Problems on Calendars - Concept Part 1 - Aptitude Test Preparation - Definition of odd days , Leap year and Odd Days in a Century

Introduction:

Today is July 4th, 2014, and you are asked to find the day of the week for this date. This type of information is sometimes asked in competitive examinations like CAT, Staff Selection Commission exams, Railway Exams, etc. The process of finding it involves calculating the number of odd days. Therefore, we should be very familiar with the concept of odd days.

 

Odd days:

What are odd days?

The number of extra days beyond the complete weeks in a given period is called the number of odd days in that period.

For example, 87 days = 12 weeks + 3 days.

These 3 extra days are the odd days in 87 days.

Days in a Week: There are 7 days in a week, so when you divide any number of days by 7, the remainder will give you the number of odd days.

The number of odd days in a given period is the remainder obtained by dividing the given period by 7.

Examples :  How many odd days are there in 200 days?

Solution :   7)200(28

                     196

                        4

           The remainder when 200 days is divided by 7 is 4.

So, there are 4 odd days in 200 days.

What is a leap year and an Ordinary Year?

Any year that is exactly divisible by 4 is a leap year.

For example, 2016, 1924, and 1080 are leap years because they are exactly divisible by 4.

Century years: A century year is a year that ends in "00" (two zeroes).

For a century year to be a leap year, it must be exactly divisible by 400.

Also, every 4th century is a leap year. For example, 400, 800, 1200, 1600, and 2000 are all leap years because they are divisible by 400.

 The other centuries, although divisible by 4, are not leap years. For example, 1700, 1800, 1900, etc., are not leap years even though they are divisible by 4 because they are not divisible by 400.

The number of odd days in a leap year:

A leap year has 366 days.

When 366 is divided by 7, the remainder obtained is 2. Therefore, a leap year has 2 odd days.

Therefore, a leap year has 2 odd days.

 The number of odd days in an ordinary year:

An ordinary year has 365 days.

When 365 is divided by 7, the quotient is 52 and the remainder is 1. Therefore, an ordinary year has 1 odd day.

 Number of odd days in a century ( 100 Consecutive years ):

 The number of odd days in a century is calculated as follows:

A century comprises 100 years.

100 years when divided by 4 gives the quotient 25.

But the 100th year is not divisible by 400, so it's not a leap year. Hence, we have only 24 leap years.

=>100 years = 24 leap years  and 76 ordinary years.

The total number of odd days is calculated as:

Odd days from leap years: 24 leap years × (2 odd days/leap year) = 48 odd days.

Odd days from ordinary years: 76 ordinary years × (1 odd day/ordinary year) = 76 odd days.

Total odd days: 48 + 76 = 124 odd days.

However, the number of odd days cannot exceed 6.

Therefore, 124 odd days % 7 = 5 odd days.

Therefore, a century has 5 odd days.

 

The Number of odd days in 200 consecutive years :

200 years contain 2 centuries.

Each century has 5 odd days =.2 x 5 = 10 i.e, 3 odd days

 

The Number of odd days in 300 consecutive years:

300 years contains 3 centuries.

Each period of 100 years has 5 odd days =>5 x 3 = 15 i.e, 1 odd days

 

The number of odd days in 400 consecutive years :

400 years contain 4 centuries.

Each century has 5 odd days. Therefore, 4 x 5 = 20 odd days.

But the 400th year is a multiple of 400, which is a leap year, so 1 extra day is added.

20 + 1 = 21 odd days, which equals 0 odd days.

Similarly, 800, 1200, 1600, etc., contain zero odd days or no odd days.