) 1000
(b) 1200
(c) 1400
(d) 1800
(e) 2100
11. Which of the following is a non-leap year?
(a) 1972
(b) 1964
(c) 1976
(d) 1990
(e) 1984
12. On what dates of June 2012 did Monday full?
(a) 4th, 11th, 18th, 25th
(b) 6th, 13th, 20th, 27th
(c) 3rd, 10th, 17th, 24th
(d) 7th, 14th, 21st, 28th
(e) None of the above
13. The last day of a century cannot be____
(a) Monday
(b) Wednesday
(c) Tuesday
(d) Friday
(e) Sunday
14. What was the day of the week on 14th march 1969?
(a) Saturday
(b) Friday
(c) Wednesday
(d) Monday
(e) Sunday
15. The Calendar for the year 2014 will be the same for the year?
(a) 2020
(b) 2021
(c) 2024
(d) 2025
(e) 2026
16. The calendar for the year 2013 will be the same for the year?
(a) 2019
(b) 2018
(c) 2025
(d) 2026
(d) 2026
(e) 2024
17. The calendar for the year 1996 will be same for the year?
(a) 2007
(b) 2024
(c) 2020
(d) 2016
(e) 2012
18. The calendar for the year 1999 will be same for the year?
(a) 2008
(b) 2015
(c) 2010
(d) 2009
(e) 2027
19. On what dates of jan 2015 did Friday fall?
(a) 2nd, 9th, 16th, 23rd, 30th
(b) 2nd, 9th, 16th, 23rd
(c) 4th, 11th, 18th, 25th
(d) 3rd, 10th, 17th, 24th
(e) 7th, 14th, 21st, 28th
20. What was the day of the week on 29th Oct 1986?
(a) Tuesday
(b) Wednesday
(c) Monday
(d) Sunday
(e) Friday
21. On 8th march, 2005 it was Tuesday, what was the day of the week on 8th Feb 2004?
(a) Sunday
(b) Monday
(c) Saturday
(d) Friday
(e) Tuesday
22. If 10-May- 2012 is a Friday, What will be the day as 26-August-2016?
(a) Monday
(b) Sunday
(c) Saturday
(d) Wednesday
(e) Tuesday
23. On what days of May 2015 did Friday fall?
(a) 1St, 8th, 15th, 22nd, 29th
(b) 6th, 13th, 20th, 27th
(c) 5th, 12nd, 19th, 26th
(d) 3rd, 10th, 17th, 24th, 31st
24. What was the day of the week on 26th Jan 2015?
(a) Sunday
(b) Tuesday
(c) Wednesday
(d) Monday
(e) Friday
25. What was the day of the week on 6th June 2012?
(a) Wednesday
(b) Sunday
(c) Monday
(d) Friday
(e) Saturday
26. If today is Friday then what will be the day after 189 days?
(a) Sunday
(b) Thursday
(c) Friday
(d) Saturday
(e) Wednesday
27. If today 17-March 2012 is Saturday then what will be the day after 69 days?
(a) Friday
(b) Monday
(c) Sunday
(d) Tuesday
(e) Saturday
28. If today 26-April 2014 was Monday then what will be the day of the date 26-Nov 2013?
(a) Friday
(b) Thursday
(c) Wednesday
(d) Sunday
(e) Monday
29. On 8th dec, 2007 Saturday falls. What day of the week was it on 8th Dec, 2006?
(a) Sunday
(b) Thursday
(c) Tuesday
(d) Friday
(e) Saturday
30. Jan I, 2007 was Monday, what day of the week lies on Jan 1, 2008?
(a) Monday
(b) Tuesday
(c) Wednesday
(d) Sunday
(e) Thursday

 ANSWER KEY 1 a 11 d 21 a 2 d 12 a 22 c 3 b 13 c 23 a 4 a 14 b 24 d 5 e 15 d 25 a 6 a 16 a 26 c 7 b 17 b 27 a 8 a 18 d 28 b 9 c 19 a 29 d 10 b 20 b 30 b
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