Clock

Clock

Concepts and formulas on clock:-

(1) Minute Spaces:-

        The dial of watch is a circle whose circumference is divided into 60 equal parts.

        The space between two parts is called minute spaces.

(2) Hour Hand and Minute Hand:-

        A clock has two hands the smaller one is called the hour hand  while

        the larger one is called minute hand.

(3) Angle made by hour hand in 12 hrs = 360°.

(4) Angle made by minute hand in 60 min. = 360°.

(5) The hands of a clock coincide 22 times in a day.

(6) In 60 minutes, the minute hand gains 55 minutes on the hour on the hour hand.

(7) Important formula:-

     Angle between minute hand and hour hand

                                          𝛳 = ｜{30 H - 5.5 M}⁰｜ ,where H is Complete hour.M is minute.
                                                    ｜｜means Absolute value i.e.always take positive value.

Example 1 :-Find the angle between minute hand and hour hand of a clock when the
               time is 7:20.

Explanation:-     𝛳 = ｜{30 H - 5.5 M}⁰ ｜                                 

                           𝛳 =｜ {30*7 - 5.5*20}⁰｜ = {210 - 110}⁰ = 100⁰ 

Example 2 :-At what time between 2 and 3 O'clock will the hands of a clock together.

Explanation:-     𝛳 = ｜ {30 H - 5.5 M}⁰ ｜                                 

                           0 = ｜{30*2 - 5.5M}⁰｜ = {60 - 5.5M}⁰ 

                           5.5M = 60

                                M = 120/11 = 10 and 10/11
                 Hence 10 and 10/11 minute past 2 O'clock.

Example 3 :-At what time between 8 and 9 O'clock will the hands of a clock  be 
                  in the same straight line  but not together.

Explanation:-     𝛳 = ｜{30 H - 5.5 M}⁰ ｜       
                                 𝛳 = 180⁰  , H = complete hour = 8
                                 180  = ｜{30 *8 - 5.5 M}⁰ ｜
                                 180 = 240 - 5.5 M
                                 5.5M = 240-180
                                 5.5M = 60
                                  M = 60/5.5 = 10 and 10/11.
                              Hence 10 and 10/11 minute past 8 O'clock.

Example 4 :-A clock is set right at 5 a.m.The clock loses 16 minutes in 24 hours.
                  What  will be the true time when the clock indicates 10 p.m.on 4^th day.

Explanation:-    Time from  5 a.m.on a day to 10 p.m.on 4^th day = 89 hours.
                          Now 24 hours correct clock = 23 hours 44 minutes of this clock
                           (sine clock loses 16 minutes in 24 hours.)
                           24 hours correct clock = 23 and 44/60  = 356/15 of this clock.
                           89 hours of this clock  = (24*15/356)*89 hours of correct clock

                                                                = 90 hours of correct clock

                                Hence correct time is 11 p.m..

Example 5 :-A watch which gains uniformly 2 minutes low at noon on Monday
                 and is  4 min.48 seconds fast at 2 p.m. on the following Monday.
                 When was is correct.              

Explanation:-     Time from 12 p.m.on Monday to 2 p.m. on the following Monday
                                                                   . = 7 days 2 hours.
                                                                                   = 170 hours
                                 The watch gains (2 + 4 and 48/60) = (2+4 and 4/5)
                                                                                              = 34/5 minutes in 170 hours
                                 2 minutes are gained in (170*5/34)*2 = 50 hrs
                              Hence watch was correct 2 days 2 hrs after 12 p.m. on Monday.