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CLOCK

A clock has three hands – Hour Hand (Smallest in size), Minute Hand, and Second Hand.

For hour hand,

In short,

For minute hand,

In short,

From the above conclusions, if we intend to find the relative speed of the minute hand with respect to the hour hand h 1 min, then it will be 

That means, 

In 1 min, the gap between the minute hand and the hour hand will increase or decrease by 

Or,

Relative Speed of Minute Hand (faster) with respect to Hour Hand 

There are three types of questions that come in SSC state level exams and other competitive exams having general intelligence –

# Calculation of Angle between Hour and Minute Hands from a given time

# Calculation of Exact Time within a given time interval and a given Angle between Hour and Minute Hands

# Miscellaneous 

CALCULATION OF ANGLE BETWEEN HOUR AND MINUTE HANDS FROM GIVEN TIME

There are two approaches to solve this type of question –

1. Ordinary Method

2. Formula Method

Suppose the given time is 8:15 PM

1. Ordinary Method

The minute hand is on ‘3’ and the hour hand is on ‘8’. But actually, the hour hand is slightly away from 8. The reason is that when the minute hand on 8.00 PM traveled towards ‘3’, then the hour hand moved slightly away from ‘8’.

So, the angle subtended between ‘8’ and ‘3’ is –

2. Formula Method

This is a very important formula required in solving most of the clock problems.

Here, H = 8 and M = 15, A = Angle

Questions:

Q1) The reflex angle between the hands of a clock at 10 : 30 is:

Solution:

By Ordinary Method:

As the hour hand moves away from ‘10’ when the minute hand travel from 10 : 00 to 10 : 30. 

In addition to that, the hour hand moves away from the minute hand, 

By how much greater?

Option (c).

By Formula Method

Q2) The reflex angle between the hands of a clock at 4 : 40 is:

Solution:

By Ordinary Method:

Here the hour hand moves towards the minute hand, 

By how much?

Option (b).

By Formula Method

Here, the formula method is the time-saving method, which is essential for any competitive exam. Hence, from now on we will avoid the conventional approach to save our valuable time.

CALCULATION OF EXACT TIME WITHIN A GIVEN TIME INTERVAL AND A GIVEN ANGLE BETWEEN HOUR AND MINUTE HANDS

Then,

A = θ for the first time and second time. 

For the first θ angle,

For the second θ angle,

‘Subtraction’ for first θ

‘Addition’ for second θ

Questions:

Q1) At what time between 6 : 00 to 7 : 00  o’clock both hands of the clock coincides?

Solution:

Option (c)

Q2) At what time between 7 : 00 to 8 : 00  o’clock both hands of the clock be in a straight line but not together?

Solution:

Option (a)

MISCELLANEOUS

In this type, questions come on the frequency of occurring a particular situation in which the two hands either coincide or make an angle in 12 hrs or in a day, i.e. 24 hrs.

To do this, the below table must be referred:

From the above chart, some important points are noted here as follows:

# Both hands of the clock coincide one time in one hour but 11 times in 12 hours or 22 times in 24 hours.

# Both hands of the clock opposite one time in one hour but 11 times in 12 hours or 22 times in 24 hours.

# Both hands of the clock right-angled twice in one hour but 22 times in 12 hours or 44 times in 24 hours.

# Both hands of the clock acute-angled or reflex – angled twice in one hour but 22 times in 12 hours or 44 times in 24 hours.

# In 24 hours, both hands of the clock lie 44 times in a straight line,