# Clocks & Calendar - Vikas Saini

• Calender

How to find the day of any century

In a non leap year 365 days.

365 = 7 x 52 + 1.

It means one odd day in a non leap year.

In a year every single day of week repeats by 52 times at least, whereas 1 odd day will be remain in a year itself.

For a leap year 2 odd days will be remain.

In a century we find 76 non leap years and 24 leap years.

So total no of odd days in a century = 76 x 1 + 24 x 2 = 124 days.

124 mod 7 = 5 days.

In a century there is 5 odd days.

100 years = 5 odd days.

200 years = 5 x 2 mod 7 = 3 odd days

300 years = 5 x 3 mod 7 = 1odd day.

400 years = (5 x 4 +1) mod 7 = 0 odd.

It means after 400 years no odd days.

For month odd days code

January - 0

February - 3

March - 3

April - 6

May - 1

June - 4

July - 6

August - 2

September - 5

October - 0

November - 3

December - 5

And code for century

100 years = +4.

200 years = +2.

300 years = 0.

400 years = -1.

 An awesome formula to find the day if you know the date : Day = ( Date + Month code + year + [year/4]+ century code ) mod 7

Q. what was the day on 15th Aug 1947 ?

Date = 15

Month code = 2

Year = 47

century code = 0(1900 = +300)

= (15 + 2 + 47 + [47/4] + 0 ) mod 7

= (15 + 2 + 47 + 11) mod 7

= 75 mod 7

= 5.

5th day of week is Friday.

Q. What was the day 26th January 1950.

Day = 26

Month code = 0

Year = 50

century code = 0.

Day = (26+0+50+[50/4]+0) mod 7

= (26+50+12+0) mod 7

= 88 mod 7

= 4.

4th day of week = Thursday.

Q. What will be the day on 30th September 2016.

Date = 30.

Month code = 5.

Year = 16.

Century code = -1 ( +400)

Day = (30 + 5 + 16 + [16/4] -1) mod 7

= (30+5+16+4-1) mod 7

= 5

5th day of week is Friday.

Q. What was the day on 12 December 1729.

Date = 12.

Month code = 5

Year = 29.

Century code = +4 (+100 year)

Day = (12+5+29+[29/4]+4) mod 7

= (12+5+29+7+4) mod 7

= 57 mod 7

= 1

1st day of week is Monday.

Clocks

Finding angle between Hands

12 h = 360°

1 h = 30°

60 m = 30°

1 m = 0.5°

 Angle = | 5.5 M - 30 H | M = minute H = Hour

Q. What is the angle between 4:45 PM ?

solution :-

5 hours = 5 x 30° = 150°

45 min = 45 x 0.5° = 22.5°

Total = 150° - 22.5° = 127.5°

using Shortcut :-

Angle = | 5.5 M - 30 H |

= | 5.5 x 45 - 30 x 4 |

= 127.5 °

Q. What is the angle between at 3:30

solution :-

3 Hours = 30° x 3 = 90°

30 min = 0.5° x 30 = 15°

Total = 90° - 15° = 75°.

Shortcut :-

Angle = | 5.5 M - 30 H |

= | 5.5 x 30 - 30 x 3 |

= 75°

Relative speed of 2 clock hands

12 H = 60 div

1 H = 5 div.

SH = 60 M = 50 div.

SH = 5/60 div/min.

SH = 1/12 div/min.

SM = 60 div / 60 min = 1div/1min

Relative speed = 1 - 1/12 = 11/12 div/min.

T = 60 / (11/12)

T = 65 + (5/11)

T = 1:05:5/11

Both hand will coincide after every 1:05:5/11.

Q. By what time both hands will coincide in between 5-6 PM ?

solution :-

(1:05:5/11) x 5

= 5:25:25/11

= 5:27:3/11

Q. By what time both hand will coincide in between 9 - 10 PM ?

solution :-

(1:05:5/11) x 9

= 9:45:45/11

= 9:49:1/11.

Q. By what time in between 2-3 PM both hand will make angle of 30°.

solution :-

angle = | 5.5 M - 30 H |

30° = | 5.5 x M - 30 x 2 |

30° = | 5.5 M - 60° |

90° = 5.5 M

M = 900 / 55 = 180/11 = 16 (4/11).

Time = 2:16:4/11.

Q. By what time in between 5-6 PM both hands will make angle of 60°.

solution :-

angle = | 5.5 M - 30 H |

60° = | 5.5 M - 30 x 5 |

60° = 5.5 M - 150

210° = 5.5 M

M = 2100/55 = 420/11 = 38(2/11)

Time = 5:38:2/11.

Q. A watch gains 5 sec in every 3 minutes and was set right at 8 AM,so what time will it show at 10 PM on the same day ?

solution :-

3 Min = 5 sec

1 min = 5/3 sec.

60 min = 100 sec.

1 Hour = 100 sec.

14 Hour = 1400 sec.

1400 sec = 23 min 20 seconds.

Time = 10:23:20 sec.

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