Clock

By Digambar Singh
 Introduction
 A Clock is a circular device provided with three hands viz. an
hour hand, minute and second hand. The study of the clock is
known as “horology”.
Structure of a Clock
 A clock is composed of 360 degrees and divided into 12 equal
divisions. The angle between the consecutive divisions is obtained
by dividing the total angle of clock 360° by the number of
divisions i.e. 12.
Angles in Clock
 The angle between any two consecutive divisions =
(360°)/12= 30°
 Angle equivalence of a minute
Minute(s) Angular values

1 6°

2 12°

3 18°

4 24°

5 30°

6 36°

7 42°

8 48°

9 54°

10 60°
Speed of the hands
 A clock has three hooks and all three move at different rates. The
speed of moving object depends on the distance travelled and the
time taken to cover a specific range.
 The speed is calculated by:
 Speed = Distance/(Time taken)
 The speed of a minute hand:
 A minute hand travels 360° in one hour. i.e. it travels through all
the 12 divisions around the clock every hour. (1 hour = 60
minutes)
 Speed of a minute hand = (360°)/(60 minutes)
 Speed of a minute hand = 6° per minute.
Speed of the hands
 The speed of an hour hand:
 An hour hand travels 30° in an hour. i.e. it covers a distance
of 5 minutes (the gap between consecutive divisions) in 60
minutes.
 Speed of an hour hand = (30°)/(60 minutes)
 Speed of an hour hand =1/2 ° per minute.
Comparison of Speed of hands
 The difference in the speed = 6°– (1/2°) = 5.5° per minute
 Comparing the speed of the minute hand and an hour hand,
one can conclude that the minute hand is always faster than
the hour hand by 5.5° or an hour hand is always slower than
the minute hand by 5.5°
 Note: The evaluation of the speed of second hands is not necessary
as it travels a corresponding distance of 1 second in a second.
 Frequency of coincidence and collision
of hands of a clock:
As we know the hands of clock moves at different speeds, they coincide and
collide and also make different angle formations among themselves at various
times in a day.
Q.1 How many times in a day do the minute and hour hands of a clock
coincide (Angle between them is zero) with each other?
Logical explanation:
 A day starts at midnight, and hence the first-ever coinciding of
hands happen at midnight. Now observing the clock, the next
coincidence will occur at approximately 1’ o’clock and 5 minutes.
Thus, one can conclude that every hour, there is one coincidence
of hands. Therefore, the answer should be 24 times for 24 hours.
But it is not the correct answer and right logic.
 Now observe the time between 11 to 12, either it can either be
A.M or P.M, the hands are not coinciding between 11 o’clock and
12 o’clock. The coinciding of hands at 12’ o clock is the
coincidence between 12 and 1 and 11 and 12. Hence, in 12 hours,
there will only be 11 coincidence, extending the logic for 24
hours of the day, there will be 22 coincidences.
Logical calculation:
 We know in 12 hours there will be 11 coincidences.
Therefore, one collision will happen at:
 Frequency of one collision = (12 hours)/11
 Frequency of one collision = (12*60 mins)/11
 Frequency of one collision = (720 mins)/11
 Frequency of one collision = 65(5/11)
 The value 65(5/11) indicates that the hands of a clock
coincide after every 65 minutes 5/11 of a minute. i.e. if
12:0:0 is the first collision, then the exact time of the next
collision will be obtained by adding 65(5/11) to 12 o’clock.
 The below table denotes the time at
which both the hands of a clock collide:
Frequency of collision Time in mixed fraction Exact time

