Quants 2 - Calendar

Key Points to Remember:

The main objective of this exercise is to find the day of the week on a particular given date. The process involves in getting the number of odd days.

Odd Days - Number of days more than one complete number of weeks in a given period.
Leap Year - Every year which is divisible by 4. Eg - 1992, 1996, 2004, 2006, 2008 etc.
Every 4th century such as 400, 800, 1200, 1600, 2000 etc are leap years.
But other centuries such as 100, 300, 500, 900, 1000, 1900 etc are not leap years.
An year which is not a Leap year is an Ordinary Year.

Number of Days in Leap year - 366; Number of days in Ordinary year - 365

Counting of Odd Days 
1 Ordinary Year - 52 weeks + 1 day. So, an Ordinary year has 1 Odd day.
1 Leap Year - 52 weeks + 2 days. So, a Leap Year has 2 Odd days.
100 Years = 76 Ordinary years + 24 Leap Years 
                 = [ (76 * 52) weeks + 76 days ] + [ (24 * 52) weeks + 48 days ]
                 = 5200 weeks + 124 days = 5217 weeks + 5 days
100 years has 5 Odd Days
200 years has 10 Odd Days = 3 Odd Days (10 = 7 + 3; 7 days = 1 week)
300 years has 15 Odd Days = 1 Odd Day
400 years has 20 + 1 Odd Days = 21 = 0 Odd Days.
Similarly, 800, 1200, 1600, 2000 etc have 0 Odd Day.

Odd Days         Day
     0                   Sunday
     1                   Monday
     2                   Tuesday
     3                   Wednesday
     4                   Thursday
     5                   Friday
     6                   Saturday

Practice Questions:

Question 1:

What was the day of the week on 16th July, 1776?

16th July 1776 = 1775 + (Jan 1, 1776 to Jul 16, 1776)
Odd Days :
1600 = 0 Odd Days
100 Years = 5 Odd Days
75 Years = 18 Leap Years + 57 Ordinary Years
               = [ (18 * 2) + (57 * 1) ] = 93 Days = 13 weeks + 2 days = 2 Odd Days
So, 0 + 5 + 2 = 7 = 0 Odd Day till 1775.
For 1776, counting days from Jan to Jul,
31 + 29 + 31 + 30 + 31 + 30 + 16 = 198 Days = 28 weeks + 2 days = 2 Odd Days
As Odd day 2 corresponds to Tuesday, 16th July, 1776 is TUESDAY.

Question 2:

What was the day of the week on 15th August, 1947?

15th August 1947 = 1946 years + (Jan 1, 1947 to Aug 15, 1947)
Odd Days:
1600 = 0 Odd Days
300 Years = 1 Odd Day
46 Years = 11 Leap Years + 35 Ordinary Years
               = (11 * 2) + (35 * 1) = 57 days = 8 weeks + 1 day = 1 Odd Day
So, 0+1+1 = 2 Odd Days till 1946.
For 1947, counting days from Jan to Aug,
31 + 28 + 31 + 30 + 31 + 30 + 31 + 15 = 227 days = 32 weeks + 3 days = 3 Odd Days
So, 2 + 3 = 5 Odd Days
As Odd day 5 corresponds to Friday, 15th August, 1947 is FRIDAY.


Question 3:
What was the day of the week on 16th April, 2000?

16th April 2000 = 1999 years + (Jan 1, 2000 to Apr 16, 2000)
Odd Days:
1600 = 0 Odd Days
300 Years = 1 Odd Day
99 Years = 24 Leap Years + 75 Ordinary Years
               = [ ( 24 * 2 ) + ( 75 * 1 ) ] = 123 days = 17 weeks + 4 days = 4 Odd Days
So, 0 + 1 + 4 = 5 Odd Days till 1999.
For 2000, counting days from Jan to Apr,
31 + 29 + 31 + 16 = 107 days = 15 weeks + 2 days = 2 Odd Days
So, 5 + 2 = 7 = 0 Odd Days
As Odd day 0 corresponds to Sunday, 16th April, 2000 is SUNDAY.


Question 4:
What was the day of the week on 7th July, 1981?

7th July 1981 = 1980 years + (Jan 1, 1981 to Jul 7, 1981)
Odd Days:
1600 = 0 Odd Days
300 Years = 1 Odd Day
80 Years = 20 Leap Years + 60 Ordinary Years
               = [ ( 20 * 2) + ( 60 * 1 ) ] = 100 days = 14 weeks + 2 days = 2 Odd Days
So, 0 + 1 + 2 = 3 Odd Days till 1980.
For 1981, counting days from Jan to Jul,
31 + 28 + 31 + 30 + 31 + 30 + 7 = 188 days = 26 weeks + 6 days = 6 Odd Days
So, 3 + 6 = 9 = 2 Odd Days (9 - 7 = 2 )
As Odd day 2 corresponds to Tuesday, 7th July, 1981 is TUESDAY.


Question 5:
What was the day of the week on 9th August, 1997?

9th August 1997 = 1996 years + (Jan 1, 1997 to Aug 9, 1997)
Odd Days:
1600 = 0 Odd Days
300 Years = 1 Odd Day
96 Years = 24 Leap Years + 72 Ordinary Years
               = [ ( 24 * 2) + ( 72 * 1 ) ] = 120 days = 17 weeks + 1 days = 1 Odd Day
So, 0 + 1 + 1 = 2 Odd Days till 1996.
For 1997, counting days from Jan to Aug,
31 + 28 + 31 + 30 + 31 + 30 + 31 + 9 = 221 days = 31 weeks + 4 days = 4 Odd Days
So, 2 + 4 = 6 = 6 Odd Days
As Odd day 6 corresponds to Saturday, 9th August, 1997 is SATURDAY.


Time to Think:

What was the day of the week on 6th February, 2015?