Quants 3 - Calendar

Practice Questions:

Question 1:

On what dates of July 2004, did Monday fall?

Let's find the day of July 1, 2004.
2000 = 0 Odd Days
3 Years = 3 Ordinary Years
             = (3 * 1) = 3 Odd Days
For 2004, counting days from Jan to Jul,
31 + 29 + 31 + 30 + 31 + 30 + 1 = 26 weeks + 1 day = 1 Odd Day
So, 3 + 1 = 4 Odd Days = Thursday.
Hence, 1st Monday in July 2004 was July 5.
Other Mondays were 12, 19, 26.
Thus, July 5, July 12, July 19 and July 26 of 2004 were Mondays.

Question 2:

Prove that the Calendar for the year 2003 will serve for the year 2014.

In order for both the calendars of 2003 and 2014 to be same, both the Jan 1st must be on same date and they both must be either Leap years or Ordinary Years.
Here, both are Ordinary Years.
For the first days to be the same, the odd days between Dec 31, 2002 and Dec 31, 2013 must be zero.
During this period, there were, 3 Leap Years and 8 Ordinary Years.
So, the Odd Days are (3 * 2) + ( 8 * 1) = 14 Days = 2 Weeks = 0 Odd Days.
Thus, it is proven that the Calendar for 2004 will serve for 2014.

Question 3:
Prove that any date in March of a year is the same day of the week as the corresponding date in November that year.

The number of Odd days between last day of February and last day of October should be zero for both the months to be same.
Counting days from March to October,
31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 = 245 = 35 weeks = 0 Odd Days
As the period between February and November has 0 Odd Days,
It is proven that any date in March of a year is the same day of the week as the corresponding date in November that year.

Question 4:
January 1, 2004 was a Thursday. What day of the week lies on January 1, 2005?

The year 2004 being a Leap year, it has 2 Odd days. So, the first day of 2005 will be 2 days beyond Thursday, and so it is a SATURDAY.


Question 5:

The first Republic day of India was celebrated on January 26, 1950. It was?

26th January 1950 = 1949 years + (Jan 1, 1950 to Jan 26, 1950)

Odd Days:

1600 = 0 Odd Days

300 Years = 1 Odd Day

49 Years = 12 Leap Years + 37 Ordinary Years

               = (12 * 2) + (37 * 1) = 61 days = 8 weeks + 5 day = 5 Odd Days

So, 0+1+5 = 6 Odd Days till 1949.

For 1950, counting days from Jan 1 to Jan 26 = 26 days = 3 weeks + 5 days = 5 Odd Days

So, 6 + 5 = 11 = 4 Odd Days

As Odd day 4 corresponds to Tuesday, 26th January, 1950 is THURSDAY.

Time to Think:

On 8th March, 2005, Wednesday falls. What day of the week was it on 8th March, 2004?