## Tuesday, 31 January 2017

### Clocks

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).

4. An angle of 180˚ between the two hands is formed when they are opposite each other. This phenomenon occurs once every hour except 5 – 7 (during this 2 hour interval, the two hands of the clock are opposite each other only once at exactly 6). Hence in a span of 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.

Angle of Minute hand 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˚

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˚

Hence the angle of Hour hand with 12 at “h” hours and “m” minutes will be 30h + m/2.

Angle between Minute & Hour hands: 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˚

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.

An angle forming more than once in an hour: 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.

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