Mental Ability

CHAPTER 14 Calendar
This is another important chapter in reasoning. Questions based on calendar are of set pattern and hence follow set
rules. We are giving below the definition/meaning of certain words which are used in solving questions on calendar.
Odd days : We know that a week contains 7 days counting from Monday to Sunday. So, any number of days,
which are more than complete number of a week in a given period are called odd days. For example, a period
of 10 days contains 3 odd days, 11 days contains 4 odd days, 12 days contains 5 odd days. But period of 14 days
contains zero odd day.
Therefore, in finding number of odd days in a given period of time, one has to divide that period by 7. If it is
completely divisible by 7, it contains zero odd day and if it is not divisible by 7, then remaining number of days
are the odd days.
1. A solar year(Gregorian Calendar), is slightly shorter than 365 days and 6 hours (365.25 days). As derived
from the Alfonsine tables, the Earth makes a complete revolution around the sun in 365 days, 5 hours, 49
minutes, and 16 seconds (365.2425 days).
i.e., 0.2425 * 4 = 0.97 days
2. Adding 1 calendar day every four years would thus add an extra 0.03 day in every 4 years(43 minutes and
12 seconds to the calendar),or 3 days every 400 years.
3. To compensate for this, three leap days must be omitted every 400 years. The Gregorian calendar reform
implemented this adjustment by making an exception to the general rule that there is a leap year every
four years.
4. Instead, a year divisible by 100 would not be a leap year unless that year was also exactly divisible by 400.
This means that the years 1600 and 2000 were leap years, as will be 2400 and 2800, but the years 1700,
1800, and 1900 were not, nor will 2100, 2200, and 2300.
The Gregorian c alendar Rules
• Every fourth year is a leap year. 2004, 2008, and 2012 are leap years.
• However, every hundredth year is not a leap year. 1900 and 2100 are not leap years.
• Every four hundred years, there's a leap year after all. 2000 and 2400 are leap years.
Working rule to find the day of the week on a part icular date when reference day is given
Step 1: Find the net number of odd days for the period between the reference date and the given date (exclude
the reference day but count the given date for counting the number of net odd days).
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Step 2: The day of the week on the particular date is equal to the number of net odd days ahead of the
reference day (if the reference day was before this date) but behind the reference day (if this date was behind the
reference day).
E x . 1 If 11th January 1997 was a Sunday then what day of the week was on 10th January 2000?
S o l . Reference day is before the given date i.e. 10-01-2000.
Total number of days between 11th January 1997 and 10th January 2000
= (365 - 11) in 1997 + 365 in 1998 + 365 in 1999 + 10 days in 2000
= (50 weeks + 4 odd days) + (52 weeks + 1 odd day) +
(52 weeks + 1 odd day) + (1 week + 3 odd days)
Total number of odd days = 4 + 1 + 1 + 3 = 9 days = 1 week + 2 days
Hence, 10th January, 2000 would be 2 days ahead of Sunday i.e. it was on Tuesday.

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E x . 2 Today is Wednesday what will be the day after 94 days ?
A. Monday B. Tuesday C. Wednesday D. Sunday
S o l . 94/7=3 is the remainder, therefore 3 days after Wednesday=Sunday.
E x . 3 Today is Thursday. The day after 59 days wil be?
A. Sunday B. Monday C. Tuesday D. Wednesday
S o l . Similar as above, 3 is the remainder, therefore 3 days after Thursday=Monday
Working rule to find the day of the week on a particular date when no reference day is given
Summarising the steps followed:
Step 1: Count the net number of odd days on the given date
Step 2: Write the day for the given number of odd days as per the table below.

Day : Sunday Monday Tuesday Wednesday Thursday Friday Saturday

Odd Day : 0 1 2 3 4 5 6

E x . 4 How often in years do calendars repeat with the same day-date combinations?
S o l . Calendars repeat in a regular cycle of 28 years, within a century.
For NON-LEAP YEARS, a given arrangement of days will repeat in 6 years, then 11, then 11 years, then begin
a new cycle of 6-11-11(6+11+11=28) eg: 1989(+6), =1995(+11),= 2006(+11),=2017.
Crossing a century changes this because only every 4th century year (e.g. 2000 but not 2100, 2200, or 2300)
is a leap year.
LEAP YEAR calendars repeat every 28 years.e.g. 1804,1832,1860,1888,1928,1956
Repeating calendars in a given century:
There is a very simple pattern for determining "when" calendars repeat with the same day/date combinations.
1) Define any year as one of four things:
(a) a "leap year", or
(b) the "1st year after a leap year", or
(c) the "2nd year after a leap year", or
(d) the "3rd year after a leap year".
2) Add 28 to a "leap year" to get the next year that it will repeat.
3) Add 6 to the "1st year after a leap year" to get the next repeating calendar year.
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4) Add 11 to the "2nd year after a leap year" to get the next repeating calendar year.
5) Add 11 to the "3rd year after a leap year" to get the next repeating calendar year.
Ex.5 In the century 2000-2100
S o l . 2010 is the 2nd yr after a leap year 2008 and will repeat in 2010 + 11 = 2021.
2011 is the 3rd year after a leap year and repeats in 2022;
2012 a leap year repeats in 2040;
2013 1st year after leap year repeats in 2019;
2014 2nd year after leap year repeats in 2025;
2015 3rd year after leap year repeats in 2026;
2016 leap year repeats in 2044;
2017 repeats in 2023; and so on.
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 Mental Ability
E x . 6 Find the day of the week on 27th Dec 1985 ?
S ol . ( A ) 27th Dec 1985 has (1984 years, 11 months and 27 days).
Now, 1600 years have 0 odd day.
300 years have 1 odd day
84 years contains
= (21 leap years and 63 ordinary years) = (21 × 366 + 63 × 365) days
30681
= (7686 + 22995) = 30681 days =
7
= 4383 weeks.
i.e., 84 years contain 0 odd day
11 months and 27 days
= 361 days
= 51 weeks + 4 days
= 4 odd days.
=4 + 1 = 5
Therefore,
27th Dec 1985 has (4 + 1) = 5 odd days.
Now counting Sunday as 0 odd day, Tuesday as 2 odd days and so on, Friday will have 5 odd days.
Therefore, 27th Dec 1985 will be Friday.
[Short trick :

