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Examples 1 to 4 of Clocks

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(1) Find the angle between the hour hand and the minute hand of  a clock when the time is 3 : 30.

Soln. Hour hand covers a distance  of  3 hrs. and  30 minutes  =  210 minutes and Minute hand covers a distance of  30 minutes.

Hour Hand

Minutes           then              Angle

60  minutes        \LARGE \longrightarrow            30\degree

210 minutes    \LARGE \longrightarrow                 (?)

= \frac{210\times 30}{60}= 105\degree

Minute Hand

Minutes           then              Angle

60  minutes        \LARGE \longrightarrow          360\degree

210 minutes    \LARGE \longrightarrow                 (?)

= \frac{300\times 360}{60}= 180\degree

∴  Required angle  =  1800  –  1050  =   850

 (2) At what time between 5 and 6 are the hands of  a clock 8 minutes apart?

Soln. Let it is 5 : 00 o’clock.

∴   Imaginary duration  =  5 : 17             or       Imaginary duration  =  5 : 33

Imaginary Duration     then      Real Duration

55  minutes               \LARGE \longrightarrow                   60  minutes

17 minutes               \LARGE \longrightarrow                   (?)

= \frac{60\times 17}{55}= \frac{204}{11} \textup{ min.} = 18\ \frac{6} {11}\textup{ min.}

Imaginary Duration     then      Real Duration

55  minutes               \LARGE \longrightarrow                   60  minutes

33 minutes               \LARGE \longrightarrow                   (?)

= \frac{60\times 33}{55}= 36\textup{ min.}

Hence, the hands are 8 minute apart at 5 hours 18 \frac{6}{11}  min. and  5 hours 36  min.

 (3) At what time between 3 : 30  and  4 : 00 p.m.  will both the hands of  a clock be at right angles ?

Soln. Let it is 3 : 00 p.m.

∴   Imaginary duration  =  3 : 30

Imaginary Duration     then      Real Duration

55  minutes               \LARGE \longrightarrow                   60  minutes

30 minutes               \LARGE \longrightarrow                   (?)

= \frac{30\times 60}{55}= \frac{360}{11}\textup{ min.}

∴  Real  Duration   =  \frac{360}{11}  =   32\frac{8}{11}   minutes.

Hence, both the hands of  a clock be at right angles at  32\frac{8}{11}  minutes past 3 p.m.

 (4) A clock is set right at 1 p.m. If it gains 2 minute an hour, what is the true time when the clock indicates 7 p.m. the same day ?

Soln. A gain  of  2 minutes an hour means the  incorrect clock covers 62 minutes when the correct clock covers 60 minutes.

Here, incorrect clock covers 6 hrs. ( 7 – 1 = 6 ).

Imaginary Duration     then      Real Duration

62  minutes               \LARGE \longrightarrow                   60  minutes

360 minutes               \LARGE \longrightarrow                   (?)

= \frac{360\times 60}{62}= 348 \ \frac{12}{31}\textup{ min.}

∴  Correct  time   =  1 : 00 p.m.  +  348 \frac{12}{31} min   = 48 \frac{12}{31}   minutes past to 6 p.m

 

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