Calendar maths problems

Calendar maths problems
Calendar's Problems for Competitive Exam
Calendar’s questions and answers                                 Part-I
Calendar Maths Problems

Syllabus:
1.   Normal Years/Ordinary Years
2.   Leap Years
3.   Odd day/Extra day
4.   Odd days of every month
5.   Normal Century Year
6.   Leap Century Year
7.   Odd day for normal Century Year
8.   Odd day for Leap Century Year
9.   How to find the day when dating different month and Year same?
10. How to find a day when dating and month different year same?
11.  How to find a day when dating and month same but year different?
12.  Reverse questions.
13.  When will repeat same year/ same calendar?
14.  How to find a day on a fixed date?



TRICKS:

What is Normal or Ordinary Year?

# A year which has total 365 days called Normal year. In a normal year, there is February Month 28 days. Another characteristic is it will never complete divisible by 4. A normal year consists of  52 weeks +1 extra day.
Eg. 1917,1943,2013,2001,2003,2002 etc are normal year.Let us checking 1917 is a leap year or not:

Last, of all, we get remainder 01, which is not equal to zero. Since the remainder is not zero, 1917 is not complete divisible by 4.Hence 1917 is not a Leap year.

What is Leap Year?

 A year which is completely divisible by 4 or 400 is called Leap year. Leap Year consist of 52 weeks +2 extra days.

Eg. 2000,2004,2008,2012 etc.Let us checking 2012 is a leap year or not:

Last, of all, we get the remainder 0,  Since the remainder is zero, 2012 is completely divisible by 4.Hence 2012 is a Leap year. What is Days?# There are total 7 days in a weak, that is Monday, Tuesday, Wednesday, Thursday, Friday and Sunday. Hence in the calendar's problems, total days always be 7. 

What is ODD Day /Extra Day ?: 

In a given period, the number of days more than the number of the complete week is called ODD day. Here, complete weak means complete 7days. We must divide by 7 to the given period ( total numbers of days ) and the remainder will be the odd day. Extra days always lies between 0 to 6. If the total days are multiples of 7 then extra days is 0.eg: If the given period is 7,15,23,778 days then the odd day will be respectively 0,1,2,1. 
Table-IThe trick for Normal & Leap Year:-      
Type of YearNos of DysFebruaryOdd Day
Leap Year366 days292
Normal Year365 days281

If we divide 366 by 7 then the remainder we get 2. So, odd day or an extra day of a leap year is 2.If we divide 365 by 7 then the remainder we get 1 So, odd day or an extra day of a normal year is1.Table-II

ODD Day in month wise:-













MonthTotal DayOdd days
January31 (÷7)3 (Remainder)
FebruaryNY=28/LY=290/1
March313
April302
May313
June302
July313
August313
September302
October313
November302
December313





.









NY= Normal year, LY= Leap year.



What is Century Year?


# The years which are completely divisible by 100 is called Century year.
Eg:1400,1700,1600,1900,2000....etc are century year.

What is Leap Century Year?

# The century years which are divisible by only 400 is called Leap Century year.
Eg. 400,800,1200,1600,2000,2800 .....etc are Leap century year.

What is Normal / Ordinary Century Year?

# The century years which are not divisible by 400 is called Ordinary Century year.
Eg. 1400,1800,1300,1700,2100,2300 .....etc are Ordinary century year.

Note*: To solve most of the questions asking in the exam from the tropic calendar 100% depends on the extra day or odd day. So, we must have knowledge about extra day 100%.
Q1. Find the Extra day of the following periods-
14days, 77days, 100days ,365days .
Sol: 14÷7,77÷7, 100÷7, 365÷7 gives the following remainders as
 0, 0, 2 and 1 respectively.




Hence, extra days of 14days is 0, 77days is 0,100days is 2 and 365days is 1.


Question on`'Day after, Day after tomorrow, Yesterday, Day before Yesterday'.

