## Clock & Calendar

Odd Days:
We are supposed to find the day of the week on a given date.
For this, we use the concept of 'odd days'.
In a given period, the number of days more than the complete weeks are called odd days.

Leap Year:
(i). Every year divisible by 4 is a leap year, if it is not a century.
(ii). Every 4th century is a leap year and no other century is a leap year.
Note: A leap year has 366 days.
Examples:
Each of the years 1948, 2004, 1676 etc. is a leap year.
Each of the years 400, 800, 1200, 1600, 2000 etc. is a leap year.
None of the years 2001, 2002, 2003, 2005, 1800, 2100 is a leap year.
Ordinary Year:
The year which is not a leap year is called an ordinary years. An ordinary year has 365 days.
Counting of Odd Days:
1 ordinary year = 365 days = (52 weeks + 1 day.)
1 ordinary year has 1 odd day.
1 leap year = 366 days = (52 weeks + 2 days)
1 leap year has 2 odd days.
100 years = 76 ordinary years + 24 leap years
= (76 x 1 + 24 x 2) odd days = 124 odd days.
= (17 weeks + days) 5 odd days.
Number of odd days in 100 years = 5.
Number of odd days in 200 years = (5 x 2) 3 odd days.
Number of odd days in 300 years = (5 x 3) 1 odd day.
Number of odd days in 400 years = (5 x 4 + 1) 0 odd day.
Similarly, each one of 800 years, 1200 years, 1600 years, 2000 years etc. has 0 odd days.
Day of the Week Related to Odd Days:
 No. of days: 0 1 2 3 4 5 6 Day: Sun. Mon. Tues. Wed. Thurs. Fri. Sat.

### CLOCK

Minute Spaces:
The face or dial of watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.
Hour Hand and Minute Hand:
A clock has two hands, the smaller one is called the hour hand or short hand while the larger one is called minute hand or long hand.

In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.
In every hour, both the hands coincide once.
The hands are in the same straight line when they are coincident or opposite to each other.
When the two hands are at right angles, they are 15 minute spaces apart.
When the hands are in opposite directions, they are 30 minute spaces apart.
Angle traced by hour hand in 12 hrs = 360°
Angle traced by minute hand in 60 min. = 360°.
If a watch or a clock indicates 8.15, when the correct time is 8, it is said to be 15 minutes too fast.
On the other hand, if it indicates 7.45, when the correct time is 8, it is said to be 15 minutes too slow.

SOLVED EXAMPLES
1.
It was Sunday on Jan 1, 2006. What was the day of the week Jan 1, 2010?

 Sunday Saturday Friday Wednesday
Explanation:
On 31st December, 2005 it was Saturday.
Number of odd days from the year 2006 to the year 2009 = (1 + 1 + 2 + 1) = 5 days.
On 31st December 2009, it was Thursday.
Thus, on 1st Jan, 2010 it is Friday.
2.
What was the day of the week on 28th May, 2006?

 Thursday Friday Saturday Sunday
Explanation:
28 May, 2006 = (2005 years + Period from 1.1.2006 to 28.5.2006)
Odd days in 1600 years = 0
Odd days in 400 years = 0
5 years = (4 ordinary years + 1 leap year) = (4 x 1 + 1 x 2) 6 odd days
Jan. Feb. March April May
(31 + 28 + 31 + 30 + 28 ) = 148 days
148 days = (21 weeks + 1 day) 1 odd day.
Total number of odd days = (0 + 0 + 6 + 1) = 7 0 odd day.
Given day is Sunday.
3.
What was the day of the week on 17th June, 1998?

 Monday Tuesday Wednesday Thursday
Explanation:
17th June, 1998 = (1997 years + Period from 1.1.1998 to 17.6.1998)
Odd days in 1600 years = 0
Odd days in 300 years = (5 x 3) 1
97 years has 24 leap years + 73 ordinary years.
Number of odd days in 97 years ( 24 x 2 + 73) = 121 = 2 odd days.
Jan. Feb. March April May June
(31 + 28 + 31 + 30 + 31 + 17) = 168 days
168 days = 24 weeks = 0 odd day.
Total number of odd days = (0 + 1 + 2 + 0) = 3.
Given day is Wednesday.

