Clock

A clock is an instrument having the numbers 1 to 12 or equivalent roman numerals around its face which display time divided into hours, minutes and seconds.

Hour Hand

The smaller or slower hand of a clock is called the hour hand. It takes two revolution in a day.

Minute Hand

The bigger or faster hand of a clock is called the minute hand.
It takes one revolution in every hour.

Second Hand

Second hand bigger or faster then minute hand. It makes one revolution per minute.

Important Points Related to Clock

Angle makes by hour hand and minute hand

Minute Hand:- Minute hand complete one revolution i,e, 360° in 60 minutes.
360° = 60 minutes
180° = 30 minutes
90° = 15 minutes
30° = 5 minutes
6° = 1 minute
∴ 1 minute = 6°
Clearly, minute hand make 6° angle in 1 minute.

Hour hand:- Hour hand complete one revolution i.e, 360° in 12 hours.
12 h = 360°
1 h = 30°
60 min = 30°

∴ 1 min =	1°
	2

Clearly, hour hand make	1°	in 1 minute .
	2

Relative Speed between Minute hand and hour hand = 6 -	1°	= 5	1°	degree per minute.
	2		2

Note :- In every minute, minute hand goes 5¹/₂ degree more than hour hand.

Speed of Hour hand and Minute hand

Speed of hour hand = 5 min/hour
Speed of minute hand = 60 min/hour
Relative speed = 60 - 5 = 55 min/hour
Note- In every hour, minute hand goes 55 min more than hour hand.

Specific features of hour hand and minute hand

When they will make 90° angle

If two hands are at 90° they are 15 min apart.
Its happens twice in 1 hr.
1 h = 2 times
12 h = 22 times
∴ 1 day = 24 h = 44 times

When they will make 180° angle

If two hands are at 180° they are 30 min apart.
direction opposite to each other.
It happens once in 1 hr.

When they will make a straight line

If angle between them 180°, then 30 min apart directly opposite.
If angle between them 0°, then no difference, overlap
It happens once in 1 hr.

Angle between minute hand and hour hand

Let we have to find angle between minute hand and hour hand at 'H hours and M minutes'.

Note :- If sign is negative, then ignore the negative sign.

Ex- Find the angle between minutes hand and hour hand at 4 : 20 A.M.
Solution:- Here, H = 4 and M = 20
According to formula,

Angle between two hands h hours and m minutes =	11	M - 30H
	2

=	11	× 20 - 30 × 4
	2

= 110 - 120 = - 10°
Ignore the negative sign the required angle is 10°

Important Formula :-

(1) Two hands at Together :-

Minute hand and hour hand of a clock will be together between h and ( h + 1 ) O' clock at ( 60/11 )h minutes past h.

Ex- At what time between 2 and 3 O’clock are the two hands of the clock together?
Solution :- Here, h = 2
According to formula ,

Two hands together at =	60	h minutes
	11

=	60	× 2
	11

=	120	= 10	10	minutes.
	11		11

Hence, they will be together at 2 : 10	10	minutes.
	11

(2) To hands at Right angle :-

Minute hand and hour hand of a clock will right angle between h and (h+1) O' clock at ( 5h ± 15 ) × 12/11 minutes past h.

Ex- At what time between 8 and 9 o' clock will the hands of a clock be at right angle?
Solution:- Here, h = 8
According to formula,

Two hands at right angle = ( 5h - 15 ) ×	12
	11

= ( 5 × 8 - 15 ) ×	12	= 25 ×	12
	11		11

Two hands at right angle=	300	= 27	3
	11		11

Hence, they will be right angle at 8 : 27	3	minutes.
	11

Ex- At what time between 4 and 5 o' clock will the hands of a clock be at right angle?
Solution:- Here, h = 4
According to formula,

Two hands at right angle = ( 5h + 15 ) ×	12
	11

= ( 5 × 4 + 15 ) ×	12	= 35 ×	12
	11		11

Two hands at right angle=	420	= 38	2
	11		11

Hence, they will be right angle at 4 : 38	2	minutes.
	11

(3) Two hands at Straight angle :-

Minute hand and hour hand of a clock will straight line at 180° between h and ( h + 1 ) o' clock at
( 5h ± 30 ) × 12/11 minutes past h.

Ex- At what time between 9 and 10 o' clock will the hands of a clock be in the same straight but not together?
Solution:- Here, h = 9
According to formula,

Two hands at straight angle = ( 5h - 30 ) ×	12
	11

= ( 5 × 9 - 30 ) ×	12	= 15 ×	12
	11		11

Two hands at straight angle=	180	= 16	4
	11		11

Hence, they will be straight angle at 9 : 16	4	minutes.
	11

Ex- Find the time between 8 and 9 O’clock when the two hands of a clock are 4 minutes apart.
Solution:- Here, h = 8 and m = 4
According to formula,