Q1.If the day before yesterday was Thursday when will be Sunday?
a.Tomorrow
b.The day after Tomorrow
c.Today
d.Two days after today
Sol: a. Tomorrow 
Trick: DBY=Day before Yesterday
                Y= Yesterday
             TD= Today 
             T  = Tomorrow
First, write down DBY then Y, TD, T serially and then count Thursday to T.  T is Sunday. Hence Sunday will be Tomorrow.


  











Q2.If the day before yesterday was Friday. What will be the third day after the day after tomorrow?
a.Thursday
b.Friday
c.Saturday
d.Sunday

Sol: b. Friday
Trick:
First, write down DBY then Y, TD, T, DAT, 1, 2, 3  serially and then count from Thursday to 3.  3 is Friday. Hence the third day after the day after tomorrow is Friday











Q.If the day after tomorrow is Monday, what day was tomorrow’s day before yesterday?
a.Saturday
b.Friday
c. Thursday
d.Wednesday
Sol: c. Thursday
Trick: Here Tomorrow =Saturday and Tomorrow's day before yesterday is Thursday.

Q.Today is Sunday. After 92 days it will be-
a. Wednesday
b.Saturday
c.Monday
d.Friday

Sol: c. Monday
We know, after every 7 days the same day is repeated. So, we must divide 92 by 7 and have to take
the remainder.


Here, after 13 complete weeks again we will get Sunday and 1 more day. So, Sunday+1=Monday.


Q.Today is Monday. After 61 days it will be
a. Wednesday
b.Saturday
c.Monday
d.Friday

Sol: b.Saturday
We know, after every 7 days the same day is repeated. So, we must divide 61 by 7 and have to take
the remainder.



Here, after 8 complete weeks again we will get Monday and 5 more days. So, Monday+5=
Saturday



QIf 6 March 2018 is Saturday. Which day of the week will be on 22 March
2018?
a.Saturday
b.Friday
c.Monday
d.Wednesday
Sol: c. Monday
TRICK: Since within month (only date different month and years same), we must count the total day
after 6, that is 7 March to 22 March and then divide by7. Take remainder as Odd /Extra day.then add
the extra day with the given day. That is, 22-6=16/7, it will go 2 times and the remainder will be
2. So, given day +2=saturday+2=Monday.


Q If 2 January  2019 is Wednesday. Which day of the week will be on 22
January 2019?
a.Saturday
b.Friday
c. Monday
d.Wednesday
Sol: c. Monday
TRICKS: Since within month ( only date different month and year same), we must count the total day after 2, that is from 3 January to 22 January and then divide by 7. Take remainder as ODD /Extra day. Then add an extra day with the given day. That is, 22-2=20/7, and it will go 2 times and the remainder will be 6. Hence given day +6= Wednesday+6= Monday. 
QIf 31 March 2008 is Wednesday. Which day of the week will be on 12 July 2008?
a.Saturday
b.Friday
c.Monday
d.Wednesday
Sol: c. Monday
TRICK: Always mind its day will be count after the given date(31 March).So after 31 March, we get complete April, May, June, and July up to 12 days.
The extra day of April =2
The extra day of May=3
The extra day of June =2
The extra day of July =5 (July month has only 12 days so in 12÷7, remainder 5)
Total extra days=12 ( Which is greater than 7, so again must divide by 7 to get a total extra day)
Therefore, 12÷7 provide remainder 5. Hence 5 will be the final extra day of whole period)
Now given day+5=Wednesday+5=Monday.
NOTE: If we get an extra day ≥ (greater than equal to ) 7. we must divide it again by 7 and remainder will be a final extra day.
QIf 1 January 2024 is a Friday. Which day of the week will be on 5 April 2024?
a.Saturday
b.Friday
c.Monday
d.Tuesday
Sol: d.Tuesday
TRICK: Always check that the given year is Leap Year or normal Year. If Leap Year and February encluded then the extra/ odd day of that Feb. month will be 1.If Normal Year and February encluded then the extra/ odd day of that Feb. month will be 0. ( As per table no-II).
Here, the day will be count after the given date(1 January). So after 1 January, we get 30 days for January month and complete Feb., Mar., and April up to 5 days.
The extra day of Jan =2 (If 30÷7, the remainder will be 2)
The extra day of Feb.=1(2024 is Leap Year and so Feb's extra day is 1) SeeTable-II
The extra day of Mar =3
The extra day of April =5 (April month has only 5 days so the extra day is 5, less than 7)
Total extra days=11 ( Which is greater than 7, so again must divide by 7 to get a total extra day)
Therefore, 11÷7 provide the remainder 4. Hence 4will be the final extra day of whole period)
Now given day+4=Friday+4=Tuesday.