st
1 12:0:0 12:0:0
nd
2 1:5:5/11 1:5:27
rd
3 2:10:10/11 2:10:54
th
4 3:16:16/11 3:16:21
th
5 4:21:9/11 4:21:16
th
6 5:27:3/11 5:27:36
th
7 6:32:8/11 6:32:43
th
8 7:38:2/11 7:38:10
th
9 8:43:7/11 8:43:38
th
10 9:49:1/11 9:49:5
th
11 10:54:6/11 10:54:32
th
12 11:59:11/11 12:0:0
How many times in a day do the minute and hour
hands of clock form a 180° straight line in a day?
 The hands of the clock make one 180° straight line every
hour except between 5’o clock and 6’o clock. A precise
observation and analysis of the watch gives the idea that
between 5 and 6 the hands make straight line 180° exactly at
6 o’clock and hence, it cannot be the one which happens
between 5 o’clock and 6 o’clock. Instead it is considered as a
straight line formed between 6 and 7 o’clock.
 Therefore, the hands of a clock make 180° straight lines 11
times in 12 hours and hence generalising it for 24 hours, the
hands make 22 consecutive lines of 180° in 24 hours.
 Finding the time when the angle is known:
 When the angle between the hands are not perfect angles like 180°, 90° or
270°, the solving of the questions becomes difficult and time-consuming at
the same time. The logic below provides a trick to address problems
involving angles of hands for other than standard aspects.
T = 2/11 [H*30±A]
Where:
 T stands for the time at which the angle formed.
 H stands for an hour, which is running.
(If the question is for the duration between 4 o’clock and 5 o’clock, it’s the
4th hour which is running hence the value of H will be ‘4’.)
 A stands for the angle at which the hands are at present.
(The value of A is provided in the question generally)
The clock is divided into two parts: 1st and 2nd half as shown
above
If the time given in the question lies in the first half, then the
positive sign is considered while evaluating the time else, then
the negative sign is used.
TCS
➢ By 20 minutes past 4, the hour hand has turned through how
many degrees? If then the clock is 12 p.m.
A. 100°
B. 110°
C. 120°
D. 130°
Solution:
 At 4 o’clock the hour hand is at 4 and has an angle of
30°*4=120°
 An Hour hand travels 1/2° per minute In 20 minutes it will
travel 20 *(1/2°) = 10°. Adding both we get 120° + 10° =
130°
 Option D is the correct answer.
WIPRO
➢ At what time between 5.30 and 6 will the hands of a clock be
at right angles?
A. 44 minutes past 5
B. 44 ( 7/11) minutes past 5
C. 43 ( 7/11) minutes past 5
D. 43 minutes past 5
Solution:
 Given: H = 5 and A = 90, since 5 and 6 lies in the first half, a
positive sign is considered.
 T = 2/11 [H*30±A]
 T = 2/11 [5*30+90]
 T = 2/11 [240] = 480/11= 43( 7/11)
 Option C is the correct answer.
Example
 At what time between 3 and 4 o’clock, the
hands makes an angle of 10 degrees?
Solution:
 Given: H = 3 , A = 10
 Since both three and four lies in the first half considered a positive sign.
 Calculations:
 T = 2/11 [H*30±A]
 T = 2/11 [3*30+10]
 T = 2/11 [90+10]
 T = 2/11 [100]
 T = 200/11
 T =18 2/11
 The answer indicates that the hands of a clock will make an angle of 10
between 3 and 4 o’clock at exactly 3:18:2/11 ( 3’ o clock 18 minutes and
2/11 of minutes = 2/11*60 = 10.9 seconds)
Accenture
➢ What is the angle between the minute hand and the hour
hand of a clock at 5.30?
A. 05°
B. 15°
C. 25°
D. 35
Solution:

 At 5 ‘o’clock the hour hand is at 5 and hence has made 30° angle.
 From 5 to 5.30 its will travel for 30 minutes with a speed of ½ °
Therefore the total distance travelled will be 30 minutes* 1/2 =
15°
 The full angle made by the hour hand will be 150°+15° = 165°.
 The minute hand at 5 o’clock is at 12, and hence the angle made is
zero. In 30 minutes, it will travel a distance of 30 minutes with a
speed of 6° per minute. Therefore, the total distance travelled will
be 30 minutes*6° = 180 °.
 The angle between the minute and hour hand is 180 – 165 = 15
 Option B is the correct answer.
Puzzle: Interview Question
 Observe the monument. How many minute and hour
hand/hands does Big Ben tower clock have?
Answer:
It has 4-minute hands and 4-hour hands as
there are four clocks on all the four sides of
it.
Correct clock v/s Wrong clock:
This section involves the comparison of time in the accurate
clock with the wrong watch. The wrong time indicates that a
clock is either slow or fast compared to the correct time. The
wrong clock can either be fast or delayed by a few
seconds/minutes/ hours or sometimes by a few days and
weeks.
RRB JE
 A clock gains 5 seconds for every 3 minutes. If the clock
started working at 7 a.m. in the morning, then what will be
the time in the wrong clock at 4 p.m. on the same day?
Solution:
A clock gains 5 seconds for every 3 minutes, then it will gain
50 seconds in 30 minutes, or it will acquire 100 seconds in 60
minutes. i.e. it will gain 100 seconds in 1 hour. Since the clock
was started at 7 a.m. in the morning and right now the correct
time is 4 p.m. the total time the clock has worked is 9 hours.
We know that in 1 hour it gains 100 seconds then in 9 hours it
increases 900 seconds. The conversion of 900 seconds to
minutes will be 15 minutes. This increase indicates that a clock
is faster by 15 minutes as the clock is gaining. Hence, the time
in the watch would be 4:15 p.m.
IB 2008
A house has two wall clocks, one in kitchen and one more in
the bedroom. The time displayed on both the watches is
12.A.M right now. The clock in the bedroom gains five minutes
every hour, whereas the one in the kitchen is slower by five
minutes every hour. When will both the watches show the same
time again?
Solution:
The faster clock runs 5 minutes faster in 1 hr.
The slower clock runs 5 minutes slower in 1 hr.
Therefore, in 1 hour, the faster clock will trace 5+5=10 min more when compared to the
slower clock.
The table given below depicts the time difference between the slower and faster clock:
In 6 hours, the faster clock will trace 10×6=60 minutes (an hour) more when compared to
the slower clock.
In 6×12=72 hours, the faster clock will trace an hour more when compared to the slower
clock since the quicker clock determines 12 hours more than, the slower clock. At this point,
both the clocks will show the same time. i.e., both the clocks will show the same time after
exactly 72 hours (or 3 days).
Correct time Slower clock Faster clock
12:0:0 12:0:0 12:0:0
1:0:0 12:55:0 1:5:0
2:0:0 1:50:0 2:10:0
3:0:0 2:45:0 3:15:0
4:0:0 3:40:0 4:20:0
5:0:0 4:35:0 5:25:0
6:0:0 5:30:0 6:30:0