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Worked Examples

Example-1

If 5 January 1991 was a Saturday, what day of the week was 3 March 1992?

Solution

1991 is an ordinary year, and hence it has only one odd day. Thus, 5 January 1992 was one day after Saturday, i.e., Sunday.

 

Now, in January 1992, there are 26 days remaining (31 – 5 = 26 days), i.e., 5 odd days (26 ÷ 7 = 3 weeks and 5 odd days). In February 1992, there are 29 days, i.e., 1 odd day (29 ÷ 7 = 4 weeks and 1 odd day).

 

In March 1992, there are 3 odd days (we are calculating odd days only up to 3 March 1992).

Therefore, the total numbers of odd days after 5 January 1992 till 3 March 1992 is 5 + 1 + 3 = 9 days, which is equivalent to 2 odd days.

 

Therefore, 3rd March 1992 was 2 days after Sunday, i.e., Tuesday.

 

Answer: Tuesday
 

Alternate method:

 

We can also solve the above problem by another method. Total number of days between 5 January 1991 and 3 March 1992 can be arrived at, as under:

In 1991

360

days (365 – 5 = 360)

In 1992, January

31

 

In 1992, February

29

(Leap year)

In March

3

 

 

423

days

 

423 days means 60 weeks and 3 days.

 

So, the number of odd days from 5 January 1991 to 3 March 1992 is 3.

 

Therefore, 3 March 1992 is 3 days after Saturday, i.e., Tuesday.

 

Answer: Tuesday
 

 

Example-2

If your birthday in 1994 fell on Sunday, on what day of the week did your birthday fall in 1995?

Solution

Since 1994 is a non-leap year, it had only one odd day. Hence, your birthday in 1995 fell one day after Sunday, i.e., on Monday.

 

Answer: Monday
 

 

Example-3

If 4 April 1988 fell on Monday, what day of the week was 3 November 1987?

Solution

Number of days from 3 November 1987 to 4 April 1988 is 153, as calculated below:

 

1987 November

27

(30 – 3)

1987 December

31

 

1988 January

31

 

1988 February

29

(leap year)

1988 March

31

 

1988 April

4

 

 

153

 

 

153 days, means 21 weeks + 6 days. So, there are 6 odd days.

 

Thus, 3 November 1987 was 6 days earlier to Monday. So, the day was Tuesday.

 

Answer: Tuesday
 

 

Example-4

What day of the week did 30 June 1974 fall?

Solution

We shall first calculate the number of odd days till 31 December 1973.

 

Number of odd days in the first 1600 years 0

 

Number of odd days in the next 300 years 1

 

73 years have 18 leap years and 55 ordinary years.

 

The number of odd days in 73 years = (2 × 18) + (1 × 55) = 36 + 55 = 91 = 13 weeks and 0 odd days.

 

So, the total number of odd days till 31st December 1973 = 0 + 1 + 0 = 1 odd day.

 

Now, we shall calculate the number of odd days in 1974 till 30th June.

 

January 31 days 3 odd days
February 28 days 0 odd days
March 31 days 3 odd days
April 30 days 2 odd days
May 31 days 3 odd days
June 30 days 2 odd days

 

Total number of odd days in 1974 till June 30th = 3 + 0 + 3 + 2 + 3 + 2 = 13 = 1 week + 6 odd days.

 

So, the total numbers of odd days from the day of commencement of calendar till 30th June 1974 is 1 + 6 = 7.

 

7 odd days means 1 week + 0 odd day.

 

Therefore, 30th June 1974 was a ‘Sunday’.

 

Answer: Sunday
 

 

Example-5

If 21 August 1980 was Thursday, what was the day on the same date after 3 years?

Solution

Since 1980 was a leap year, none of the next 3 years was a leap year. So the day of the week was 3 days after Thursday, i.e., it was on Sunday.

 

Answer: Sunday
 

 

Example-6

India’s first Republic day was celebrated on 26 January 1950. What day of the week was that memorable day?

Solution

The total number of odd days till 1949 can be calculated as shown.