Q If 10 January  2013 is Thursday. Which day of the week will be on 10
January 2018?
a.Saturday
b.Friday
c.Monday
d.Wednesday
Sol:d.Wednesday
Important Note: Here date and month same but Year are different. For such type of
question, write down serially all year, Extra/Odd day of each year with a normal year and Leap
Year justification as shown below,  

Here,
From 10 Jan. 2013 to 10 Jan.2014 we get only one Feb. of 2013 and 2013 is NY, hence there are total365 days except for 10 Jan. 2013. So odd day or an extra day of 365 is 1.
From 10 Jan. 2014 to 10 Jan.2015 we get only one Feb. of 2014 and 2014 is NY, hence there are total365 days except for 10Jan. 2014. So odd day or an extra day of 365 is 1.
From 10 Jan. 2015 to 10 Jan.2016 we get only one Feb. of 2015 and 2015 is NY, hence there are total 365 days except for 10Jan. 2015. So odd day or an extra day of 365 is 1.
From 10 Jan. 2016 to 10 Jan.2017 we get only one Feb. of 2016 and 2016 is LY, hence there are total 366 days except 10 Jan .2016. So odd day or an extra day of 366 is 2.
From 10 Jan. 2017 to 10 Jan.2018 we get only one Feb. of 2017 and 2017 is NY, hence there are total 365 days except 10 Jan .2017. So odd day or an extra day of 365 is 1.
Now, 10 Jan.2013+6=Thursday+6=Wednesday

Q If 3 March  2013 is Sunday. Which day of the week will be on 3 March
2016?
a.Saturday
b.Friday
c.Monday
d.Thursday
Sol:d.Thursday
IMPORTANT NOTE: Here date and month same but Year are different.F or such
type of question, write down serially all year, Extra/Odd day of each year with normal year
and Leap Year justification as shown below,
 
Here,
From 03 March. 2013 to 03 March.2014 we get only one Feb. of 2014 and 2014 is NY, hence there are total 365 days except for 3 March. 2013. So odd day or an extra day of 365 is 1.
From 03 March. 2014 to 03 March.2015 we get only one Feb. of 2015 and 2015 is NY, hence there are total365 days except for 3 March. 2014. So odd day or an extra day of 365 is 1.
From 03 March. 2015 to 3 March.2016 we get only one Feb. of 2016 and 2016 is LY, hence there are total366 days except for 03 March 2015. So odd day or an extra day of 366 is 2.
Now, Sunday+4=Thursday.
Q If 20 July  1989 is Thursday. Which day of the week will be on 20
December  1995?
a.Saturday
b.Friday
c.Monday
d.Wednesday
Sol:d.Wednesday
Important Note:
      Here date same but Month and Year are different. For such type of question, write down serially all year up to the same date and month. Remaining duration calculation will be in the second part. So 1st we must find odd days from 20July 1989 to 20 July 1995 then 20 July to 20 December 1995 Extra/Odd day of each year with normal year and Leap Year justification as shown below,
  

Now, we will find out an extra day of 20 July 1995 to 20 December 1995
TRICK: Always mind its day will be count after the date 20 July (because we have already counted up to 20 July 1995).So after 20 July, we get 11 days for July and complete Aug, Sept,
Oct, Nov., and December up to20 days.
The extra day of July=11÷7, remainder=4 (extra day)
The extra day of Aug=3
The extra day of Sept. =2
The extra day of Oct. =3
The extra day of Nov. =2
The extra day of Dec. =20÷7, remainder=6 (extra day)
Total extra day= 20 ………………( 2)
Now (1)+(2)=7+20=27÷7, remainder=6 ( Final extra day)
Now, 20 July 1389+6=Thursday+6=Wednesday





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