The chapter on Clocks mostly pertains with the
angles created by the minute and hour hand. It is thus, important to understand
some of the following rudiments before delving into the ways to determine the
angle of the two hands at a particular time or the time at a particular angle –

1. The circle of a clock represents 360˚ or 60
minutes. The space of every minute is hence, equivalent to 6˚. Also the space
of

**5 minutes or 1 hour is (5 x 6˚) = 30˚**.
2. In an hour’s time the minute hand completes one
full revolution and covers a distance of 60 minutes or 360˚; which means it
covers

**6˚ per minute**. During the space of an hour, the hour hand covers a distance of 5 minutes or 30˚; which means it covers**½˚ per minute**.
The speed ratio of the minute hand and hour hand
with respect to the angle covered by them is 360:30 or

**12:1**.
3. An angle of 0˚ between the two hands is formed
when they cross each other. This phenomenon occurs once every hour except 11 –
1 (during this 2 hour interval, the two hands of the clock meet only once at
exactly 12). Hence in a span of

**12 hours**the**two hands meet**exactly**11 times**or they meet 22 times in a day (i.e. 24 hours).**12 hours**the

**two hands form an angle of 180˚**exactly

**11 times**or they are opposite each other 22 times in a day (i.e. 24 hours).

__Angle between the two Hands:__

The angle between the two hands of the clock at any
point of time can be found by comparing their angles with 12.

**As any hour begins with the Minute hand at 12 (i.e. 0˚ from 12), the angle between the minute hand and 12 at any time is the product of minutes passed from 12 in that hour and 6˚. Eg. At 4:23 the angle of minute hand with 12 will be (23 x 6˚) = 138˚ whereas at 4:12 the angle will be (12 x 6˚) = 72˚**

*Angle of Minute hand with 12:*
The reason for this, as mentioned above is distance
of 1 minute = 6˚. Hence if “m” number of minutes have passed in an hour, the
angle between 12 and Minute hand will be

**6m.**

*Angle of Hour hand with 12:***The distance of Hour hand from 12 is the sum of distance at the beginning of the hour and the distance covered by it in the minutes passed in that hour.**

Eg. At
4:23 the initial angle between 12 and Hour hand at 4 was (20 x 6˚) = 120˚ as
position of 4 is 20 minutes away from 12 and each minute is equivalent to 6˚.
Over and above this 120˚, further 23 minutes have passed by at 4:23 during
which the Hour hand must have further covered a distance of (23 x ½˚) = 11½˚.
Therefore angle between Hour hand and 12 at 4:23 is (120˚ + 11½˚) = 131½˚.

Similarly at 4:12 the angle would be 120˚ + (12 x ½˚) = 126˚

**30h + m/2.**

*The angle between the two hands at any time will be the difference between the angles created by the two hands with 12. In the above eg. at 4:23 pm the angle between the two hands will be (138˚ - 131½˚) = 6½˚ and at 4:12 the angle between the two hands will be (126˚ - 72˚) = 54˚***Angle between Minute & Hour hands:****Combining the above two measures of angles created by the two hands with 12, the angle between the two hands at “h” hours and “m” minutes can be –**

6m – (30h + m/2) i.e.

**11/2m – 30h ………. (i)**

**Or**

**(30h + m/2) – 6m i.e.**

**30h – 11/2m ………. (ii)**

Depending on whether Minute hand is ahead of
Hour hand or the Hour hand is ahead of Minute hand equation

**(i)**and**(ii)**are to be used respectively.*As mentioned above, just like angles 0˚ and 180˚ all other angles also have a similar pattern of forming not more than once in an hour and forming exactly eleven times in twelve hours or 22 times a day. However angles above 180˚ are converted into obtuse angles (< 180˚ | > 90˚) and thus their number of occurrences increase. So while 160˚ is formed 11 times in 12 hours so does 200˚ which is also equivalent (360˚-200˚=160˚). Hence number of times any angle other than 0˚ and 180˚ is formed between the two hands are (11 x 2) = 22 times in 11 hours or (22 x 2) = 44 times in a day.*

**An angle forming more than once in an hour:****Sample Question**

#1. What is the angle between the two hands at 7:25pm?
Does this angle form again before 8pm? If yes, when?

At 7:25 we can visualize that Hour hand will be
ahead of Minute hand, hence we can apply equation

**(ii)**mentioned above –
(30 x 7) – (11/2 x 25) = 72.5˚

Once the Minute hand crosses Hour hand, the maximum
angle that can be attained is at 8pm, which is (20 x 6˚) = 120˚. As 72.5˚ is less
than that, it can form again.

Using equation

**(i)**as described above –
(11/2 x m) – (30 x 7) = 72.5

or m = 282.5 x 2/11 = 51 4/11

The angle of 72.5˚ will also form at 7 hrs 51 4/11
mins or approximately 7:51:21pm

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