Clocks

Clocks

The circumference of a dial of a clock or watch is divided into 60 equal parts called minute spaces. The clock has two hands - the hour hand and the minute hand. The hour hand (or short hand) indicates time in hours and the minute hand (or long hand) indicates time in minutes.

In an hour, the hour hand covers 5 minutes spaces while the minute hand covers 60 minutes spaces. Thus, in hour or 60 minutes, the minute hand gains 55 minutes spaces over the hour hand.

1. In every hour:

  • Both the hands coincide once.
  • The hands are straight (point in opposite directions) once. In this position, the hands are 30 minutes spaces apart.
  • The hands are twice at right angles. In this position, the hands are 15 minutes spaces apart.

2. The minute hand moves through 6° in each minute whereas the hour hand moves through 1/2 degree in each minute. Thus, in one minute, the minute hand gains 5 and half degree than the hour hand.

3. When the hands are coincident, the angle between them is 0°. When the hands point in opposite directions, the angle between them is 180°. The hands are in the same straight line, when they are coincident or opposite to each other. So, the angle between the two hands is 0° or 180°.

4. The minute hand moves 12 times as fast as the hour hand.

5. If a clock indicates 6.10, when the correct time is 6, it is said to be 10 minutes too fast. If it indicates 5.50, when the correct time is 6, it is said to be 10 minutes too slow.

6. The two hands of the clock will be together between H and (H + 1) O’clock at 60H/11 minutes past H O’clock.


Examples

Example 1: At what time between 5 and 6 O’clock are the hands of a clock together?

Here, H = 5.

60H/11 = (60 × 5)/11

= 300/11

At 5 O’clock, the minute hand is 25 min spaces behind the hour hand.

The minute hand gains 55 minutes spaces in 60 minutes, so the minute hand will gain 25 minutes spaces in 60/55 × 25 minutes.

Thus, the two hands of clock will be together at 300/11 minutes past 5 O’clock.

 

Example 2: At what time between 5 and 6 O’clock will the hands of a clock be at right angle?

Case 1: When the minute hand is 15 minutes spaces behind the hour hand, to be in this position, the minute hand will have to gain (5H - 15) minutes spaces over the hour hand.

Here, H = 5.

Now, 55 minutes spaces are gained in 60 minutes. So, 10 minute spaces are gained in 60/55 × 10 minutes. Thus, the two hands will be at right angle at 120/11 minutes past 5 O’clock.

Case 2: When the minute hand is 15 minutes spaces ahead of the hour hand, to be in this position, the minute hand will have to gain (5H + 15) minutes spaces over the hour hand. So, 35 minute spaces are gained in 60/55 × 35 minutes. Thus, the two hands will be at right angle at 420/11 minutes past 5 O’clock.

Example 3: Find at what time between 2 and 3 O’clock will the hands of a clock be in the same straight line but not together?

Hands of the clock will be in straight line when the minute hand is 30 minute spaces ahead of the hour hand. To be in this position, the minute hand will have to gain (5H + 30) minutes spaces over the hour hand.

Here, H = 2.

Now, 55 minutes spaces are gained in 60 minutes. So, 40 minute spaces are gained in 60/55 × 40 minutes. Thus, the two hands will be in the same straight line but not together at 480/11 minutes past 2 O’clock.

In case H > 6, the minute hand would have to gain (5H - 30) minutes spaces over the hour hand.

Example 4: Find the angle between the two hands of a clock at 15 minutes past 4 O’clock?

Angle traced by hour hand in 15 minutes = 7.5° (half degree per minute)

Angle traced by minute hand in 15 minutes = 90° (six degrees per minute)

Angle between two hands at 4 O’clock = 120°

Required angle = 120° - 90° + 7.5° = 37.5°