 

1600 years had 0 odd day; 300 years had 1 odd day.

 

49 years had 12 leap years and 37 ordinary years.

 

So, the number of odd days in 49 years = (12 × 2) + (37 × 1) = 24 + 37 = 61 = 8 weeks and 5 odd days.

 

26 days of January had 5 odd days.

 

So, the total number of odd days till 26 January 1950 = 0 + 1 + 5 + 5 = 11 days = 1 week + 4 odd days.

 

So, the day was Thursday (count four days starting from Monday).

 

Answer: Thursday
 

 

Example-7

If 2 February 2004 fell on Monday, what was the day on 10 November of the same year?

Solution

Since 2004 was a leap year, February had 29 days.

 

Month
February
March
April
May
June
July
August
September
October
November
 
27
31
30
31
30
31
31
30
31
10
Days
(29 – 2)
 
 
 
 
 
 
 
 

 

 

Total number of days from 2 February 2004 to 10 November 2004 = 27 + 31 + 30 + 31 + 30 + 31 + 31 + 30 + 31 + 10 = 282 = 40 weeks and 2 days

 

So, 10 November fell 2 days after Monday, i.e., on Wednesday.

 

Answer: Wednesday
 

 

Example-8

India got its Independence on 15 August 1947. What was the day of the week?

Solution

We shall first calculate the number of odd days till 31 December 1946.

 

Number of odd days in the first 1600 years 0

 

Number of odd days in the next 300 years 1

 

46 years had 11 leap years and 35 ordinary years.

 

The number of odd days in 46 years = (2 × 11) + (1 × 35) = 22 + 35 = 57 = 8 weeks and 1 odd day.

 

Now, we shall calculate the number of odd days in 1947 till 15 August.

 

Month
January
February
March
April
May
June
July
August
 
Days
31
28
31
30
31
30
31
15
= 227

 

227 days = 32 weeks and 3 odd days.

 

So, the total number of odd days till 15 August 1947 = 0 + 1 + 1 + 3 = 5.

 

On counting five days from Monday, we get Friday.

 

Therefore, 15 August 1947 was a Friday.

 

Answer: Friday
 

 

Example-9

Show that the calendars for the years 1990 and 2001 were identical.

Solution

Find the number of odd days between 31 December 1989 and 31 December 2000. The number of odd days should be zero in order to have the same calendar.

 

Odd days can be calculated as follows:

 

Years
1990
1991
1992
1993
1994
1995
1996
1997
1998
1999
2000
Odd days
1
1
2
1
1
1
2
1
1
1
2

 

The numbers of odd days from 31 December 1989 to 31 December 2000 = 1 + 1 + 2 + 1 + 1 + 1 + 2 + 1 + 1 + 1 + 2 = 14 = 2 weeks and 0 odd days.

 

Further, both 1990 and 2001 are ordinary years.

 

Hence, the years 1990 and 2001 had the same calendar.
 

 
From 31 December 1989 to 31 December 1995, the number of odd days was 0. But, 1996 was a leap year where as 1990 was an ordinary year. So, the calendar of 1990 did not hold for the year 1996. However, the calendars of the two years were identical only up to 28 February.
 

Example-10

On what dates of July 1776 did Sundays fall?

Solution

First, we shall find out what day was 1 July 1776.

 

1600 years have 0 odd days.

 

100 years have 5 odd days.

 

75 years comprise of 18 leap years and 57 ordinary years.

 

So, the number of odd days in 75 years = 18 × 2 + 57 = 93 = 13 weeks, 2 odd days.

 

Thus, 1775 years have (0 + 5 + 2) = 7 odd days. So, 0 odd days.

 

In 1776, till 1 July, the number of odd days is calculated as follows:

 

Month
January
February
March
April
May
June
July
Days
31
29
31
30
31
30
1

 

183 days = 26 weeks + 1day = 1 odd day.

 

So, 1 July 1776 was Monday, and hence the first Sunday in July 1776 was on the seventh.

 

The other Sundays in July 1776 were 14th, 21st and 28th.
 





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