One of the most important sections to learn is the "clock". We need to understand this in order to answer aptitude questions that come up in placement exams and other entrance examinations.

It is an essential part of logical reasoning that requires in-depth logical analysis. Also, it requires a detailed calculation level to solve the questions correctly.

In this blog, we will cover the detailed concept of a clock and the questions asked in the exams. Also, you can practice some clock questions given in this blog to get a better approach to this topic.

What is a Clock

A “clock” is an analog/digital device. In analog, it has three hands:

hour hand

minute hand

second hand

Generally, a clock has a round shape. Horology is the study of clocks.

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Clock Structure

A clock contains 360 degrees which are divided into 12 equal divisions.

After 360 degrees are divided by 12 divisions, the angle between any two consecutive divisions is determined.

Formed angle by the 2 consecutive divisions is equal to (360 degrees)/12, i.e., 30°.

Dividing Clock into Angles

Upon closely observing a clock, it is revealed that an angular space between any consecutive divisions can have five more divisions.

Two consecutive divisions area = 5 minutes

Thus, we can deduce that the angular value of a minute is obtained by (30 degrees)/5 = 6 degrees. This is also known as the angle equivalence of a minute.

The angular values for the first 10 minutes of the clock are depicted in the table below:

Minutes(In Clock)

Angular Value (In Degree)

1

6

2

12

3

18

4

24

5

30

6

36

7

42

8

48

9

54

10

60

Let us look at the speed of a minute and hour hand in a clock.

Varying Speed of Clock Hand

A clock has three hands, and each move at different rates. The distance covered and the amount of time required to cover that distance define the speed of moving hands.

It is calculated by the following formula:

Speed of Moving Hand = Distance Covered/Time taken

Let us look further into the varying speed of hour and minute hands.

For Hour Hand

Covers one divisionin an hour. i.e., we can say that it covers a distance of 5 minutes in 60 minutes with respect to the minute hand.

Speed = (30 degrees)/(60 minutes).

Thus, speed = ½ degree per minute.

For Minute Hand

Travels 12 divisions around the clock every hour.

Speed = (360 degrees)/(60 minutes).

Thus, speed = 6 degrees per minute.

Comparison

The difference in the speed of the minute hand and the hour hand

= (6 - ½) degrees per minute

= 5 ½ degrees per minute.

It is clear from the comparison of the minute and hour hands' speeds that the minute hand is almost always 5 ½ degrees quicker than the hour hand.

Collision and Coincidence

As we know, the hands of the clock move at different speeds. They sometimes collide and coincide to make different angle formations among themselves several times a day.

Important Problems of Clocks

Example 1: Find the angle produced by the clock's two hands at 30 minutes past nine in the night.

Solution:

Explanation:

30 minutes past 9 in the night means 9:30 pm.

9:30 means 9 hours 30 minutes = 9 + 30/60 hours

= 570/60 hours

= 9 ½ hours

We know 12 hours = 360 degrees

Therefore, 1 hour = 30 degrees

So, 9 ½ hours * 30 degrees = 285 degrees

Also, 60 minutes = 360 degrees

Therefore, 1 minute = 6 degrees

∴ 30 minutes means (6 * 30) degrees = 180 degrees

Angle between the two hands = 285 - 180 = 105°

Example 2: Compute how many times a clock's minute and hour hands coincide in a day(such that angle between an hour and the minute hand is zero) with each other.

Solution:

The first collision of the hour and minute hands in a clock occurs at midnight.

Explanation:

Since we know the day starts at midnight, and hence we can say that the first-ever coinciding of hands takes place at midnight.

By observing the clock, we can conclude the next coincidence will occur approximately at 1 o’clock and 5 minutes.

Thus, we can say that there is one collision of hands every hour. Therefore, the answer would be 24 times for 24 hours. But this is not the correct answer and not the right logic.

Now let us observe the time between 11 to 12. We can say that it can either be P.M. or A.M.; thus, the hands are not coinciding between 12 o’clock and 11 o’clock. Hence, we can say that the coinciding of hands at 12 o’clock is the coincidence between 11 and 12 and 12 and 1.

Therefore, now we can conclude that in 12 hours, there will only be 11 coincidences occurring, extending the logic for the whole day(i.e., 24 hours), so we can say that there would be 22(as (24/12) * 11) coincidences.

Logical Calculation:

We know that in 12 hours, there will be 11 coincidences. Therefore, we can conclude that one collision will happen at:

Frequency of 1 collision w.r.t hour = (12 hours)/11,

Frequency of 1 collision w.r.t minute = (12*60 mins)/11,

= (720 mins)/11,

Thus, the frequency of one collision = 65+(5/11).

The value 65(5/11) refers to the hands of a clock coinciding after every 65 minutes + 5/11 of a minute. So, if the first collision occurs at 12:00:00, we can calculate the particular time of the next collision by adding 65(5/11) to 12 o'clock.