4.
What will be the day of the week 15th August, 2010?

 Sunday Monday Tuesday Friday
Explanation:
15th August, 2010 = (2009 years + Period 1.1.2010 to 15.8.2010)
Odd days in 1600 years = 0
Odd days in 400 years = 0
9 years = (2 leap years + 7 ordinary years) = (2 x 2 + 7 x 1) = 11 odd days 4 odd days.
Jan. Feb. March April May June July Aug.
(31 + 28 + 31 + 30 + 31 + 30 + 31 + 15) = 227 days
227 days = (32 weeks + 3 days) 3 odd days.
Total number of odd days = (0 + 0 + 4 + 3) = 7 0 odd days.
Given day is Sunday.

5.
Today is Monday. After 61 days, it will be:

 Wednesday Saturday Tuesday Thursday
Explanation:
Each day of the week is repeated after 7 days.
So, after 63 days, it will be Monday.
After 61 days, it will be Saturday.

6.
If 6th March, 2005 is Monday, what was the day of the week on 6th March, 2004?

 Sunday Saturday Tuesday Wednesday
Explanation:
The year 2004 is a leap year. So, it has 2 odd days.
But, Feb 2004 not included because we are calculating from March 2004 to March 2005. So it has 1 odd day only.
The day on 6th March, 2005 will be 1 day beyond the day on 6th March, 2004.
Given that, 6th March, 2005 is Monday.
6th March, 2004 is Sunday (1 day before to 6th March, 2005).

7.
On what dates of April, 2001 did Wednesday fall?

 1st, 8th, 15th, 22nd, 29th 2nd, 9th, 16th, 23rd, 30th 3rd, 10th, 17th, 24th 4th, 11th, 18th, 25th
Explanation:
We shall find the day on 1st April, 2001.
1st April, 2001 = (2000 years + Period from 1.1.2001 to 1.4.2001)
Odd days in 1600 years = 0
Odd days in 400 years = 0
Jan. Feb. March April
(31 + 28 + 31 + 1) = 91 days 0 odd days.
Total number of odd days = (0 + 0 + 0) = 0
On 1st April, 2001 it was Sunday.
In April, 2001 Wednesday falls on 4th, 11th, 18th and 25th.

8.
How many days are there in x weeks x days?

 7x2 8x 14x 7
Explanation:
x weeks x days = (7x + x) days = 8x days.

9.
The last day of a century cannot be

 Monday Wednesday Tuesday Friday
Explanation:
100 years contain 5 odd days.
Last day of 1st century is Friday.
200 years contain (5 x 2) 3 odd days.
Last day of 2nd century is Wednesday.
300 years contain (5 x 3) = 15 1 odd day.
Last day of 3rd century is Monday.
400 years contain 0 odd day.
Last day of 4th century is Sunday.
This cycle is repeated.
Last day of a century cannot be Tuesday or Thursday or Saturday.

10.
On 8th Feb, 2005 it was Tuesday. What was the day of the week on 8th Feb, 2004?

 Tuesday Monday Sunday Wednesday
Explanation:
The year 2004 is a leap year. It has 2 odd days.
The day on 8th Feb, 2004 is 2 days before the day on 8th Feb, 2005.
Hence, this day is Sunday.
11.
The calendar for the year 2007 will be the same for the year:

 2014 2016 2017 2018
Explanation:
Count the number of odd days from the year 2007 onwards to get the sum equal to 0 odd day.
Year : 2007 2008 2009 2010 2011 2012 2013 2014 2015 2016 2017
Odd day : 1 2 1 1 1 2 1 1 1 2 1
Sum = 14 odd days 0 odd days.
Calendar for the year 2018 will be the same as for the year 2007.
12.
Which of the following is not a leap year?

 700 800 1200 2000
Explanation:
The century divisible by 400 is a leap year.
The year 700 is not a leap year.

13.
On 8th Dec, 2007 Saturday falls. What day of the week was it on 8th Dec, 2006?

 Sunday Thursday Tuesday Friday

14.
January 1, 2008 is Tuesday. What day of the week lies on Jan 1, 2009?

 Monday Wednesday Thursday Sunday
Explanation:
The year 2008 is a leap year. So, it has 2 odd days.
1st day of the year 2008 is Tuesday (Given)
So, 1st day of the year 2009 is 2 days beyond Tuesday.
Hence, it will be Thursday.

15.
January 1, 2007 was Monday. What day of the week lies on Jan. 1, 2008?

 Monday Tuesday Wednesday Sunday
Explanation:
The year 2007 is an ordinary year. So, it has 1 odd day.
1st day of the year 2007 was Monday.
1st day of the year 2008 will be 1 day beyond Monday.
Hence, it will be Tuesday.
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