( 5h ± m ) ×	12
	11

= ( 5 × 8 ± 4 ) ×	12	= ( 40 ± 4 ) ×	12
	11		11

Taking + ve and - ve sign , we get

= 44 ×	12	and 36 ×	12
	11		11

= 48 and	432
	11

∴ Between 8 and 9 O’clock the two hands of a clock will be 4 minutes

apart at 39	3	and 48 minutes past 8 O’clock
	11

Ex- The minutes hand of a clock overtakes the hour hand at intervals of 70 min of the correct time. how much in a day does the clock gain or loss?
Solution:- Here, m = 60 minutes
According to formula,

	720	- m			( 60 × 24 )
	11				m

	720	- 70			( 60 × 24 )
	11				70

= -	(720 - 770)	×	( 6 × 24 )
	11		7

= -	50	×	( 6 × 24 )
	11		7

=	-7200	minutes
	77

=	7200	minutes loss .
	77

Calendar :-

A calendar is a chart which show the day, week and months of a particular year. A calendar consist of 365 or 366 days divide into 12 months.

Ordinary Year :-

A year in which having 365 days is called an ordinary year. For example- 2011, 2015, 2019 etc.

Leap Year

A year in which having 366 days is called leap year. For example- 2000, 2012, 2016 etc.

If a normal year is divisible by four is called leap year but in the case of century it must be divisible by 400.
For example- 2000, 1600, 2400, are leap year ( divisible by 400 )
1700, 1800, 1900 are not leap year ( not divisible by 400 )

In a century, there are 76 ordinary year and 24 leap years.

Ex- Right now how many century leap year crossed?
Solution:- 400, 800, 1200, 1600, 2000 = 5.

Ex- In 100 years how many leap year are there?

Solution:-	100	- 1 = 25 - 1 = 24 years .
	4

Leap years in century years :-

In 100 years = 24 leap years
In 200 Years = 24 × 2 = 48
In 300 years = 24 × 3 = 72
In 400 years = 24 × 4 = 96 + 1 = 97
In 500 years = 24 × 5 = 120 + 1 = 121

Ex- In 400 years how many times we will get the date of 29th?
Solution:- In ordinary year 29th comes 11 times
11 × 400 = 4400
In 400 years 97 leap years
∴ Total 29th in 400 years = 4400 + 97 = 4497

Ex- In 400 years how many times we will get the date of 29th feb?
Solution:- We know that, In 400 years 97 leap years
∴ In 400 years 97 29th feb get.

Ex- In between two consecutive leap year how many normal year?Solution:- Consider two consecutive leap year 1984 and 1988.
1984, 85, 86, 87, 1988 Means 3 normal year.
Now consider two consecutive leap year 1896 and 1904.
1896, 97,98,99, 1900, 1901,02, 03, 1904 Means 7 normal leap year.
∴ Between two consecutive leap year 3 and 7 normal year.

Odd days :-

When we divide the number of days by 7, if remainder left that remainder is called odd days.
For example- How many odd days in 1 normal year.
In normal year 365 days,

odd days =	365	= 1 odd days
	7

1 leap year = 366 days,

odd days =	366	= 2 odd days
	7

Ex- In 100 years, how many odd days are there?
Solution:- We know that
In 100 years 76 normal year = 76 × 1 = 76
In 100 years 24 leap year = 24 × 2 = 48
Odd days = 76 + 48

odd days =	124	= 5 odd days
	7

Fast Trick to Find number of odd days in given year :-

Ex- In 50 years, how many odd days are there?
Solution:- In 50 years 12 leap years

odd days =	50	+	12	= 1 + 5 = 6 odd days
	7		7

Ex- In 17 years, how many odd days ?
Solution:- In 17 years 4 leap year

odd days =	17	+ 4 = 3 + 4 =	7	= 0 odd days.
	7		7

Number of odd days in century years :-

We know that,
In 100 years = 5 odd days

In 200 years = 2 × 5 = 10 =	10	= 3 odd days
	7

In 300 years = 3 × 5 = 15 =	15	= 1 odd days
	7

In 400 years = 4 × 5 = 20 + 1 =	21	= 0 odd days
	7

Note :- As 400th is a leap year, therefor 1 more day has been taken
5     3     1     0
100 200 300 400
500 600 700 800
900 1000 1100 1200
1300 1400 1500 1600
Number of odd days in 100, 400, 900, 1300 = 5
Number of odd days in 200, 600, 1000, 1400 = 3
Number of odd days in 300, 700, 1100, 1500 = 1
Number of odd days in 400, 800, 1200, 1600 = 0

Ex- Find odd days in 2015.
Solution:- 2015 is a ordinary year.
We know that, in 1 ordinary year 1 odd day.
∴ In 2015 = 1 odd day

Ex- Find odd days till 2015.
Solution:- 2015 = 2000 + 15
In 2000 years, odd days = 0

In 15 years, 3 leap year =	15	+ 3 = 1 + 3 = 4
	7

∴ Odd days till 2015 = 0 + 4 = 4 odd days.

