☼ GENERAL  CONCEPTS  :  

 ↦ It is 11 : 00  o’clock. In how many minutes both the hands will coincide ?

 Soln. Our wrong answer is 55 minutes. Because we always give answer by fixing ( no movement ) an  hour hand. But the real answer is 60 minutes. Because on every movement of  minute hand there is a corresponding movement of  hour hand, as per the following diagrams.

that is .

Imaginary Duration     then      Real Duration

55  minutes                                  60  minutes

Imaginary  Time   means  Only movement of  minute hand.

[ i.e.   Fix ( or no movement of  ) hour hand at its place ]

And so Imaginary Duration 

means  minutes covered by minute hand during imaginary time.

 ⟼ The face or dial of  a watch is a circle whose circumference is divided into 60 equal parts, called minute spaces.

 ⟼ Angle traced by a minute hand in 1 hr. ( or  60 minutes )  =   360⁰

 ⟼ Angle traced by an hour hand in 12 hrs.  =   360⁰

 ⟼ It is 12 : 00  o’clock. When the hands of  the clock will coincide again ?

 Soln. Imaginary  Duration  =  60 minutes.

Imaginary Duration     then      Real Duration

55  minutes                                  60  minutes

60 minutes                                  (?)

∴  Real  Duration   =   =   65  minutes.

∴ Real  Time  = 12 : 00  o’clock   +   65   min.

=     5   min. past 1.

=      1 : 5    o’clock.

In every   minutes  ( or 65   minutes )

both the hands of  the clock coincident to each other.

In 12 hrs. both the hands of  the clock coincident to each other 11 times.

Coincident          then               No. of  times

  minutes                          1

60 minutes                              (?)

 ⟼ It is 6 : 00  o’clock. When the hands of  the clock will in opposite direction ?

 Soln. Imaginary  Duration  =  60 minutes.

⟼ Imaginary Duration     then      Real Duration

55  minutes                                  60  minutes

60 minutes                                  (?)

∴  Real  Duration   =    =   65  minutes.

∴ Real  Time  =  6 : 00  o’clock   +   65   min.

=  5    min. past  7.

=  7  :  5      o’clock.

In every   minutes  ( or 65   minutes )

both the hands of  the clock opposite to each other.

In 12 hrs. both the hands of  the clock opposite to each other 11 times.

Opposite            then               No. of  times

  minutes                          1

12 60 minutes                     (?)

 ⟼ It is 3 : 00  o’clock. When the hands of  the clock will at right angle ?

 Soln. Imaginary  Duration  =  30 minutes.

⟼ Imaginary Duration     then      Real Duration

55  minutes                                  60  minutes

30 minutes                                  (?)

∴  Real  Duration   =   =   32   minutes.

∴  Real  Time =   3 : 00  o’clock   +   32    min.

=   32    min. past  3.

=    3  :  32      o’clock.

In every    minutes  ( or  32   minutes )

both the hands of  the clock are at right angle.

In 12 hrs. both the hands of  the clock are at right angle  22  times.

Right angle           then               No. of  times

  minutes                          1

12 60 minutes                     (?)

 ⟼ Too Fast  :  If  a clock indicates 10 : 30, when the correct time is 10 : 15, it is said to be 15 minutes too fast.  (  in this case clock gain 15 minutes. )

If  both hands start moving from the same position, they coincide in 65 mins. If   the hands coincide in time less than 65, the clock is too fast.

 ⟼ Too Slow  :  If  a clock indicates 10 : 30, when the correct time is 10 : 45, it is said to be 15 minutes too slow. (  in this case clock loss 15 minutes. )

If  both hands start moving from the same position, they coincide in 65 mins. if the hands coincide in time more than 65  mins, the clock is too slow.

Click here to see Examples