Step-I. Date Month Century year Year Leap Year only Remainder
7

Step-II. Remainder – 1 = Day

He re

Months Code
J an Fe b Ma r Ce nturies Co de
1 4 4 Befo re
29 th Feb
Apr Ma y J une
1600 6 5
0 2 5
J uly Augs Sep 1700 4
1800 2
0 3 6
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1900 0
Oct Nov Dec
2000 6 5
1 4 6

Use : Find the day of the week on 27th Dec 1985

27 6 0 85 21
(i) = 6 (Remainder)
7

(ii) 6 – 1 = 5 Friday
(Note: (i) If there is remainder zero than use 7 on place of zero.
(ii) If given year is a leap year than that year is not consider as a leap year before 29th Feb. After 29th Feb. that
year consider as a leap year.) ]

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E x . 7 What was the day on 26th Jan 1950, when 1st Republic Day of India was celebrated?
(A) Monday (B) Tuesday (C) Thursday (D) Friday
S o l . 26th Jan, 1950 means
(1949 years and 26 days)
1600 years have 0 odd day
300 years have 1 odd day
49 years have (12 leap years and 37 ordinary years)
(12 x 366 + 37 x 365)
days – (4392 + 13505) days
(17897) days = 2556 weeks + 5 days
So, 49 years have 5 odd days
and 26 days have 5 odd days
Total number of odd days = 0 + 1 + 5 + 5 = 11 days = 4 odd days.
Hence, the day on 26th Jan 1950 was Thursday.

E x . 8 What is the number of odd days in a leap year?

(A) 1 (B) 2 (C) 3 (D) 4

S o l . A leap year has 366 days

Now if we divide 366 by 7, it gives 2 as remainder.

Hence, number of odd days in 366 days is 2.

Ex.9 Prove that the calendars for 1990 will serve for 2001 also.

S o l . There are two ways of proving it. One find the day on 1st Jan, 1990 and 1st Jan, 2001. Both the days should
be identical. 2nd find the number of odd days between 31st Dec 1989 and 31st Dec 2000. The sum of odd
days should be zero.

Odd days are calculated as below:

Year 1990 1991 1992 1993 1994 1995 1996 1997 1998 1999 2000
Odd Days : 1 1 2 1 1 1 2 1 1 1 2
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Sum of odd days = 14 i.e., 0 odd day.

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 Mental Ability

EXERCISE
1. What was the day on 1st Jan 1901?

(1) Monday (2) Wednesday (3) Sunday (4) Tuesday

2. What was the day on 31st Oct 1984 ?

(1) Friday (2) Sunday (3) Wednesday (4) Monday

3. What was the day on 14th Mar 1993?

(1) Friday (2) Thursday (3) Sunday (4) Saturday

4. On what dates of Aug 1980 did Monday fall ?

(1) 4th, 11th, 18th and 25th (2) 3rd, 10th, 17th and 24th

(3) 6th, 13th, 20th and 27th (4) 9th, 16th, 23rd and 30th

5. On what dates of Dec 1984 did Sunday fall ?

(1) 6th, 13th, 20th & 27th (2) 7th, 14th, 21st, & 28th

(3) 2nd, 9th, 16th, 23rd & 30th (4) 1st, 8th, 15th & 22nd

6. Today is Wednesday, what will be the day after 94 days?

(1) Monday (2) Tuesday (3) Wednesday (4) Sunday

7. Today is Thursday. The day after 59 days will be :

(1) Sunday (2) Monday (3) Tuesday (4) Wednesday

8. Today is Friday. The day after 63 days will be :

(1) Friday (2) Thursday (3) Saturday (4) Monday

9. What was the day of the week on 2nd July 1984 ?

(1) Wednesday (2) Tuesday (3) Monday (4) Thursday

10. The year next to 1990 will have the same calendar as that of the year 1990 :

(1) 2001 (2) 1997 (3) 1996 (4) 1992

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11. The year next to 1996 will have the same calendar as that of year 1996 :