The table given below denotes the time at which both the hands of a clock will collide:

Frequency of Collision

Mixed Fraction time

Exact Time

1

12:0:0

12:0:0

2

1:5:5/11

1:5:27

3

2:10:10/11

2:10:54

4

3:16:16/11

3:16:21

5

4:21:9/11

4:21:16

6

5:27:3/11

5:27:36

7

6:32:8/11

6:32:43

8

7:38:2/11

7:38:10

9

8:43:7/11

8:43:38

10

9:49:1/11

9:49:5

11

10:54:6/11

10:54:32

12

11:59:11/11

12:0:0

Here, you might need clarification about how mixed fraction time 1:5:5/11 is giving the exact time as 1:5:27. Let us make it simple for you.

60*5/11 = 27 seconds.

Hence, the 27 seconds is getting added to the 1:5:00, so the exact time becomes 1:5:27.

Example 3: Find the time in a clock between 5 am and 6 am when the hour and minute hand will coincide.

Solution:

Explanation:

In 1 hour, minute hand travels 60 minutes.

The hour hand moves in just five-minute intervals.

As a result, the minute hand gains 55 minutes over the hour hand in 60 minutes(i.e., 60 - 5 = 55).

At 5 o'clock, hour hand and minute hand difference = 5*5 = 25 minutes

So to coincide with the hour hand, the minute hand must travel 25 minutes more.

In 60 minutes, a minute hand gains 55-minute spaces.

Time(in minutes) it gains in 25 minutes = (25 x 60)/55

= 1500/55

= 300/11

= 27(3/11) minutes past 5 o'clock.

Note: 27(3/11) here means 27+(3/11) = 27.27(approx.)

Example 4: Compute how many times a day the minute and hour hands will form a 180° straight line.

Solution:

Explanation:

We know that the hands of the clock make one 180° straight line every hour in a clock except between 5 o’clock and 6 o’clock. When a precise observation and analysis of the watch is made, it gives the idea that between 5 and 6, the hands of the clock make a straight line of 180° exactly at 6 o’clock, and so, it cannot be the one that happens between 5 o’clock and 6 o’clock. Instead, it is considered a straight line formed between 6 o’clock and 7 o’clock.

As a result, a clock's hands turn 180 degrees in a straight line 11 times in a day of 12 hours and 22 times in a day of 24 hours if we expand the observation to 24 hours.

Example 5: If a clock's minute and second hand are separated by 18 minutes. What is the angle created between them?

Solution:

Explanation:

The angle covered during 60 minutes is 360°.

So for 18 minutes, let the difference angle be 'a'.

∴ 60 * a = 18 * 360

∴ a = (18/60) * 360 = 108°

Time Calculation for Known Angle

The solutions to the problems become more complicated and time-consuming when the angle between the two hands is not a perfect angle, such as 180°, 90°, or 270°.

The logic below provides a trick to solve problems involving angles of hands other than the standard aspects. Refer to the formula given below:

T = ( 2/11 ) * [H * 30 ± A]

Here:

‘T’ equals the time at which the angle formed.

‘H’ equals an hour, which is running. (If the question is given for the duration between 4 o’clock and 5 o’clock, that means it’s the fourth hour that is running; hence the value of H will be equal to ‘4’.)

‘A’ equals the angle at which a clock's hands are present. (The value of ‘A’ will be provided in the question.)

The clock is divided into two parts vertically: 1st(right-side) and 2nd(left-side) half. If the given time in the question lies in the first half of the clock, then we consider the positive sign while evaluating the time, or else, we consider the negative sign.

You can also refer here for clock problems and to understand the concepts of clocks in detail.

Frequently Asked Questions

What is a clock?

A clock is a circular device that shows time and has three hands, a second hand, an hour and a minute hand.

What is the study of clocks called?

The study of clocks is known as horology.

What are the important points on the structure of clocks?

The clock contains three hands:

Hour Hand:- It covers 360 degrees in 12 hours.

Minute Hand:- It covers 360*12 degrees in 12 hours.

Second Hand:- It covers 360*60*12 degrees in 12 hours.

What are the time formats in a clock?

There are two-time formats: 12-hour and 24-hour formats in a clock.

How many times are the minute and hour hands straight to each other in a day(overlapping or exact opposite to each other)?

The hands of a clock are opposite to one another exactly 22 times over a day, and they also coincide exactly 22 times. As a result, they occur 44 times in a 24-hour period.

Conclusion

In this article, we covered one of the aptitude's important topics, known as the clock problem. Additionally, we’ve also discussed the various conditions to be considered for solving the problem of clocks.

If you would like to learn more similar to this topic, check out our related articleson aptitude preparation-