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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