(1) 2001 (2) 2024 (3) 1997 (4) 1999

12. The first republic day of India was celebrated on 26th January, 1950. It was —

(1) Monday (2) Tuesday (3) Thursday (4) Friday

13. On January 12, 1980, it was Saturday. The day of the week on January 12, 1979 was —

(1) Saturday(2) Friday (3) Sunday (4) Thursday

14. The number of odd days in a leap year is —

(1) 1 (2) 2 (3) 3 (4) 4

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15. The year next to 1991 having the same calendar as that of 1991 is —

(1) 1998 (2) 2001 (3) 2002 (4) 2003

16. Monday falls on 4th April, 1998. What was the day 3rd November, 1987?

(1) Monday (2) Sunday (3) Thursday (4) Wednesday

17. Smt. Indira Gandhi died on 31st October, 1984. The day of the week was —

(1) Monday (2) Tuesday (3) Wednesday (4) Friday

18. What will be the day of the week on 1st January, 2010?

(1) Friday (2) Saturday (3) Sunday (4) Monday

19. The calendar for the year 2005 is the same as for the year :

(1) 2010 (2) 2011 (3) 2012 (4) 2013

20. How many times does the 29th day of the month occur in 400 consecutive years?

(1) 4497 (2) 1237 (3) 5012 (4) 4126

21. If 20th September in a year is Wednesday, the number of Fridays in that month is

(1) 6 (2) 5 (3) 4 (4) 3

22. If 14th February, 2011 falls on Monday then what will be the day on 14 th February, 2013 ?

(1) Friday (2) Tuesday (3) Wednesday (4) Thursday

23. My brother is 562 days older to me while my sister is 75 weeks older to him. If my sister was born on Tuesday,
on which day was I born :

(1) Sunday (2) Monday (3) Thursday (4) Tuesday

24. Karan was born on Saturday 22nd March 1982. On what day of the week was he 14 years 7 months and
8 days of age ?

(1) Sunday (2) Tuesday (3) Wednesday (4) Monday

25. On what dates of August 2020 will Monday fall

(1) 3, 10, 17, 24, 31 (2) 5, 12, 19, 26

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(3) 1, 8, 15, 22, 29 (4) None of these

ANSWERS
Q ue . 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
A ns . 4 3 3 1 3 4 2 3 3 1 2 3 2 2 3 3 3 1 2 1
Q ue . 21 22 23 24 25
A ns . 2 4 3 4 1

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 Mental Ability

EXERCISE
QUESTIONS RELATED TO VARIOUS OLYMPIADS
1. If 3rd January,2004 is was Monday , what will be the day on 5th april , 2004?
(1) Wednesday (2) Monday (3) Tuesday (4) Sunday
2. If 14th July of 1995 was Friday, then what was the day on 30th March of 1994?
(1) Monday (2) Sunday (3) Tuesday (4) Wednesday
3. If 4 days before it was Monday , what day will it fall after 3 days?
(1) Sunday (2) Monday (3) Tuesday (4) Wednesday
4. If 4th day of any month was Sunday , what will be the day on 27th day of the same month?
(1) Monday (2) Tuesday (3) Wednesday (4) Saturday
5. If your birthday 30th june,2003 falls on Monday on what day of the week does your birthday fall in the
year 2005?
(1) Sunday (2) Tuesday (3) Wednesday (4) Thursday
6. On what day of the week India will celebrate its Republic day on 26th January,2015?
(1) Sunday (2) Tuesday (3) Wednesday (4) Monday
7. If 1st October is Sunday ,then 1st November will be?
(1) Monday (2) Tuesday (3) Wednesday (4) Thursday
8. Which two months in a year have the same calendar?
(1) June,October (2) April,November (3) April,June (4) September,December
9. If the first day of a leap year is Monday , then what day will be on the last day of that year?
(1) Wednesday (2) Tuesday (3) Thursday (4) Sunday
10. If 14th September 2013 is Saturday , then what day w ill be 22nd December,2014?
(1) Sunday (2) Monday (3) Tuesday (4) Wednesday
11. 'Environment Day ' falls on Thursday , then on which day 'Childrens Day' will fall in the same year?
(1) Friday (2) Saturday (3) Sunday (4) Monday
12. If 5th day of a month is four days earlier than Wednesday , what day will it be on 28 day of the month?
(1) Sunday (2) Tuesday (3) Wednesday (4) Monday
13. If the 3rd of the month falls on Thursday. What day will be fourth after 22nd of the month?
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(1) Saturday (2) Tuesday (3) Thursday (4) Sunday

14. If in a month , the third day is Monday , then what will be the fourth day from 21st of that month?
(1) Monday (2) Tuesday (3) Wednesday (4) Thursday
15. If the first day of a leap year is Monday , then how many Tuesdays will be in that year?
(1) 53 (2) 52 (3) 54 (4) 55

ANSWERS
Q ue. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
A ns. 1 4 2 2 4 4 3 4 2 2 1 4 1 2 1

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Important Notes

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