**Quantitative Aptitude Clocks Study Material**

A Clock is a device which shows Time. It essentially consists of two hands — an hour hand and a minute hand. The minute hand is longer than the hour hand. For simplicity, we will not consider the third hand — the second hand which is also present in many clocks and watches. Conventionally clocks are circular in shape but we can now find them in other different shapes too. Irrespective of the shape of the dial of the clock, the tips of their hands — hour as well as minute—describe a circular path. The circumference of this circular path is divided into 12 equal parts, called Hour spaces. Each hour space is farther divided into 5 equal parts called Minute spaces. Thus the complete circumference is divided into 12 x 5 = 60 equal minute spaces.

We know that 1 hour equals 60 minutes. Therefore, the time taken by the hour-hand to move one hour-space (or 5 minute-spaces) equals the time taken by the minute-hand to move 60 minute-spaces (or 12 hour-spaces) which is equivalent to covering the complete circumference. In other words, while the hour-hand moves one hour-space, the minute-hand goes one full circle.

From the above description we notice that in 1 hour (or 60 minutes), the minute-hand passes 60 minute-spaces while the hour-hand moves 5 minute-spaces. Thus, we may conclude that in 1 hour or 60 minutes, the minute-hand gains (60 – 5) = 55 minute-spaces (or minutes) over the hour-hand. This is a very important and useful conclusion and therefore, must be always remembered while solving problems on clock.

**More Facts to be Remembered **

(i) In every one hour the two hands coincide once.

360°

(ii) Each hour-space equals 360/12 > hour-spaces are 30° apart.

(iii) Each minute-space equals =360/60= 6° i.e minute- spaces are 6° apart.

(iv) When the two hands are at right angle (90°), they are 90/6 = 15 minutes apart. This occurs twice in every hour.

(v) When the two hands are in opposite directions, they 180 are 180/6 = 30 minutes apart. This occurs only once in each hour.

(vi) When the hands are coincident, they are obviously at zero degree and zero minute apart.

(vii) The two hands are in the same straight line when they are either coincident or opposite to each other.

(viii) The hands coincide 11 times in every 12 hours, because there is a common position 12 O’clock between 11 and 1 O’clock. Hence, they coincide only 22 times in a day.

(ix) The hands of a clock are at right angles twice in every hour but in 12 hours they are at right angles only 22 times. It is so because there are two positions common in every’ 12 hours i.e., 3 O’clock and 9 O’clock.

**Too Fas**t: When the clock indicates time more than the correct time, it is said to be running too fast by the difference between the correct time and the time indicated by the clock. For example, the clock indicates 10: 20 am when the correct time is only 10:05 am, it is said to be 15 minutes fast.

**Too Slow** : When the time indicated by the clock is less than the correct time, it is said to be too slow. For example, the clock indicates 9:30 am when the correct time is 9:35 am, it is said to be slow by 5 minutes.

**Ex. 1. At what time between 2 and 3 O’clock are the two hands of the clock together?**

**Sol**. At 2 O’clock, the hour hand is at 2 and the minute hand at 12. It implies that the minute hands is 10 minute- spaces behind the hour hand at 2 O’clock. The two hands will be together when the minute hand gains 10 minute-spaces over the hour hand.

We know that,

The minute hand gains 55 minute spaces in 60 minutes.

The minute hand will gain 10 minute-spaces in

60/55 x 10 = 120/11 = 10 (10/11)

Therefore, the two hands will be together at 10 (10/11) past 2 O’clock Ans.

**Case (i)**: The minute hand is 15 minute-spaces behind the hour hand:

For the two hands to be in this position, the minute hand must gain (25 – 15) = 10 minute-spaces over the hour hand.

The minute hand gains 55 minute-spaces in 60 minutes.

The minute hand gains 10 minute-spaces in

60/55 x 10 = 120/11 = 10x(10/11) minutes

They will be at right angle at 10(10/11) minutes past 5 O’clock Ans.

**Case (ii)**: The minute hand is 15 minute-spaces ahead of the hour hand:

For the two hands to be in this position, the minute hand must gain (25 + 15) = 40 minute-spaces.

The minute hand gains 55 minute-spaces in 60 minutes.

The minute hand gains 40 minute-spaces in

60/55 x 40 = 480/11 = 43x 7/11

They will be at right angle again at 43— minutes past 5 O’clock Ans.

## Leave a Reply