## Clocks and Calendars

### 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.

1 h = 1 times
12 h = 11 times
∴ 1 day = 24 h = 22 times
,

#### 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.

1 h = 1 times
12 h = 11 times
1 day = 24 h = 22 times
,

#### 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'.

 Angle between two hands = 11 M - 30H 2

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.

 = ( 5h - 15 ) × 12 When, h > 6 11

 = ( 5h + 15 ) × 12 When, h < 6 11

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.

 = ( 5h - 30) × 12 When, h > 6 11

 = ( 5h + 30) × 12 When, h < 6 11

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

(4) The hands of a clock are m minutes apart between h and ( h + 1 )
 o' clock at ( 5h ± m ) × 12 minutes past h. 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

(5) If the minute hand of a clock overtakes the hour hand at intervals of m minutes of the correct time,

 then the clock losses or gains 720 - m ( 60 × 24 ) minutes. 11 m

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 :-

 First find number of leap years in given year, then Odd days = Given year + Leap year 7 7

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.

MonthsCode
January1
February4

March
4
April0
May2
June5
July0
August3
September6
October1
November4
December6

DaysCode
Monday1
Tuesday2
Wednesday3
Thursday4
Friday5
Saturday6
Sunday7/0

### 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