Introduction

The hands of a clock can have a maximum angle of 180o between them. When this happens, both the hands represent a straight line.

The hands of a clock make right angles twice every hour when they are 15 minutes away from each other.

The two hands of a clock coincide once in every hour.

A clock is very essential for every human being to schedule their daily tasks. The clock theory is related to everyday life.

A clock is an instrument that displays time by dividing it into hours, minutes, and seconds.

Dial

It has a circular dial numbered from 1 to 12 that indicates the hours. The circumference of a dial is further divided into 60 equal spaces, that are called the minutes’ spaces.

1 hour = 60 minutes

1 minute = 60 seconds and

1 hour = 3600 seconds

**Relative Movement of Hands of Clock:**

Clocks follow the principle of relative speed. Both hands move in the same direction i.e., (clockwise direction). The minute hand moves around the whole circumference of the clock once in 1 hour. The hour hand moves around the whole circumference of the clock once in 12 hour. Thus, the minute hand is twelve times faster than the hour hand.

The distance between two consecutive number in the clock is equal to 5 min. In an hour, the hour hand crosses 5 min spaces while the minute hand crosses 60 min spaces.

**Distance covered by hour needle & minute needle in 1 minute: **

A minute needle covers the distance of 60 minutes in 60 minute, whereas an hour needle covers the distance of 5 minutes in 60 minutes. Thus, the minute needle covers 55 minutes’ more distance in 60 minute than the hour needle.

∴ it covers the distance of 55 minute in 60 minutes.

Thus, it will cover the distance of 1 minute in min. OR min.

Some facts about clocks:

- Every hour, both the hands coincide once. In 12 hours, they will coincide 11 times. It happens due to only one such incident between 12 and 1’o clock.
- The hands are in the same straight line when they are coincident or opposite to each other.
- When the two hands are at a right angle, they are 15-minute spaces apart. In one hour, they will form two right angles and in 12 hours there are only 22 right angles. It happens due to right angles formed by the minute and hour hand at 3’o clock and 9’o clock.
- When the hands are in opposite directions, they are 30-minute spaces apart.
- If both the hour hand and minute hand move at their normal speeds, then both the hands meet after 65 minutes.

Now, let’s apply the above concept to some questions.

Type 1: Finding the time when the angle between the two hands is given.

Solved Examples

Example 1: At what time between 4 and 5, will the hands of a clock coincide?

Sol: At 4 O’clock, the hour hand has covered (4*30°) = 120°.

To catch up with the hour hand, the minute hand has to cover a relative distance of 120°, at a relative speed of 5 ° per minute.

Thus, time required = = = or 21 minutes.

Example 2: At what time between 10 and 11 will the minute and hour hand be at right angles?

Solution: At 10 O’clock, the hour hand has covered (10*30°) = 300°.

Note: There will be two right angles (clockwise and anti-clockwise)

Considering that hour hand is at 10, to make a 90-degree angle with the hour hand, the minute hand has to be at 1 or 7.

For the first right angle, minute hand has to cover a relative distance of (1*30) = 30°.

For the 2nd right angle, minute hand has to cover a relative distance of (7*30)= 210°.

We know that the relative speed between the two hands is of 5 ° per minute.

Hence, time required for the 1st right angle = = or 5 minutes.

Time required for the 2nd right angle = = = 38 minutes.

Type 2 : Finding the angle between the two hands at a given time.

**Solved Examples**

Example 1: The angle between the minute hand and the hour hand of a clock when the time is 4:20 is:

Solution: At 4:00, hour hand was at 120 degrees.

Using the concept of relative distance, the minute hand will cover = =110 degrees

The angle between the hour hand and minute hand is = 120-110 = 10 degrees.

Example 2: The angle between the minute hand and the hour hand of a clock when the time is 3:30 is:

Solution: At 4:00, hour hand was at 90 degrees.

Using the concept of relative distance, the minute hand will cover = =165 degrees

The angle between the hour hand and minute hand is = 165-90= 75 degrees.