Tuesday, 10 February 2015

14.Aptitude Questions ---- Calendar - Concept


Calendars

If someone asks us what day it was on 10th April , 1924 or what day it would be on 14th July 2475, we feel these are crazy questions. But if you know the rule how to find it , it is very easier task.
The clue to the process of finding the day of the week on a particular date  lies in finding the number of odd days , which is different from the odd numbers.

ODD DAYS : The number of days more than complete number of weeks in a given period are called odd days.

In other words , the remainder obtained when the given number of days is converted into weeks by dividing by 7.
 Example : The number of odd days in a period of 72 days  is => 2.

 Note => (7a +b) odd days , where b is less than or equal to is equivalent to number of odd days .
     17 odd days= 7 x 2 + 3  = 3 odd days


LEAP YEAR:  A year which is exactly divisible by 4 is leap year.

For example: 1984, 1624,2012, etc.
For a century year to be the leap year it must be divisible by 400. thus years 400, 1200, 1600, 2000 are leap years. And 500,1100,1300,1500 etc are not leap years ( because they are not exactly divisible by 400).
An ordinary year has 365 days. Therefore we divide 365 by 7 to get the complete number of weeks and the remainder will be the odd days: as 365=(7×52)+1.
An ordinary year has 1   odd day           .                         
 Since a leap year has 366 days, there will be 2 odd days.

Things to be remembered:

100 years = 76 ordinary years +  24 leap years.
(Note that 100/4=25, but since 100th year is not a leap year as it is not visible by 400, we have only 24 leap year)
100 years= (76×1)+(24×2)
              =  76+48 = 124 =  (7×17)+5
Thus there will be 5 odd days, in 100 years.
In 200 years there will be  -> 2×5=10=(7×1)+3 =3 odd days.
In 300 years there will be -> 3×5=15=(7×2)+1 =1 odd day
In 400 years there will be ->4×5+1 leap year=21 = 7×3+0=0 odd days.(Every 400th year is a leap year. Therefore additional day is added)
Therefore each one of 800, 1200, 1600, 2000 years will leave zero (0) odd days.
To summarize

Years
Number of odd days
Example year
Number of odd days
Ordinary
1
2009
1
Leap
2
2008
2
100
5
1700
5
200
3
1800
3
300
1
1900
1
400
0
2000
0

Symmetricity of Calendars  ( Repetition of the calendar )

For a leap Year :

 
 Let us see , for example , the case of 2004 .

Year                2004          2005      2006        2007    2008
Odd days           2               1            1             1          2      

Since , the number of odd days are 7 , so days of the year  2004  and 2009 will be same from 1st January to 28 th February . ( Because 2009 is not a leap year )
To know which year will have the same calender as the given leap year , add 28  to given leap year i.e  2004 +28 =2032 will have the same calendar like 2004 for the whole year

For any ( leap year + 1) year :

Let us take an example , 2005

Year          2005   2006     2007    2008   2009   2010
Odd days     1        1           1          2      1       1

Since the number of odd days are 7 , so calendars of 2005 and 2011 will be same for whole year.

Any ( leap year  +1) , the same calendar will happen after 6 years .    2005 + 6 =2011

For any ( leap year + 2) year :

Let us take an example , 2006

Year            2006        2007    2008      2009   2010  2011
Odd days      1             1           2          1        1        1

Since the number of odd days are 7 , so calendars of 2006 and 2012 will be same till 28th February.
Any ( leap year  + 2) , the same calendar will happen after 6 years .    2006 + 6 =2012

For any ( leap year + 3) year :

Let us take an example , 2007

Year               2007    2008     2009   2010     2011 2012     
Odd days          1          2          1         1        1      2          
Year                 2013    2014     2015     2016    2017
Odd Days           1         1           1          2         1

Since the number of odd days are 14 , so calendars of 2007 and 2018 will be same for whole year.

Any ( leap year  +3) , the same calendar will happen after 11 years .    
2007 + 11 =2018

Symmetricity of Calendars ( Months in a year)

The following data gives the months in which the same calendar can be used, i.e, the corresponding dates of the follow the same week day.

Leap Year:  January & July, February & August.

Non-Leap Year: January & October, February & March

Any year, leap or non leap year: March & November, April & July, September & December

Important Note : 

i)            The last day of a century cannot be a Thursday, Tuesday or a Saturday.
ii)          The first day of a century must be a Monday, a Tuesday, a Thursday or Saturday.



For more solved problems and examples, go through  Problems on Calendars
For TCS placement questions on calendars , visit TCS PLACEMENT QUESTIONS ON CALENDARS