Month Code :-

Months	Code
January	1
February	4
March	4
April	0
May	2
June	5
July	0
August	3
September	6
October	1
November	4
December	6

Day's Code :-

Days	Code
Monday	1
Tuesday	2
Wednesday	3
Thursday	4
Friday	5
Saturday	6
Sunday	7/0

To Find a Particular Day Without Given Date and Day :-

Year between 1900 - 1999 :-

1.If the given year is Normal year

Step -1 - Consider last 2 digits of the given year

Step-2 :-	Last 2 digits	= Quotient
	4

Step-3 - Month code
Step-4 - Date
Step-5 - Add step 1 + 2 + 3 + 4

Step-6 :-	Step-5	= Remainder will be Day's Code
	7

Ex- Find the day of the week on 15 August 1947?
Solution:- Step-1 :- 47 = 47

Step-2 :-	47	= 11
	4

Step-3 ; - Month code of August = 3
Step-4 :- Date = 15
Step-5 ;- Sum = 47 + 11 + 3 + 15 = 76
Step-6 ;- 76/7 = 6 Friday
∴ It was Friday on 15th August 1947.

2. If the given year is Leap year

In the case of	Jan	- ( -1 day ) Rest of the months i.e, March onward same as normal year.
	Feb

Ex- Find the day of the week on 7 January 1904?
Solution:- Step-1- 04

Step-2 :-	04	= 1
	4

Step-3- Month code of January = 1
Step-4- Date = 7
Step-5- Sum = 4 + 1 + 1 + 7 = 13

Step-6 :-	13	= 6
	7

Step-7- 6 - 1 = 5 Thursday
∴ It was Thursday on 7th January 1904.

Year between 2000-2099

1.If the given year is Normal year
Step-1 - Consider last 2 digits of the given year

Step-2 :-	Last 2 digits	= Quotient
	4

Step-3 - Month code
Step-4 - Date
Step-5 - Add step 1 + 2 + 3 + 4

Step-6 :-	Step-5	= Remainder
	7

Step-7- Remainder - 1 = will be Day's Code

Ex-Find the day of the week on 20 January 2018?
Solution:- Step-1- 18

Step-2 :-	18	= 4
	4

Step-3- Month code of January = 1
Step-4- Date = 20
Step-5- Sum = 18 + 4 + 1 + 20 = 43

Step-6 :-	43	= 1
	7

Step-7- 1 - 1 = 0 Saturday
∴ It was Saturday on 20 January 2018.

2.If the given year is Leap year In the case of	Jan	- ( -2 days ) .
	Feb

Rest of the months i.e, March onward same as normal year.

Ex- Find the day of the week on 8 February 2008?
Solution:- Step-1- 08

Step-2 :-	08	= 2
	4

Step-3- Month code of January = 4
Step-4- Date = 8
Step-5- Sum = 08 + 2 + 4 + 8 = 22

Step-6 :-	22	= 1 = 8
	7

Step-7- 8 - 2 = 6 Friday
∴ It was Friday on 08 February 2008.

Year between 1800-1899

1.If the given year is Normal year

Step-1 - Consider last 2 digits of the given year

Step-2 :-	Last 2 digits	= Quotient
	4

Step-3 - Month code
Step-4 - Date
Step-5 - Add step 1 + 2 + 3 + 4

Step-6 :-	Step-5	= Remainder
	7

Step-7- Remainder + 2 = will be Day's Code

Ex- Find the day of the week on 7 October 1807?
Solution:- Step-1- 07

Step-2 :-	07	= 1
	4

Step-3- Month code of January = 1
Step-4- Date = 7
Step-5- Sum = 07 + 1 + 1 + 7 = 16

Step-6 :-	16	= 2
	7

Step-7- 2 + 2 = 4 Wednesday
∴ It was Wednesday on 07 October 1807.

2.If the given year is Leap year In the case of	Jan	- ( + 1 day ) .
	Feb

Rest of the months i.e, March onward same as normal year.

Ex- Find the day of the week on 8 July 1806?
Solution:- Step-1- 06

Step-2 :-	06	= 2
	4

Step-3- Month code of July = 0
Step-4- Date = 8
Step-5- Sum = 06 + 2 + 0 + 8 = 16

Step-6 :-	16	= 2
	7

Step-7- 2 + 1 = 3 Tuesday
∴ It was Tuesday on 08 July 1806.

Day Gain/Loss :-

Ordinary Year ( ± 1 day )

1.When we proceed forward by 1 yr, then 1 day is gained.
For example- If 11th August 2013 is Sunday, then 11th August 2014 has to be Sunday + 1 = Monday.

2. When we move backward by 1 yr, then 1 day is lost.
For example- If 27th December 2013 is Friday, then 24th December 2012 has to be Friday - 1 = Thursday.

Leap Year ( ± 2 day )

1.When we proceed forward by 1 leap year, then 2 days are gained.
For example- If 29th December 2011 is Sunday, then 29th December 2012 has to be Sunday + 2 days = Tuesday.

2. When we move backward by 1 leap year, then 2 days are lost.
For example- If 22nd December 2012 is Sunday, then 22nd December 2011 has to be Sunday - 2 days = Friday.

Special Case

2 days after Monday = + 2 = Wednesday
3 days after Monday = + 3 = Thursday
2 days before Monday = -2 = Saturday
3 days before Monday = -3 = Friday