28. CLOCKS
IMPORTANT FACTS
The Face or dial of a watch is a circle
whose circumference is divided into 60
equal parts, called minute
spaces.
A clock
has two hands, the smaller one is called the hour hand or
short hand while the larger one is called the minute
hand or long hand..
i) In 60 minutes, the minute hand gains 55
minutes on the hour hand.
ii) In every hour, both the hands coincide
once.
iii) The hands are in the same straight
line when they are coincident or opposite to each other.
iv) When the two hands are at right angles,
they are 15 minute spaces apart.
v)When the hand's are in opposite
directions, they are 30 minute spaces apart.
vi)Angle traced by hour hand in 12 hrs = 360°.
vii)Angle traced by minute hand in 60 min.
= 360°.
Too Fast and Too Slow: If a watch or a
clock indicates 8.15, when the correct time , 8 is said to be 15
minutes too fast.
On the other hand, if it indicates 7.45, when the correct time is 8, it
is said to be 15 minutes too slow.
SOLVED EXAMPLES
Ex 1:Find the angle between the hour hand and the
minute hand of a clock when 3.25.
Solution:angle traced by the hour hand in 12 hours = 360°
Angle traced by it in three
hours 25 min (ie) 41/12 hrs=(360*41/12*12)° =102*1/2°
angle traced by minute hand in 60 min. = 360°.
Angle traced by it in 25 min. = (360 X
25 )/60= 150°
Required angle = 1500 – 102*1/2°= 47*1/2°
Ex 2:At what time between 2 and 3 o'clock will
the hands of a clock be together?
Solution: At 2 o'clock, the hour hand is at 2 and the minute
hand is at 12, i.e. they are 10 min
spaces apart.
To be together, the minute hand must gain
10 minutes over the hour hand.
Now, 55 minutes are gained by it in 60 min.
10 minutes will be
gained in (60 x 10)/55
min. = 120/11 min.
The hands will coincide at 120/11 min. past 2.
Ex.
3.
At what time between 4 and 5 o'clock will the hands of a clock
be
at right angle?
Sol: At 4 o'clock, the minute hand
will be 20 min. spaces behind the hour hand, Now, when the two hands are at
right angles, they are 15 min. spaces apart. So, they are at right angles in
following two cases.
Case I. When minute hand is 15 min.
spaces behind the hour hand:
In this case min.
hand will have to gain (20 - 15) = 5 minute spaces. 55 min. spaces are gained by it in 60
min.
5 min spaces will be gained by it in 60*5/55 min=60/11min.
:. They are at
right angles at 60/11min. past 4.
Case II. When the
minute hand is 15 min. spaces ahead of the hour hand:
To be in this
position, the minute hand will have to gain (20 + 15) = 35 minute spa' 55 min. spaces are
gained in 60 min.
35 min spaces are gained in (60 x 35)/55 min
=40/11
:. They are at right angles at 40/11 min.
past 4.
Ex. 4. Find at what time between 8 and 9
o'clock will the hands of a clock being the same straight line but
not together.
Sol: At 8 o'clock, the hour hand
is at 8 and the minute hand is at 12, i.e. the two hands_ are 20 min. spaces
apart.
To be in the same
straight line but not together they will be 30 minute spaces apart. So, the minute
hand will have to gain (30 - 20) = 10 minute spaces over the hour hand.
55 minute spaces
are gained. in 60 min.
10 minute spaces will be gained in (60 x
10)/55 min. = 120/11min.
:. The hands will
be in the same straight line but not together at 120/11 min.
Ex. 5. At what
time between 5 and 6 o'clock are the hands of a clock
3minapart?
.
Sol.
At 5 o'clock, the minute hand is 25 min. spaces behind the hour hand.
Case I. Minute hand is 3 min. spaces behind
the hour hand.
In this case, the
minute hand has to gain' (25 - 3) = 22 minute spaces. 55 min. are gained in 60
min.
22 min. are gaineg in (60*22)/55min.
= 24 min.
:. The hands will
be 3 min. apart at 24 min. past 5.
Case II. Minute hand is 3 min. spaces ahead
of the hour hand.
In this case, the
minute hand has to gain (25 + 3) = 28 minute spaces. 55 min. are gained in 60
min.
28 min. are gained in (60 x 28_)/55=346/11
The hands will be
3 min. apart at 346/11 min. past 5.
Ex 6. Tbe minute hand of a clock
overtakes the hour hand at intervals of 65 minutes of the correct time. How
much a day does the clock gain or lose?
Sol: In a correct clock, the
minute hand gains 55 min. spaces over the hour hand in 60 minutes.
To be together
again, the minute hand must gain 60 minutes over the hour hand. 55 min. are
gained in 60 min.
60 min are gained in 60 x 60 min =720/11 min.
55
But,
they are together after 65 min.
Gain in 65 min =720/11-65
=5/11min.
Gain in 24 hours =(5/11 * (60*24)/65)min =440/43
The clock gains
440/43 minutes in 24 hours.
Ex. 7. A watch
which gains uniformly, is 6 min. slow at 8 o'clock in the morning Sunday and it is 6 min.
48 sec. fast at 8 p.m. on following Sunday. When was it correct?
Sol. Time from 8 a.m.
on Sunday to 8 p.m. on following Sunday = 7 days 12 hours = 180 hours
The watch gains (5 + 29/5)
min. or 54/5 min. in 180 hrs.
Now 54/5 min. are gained in 180 hrs.
5 min. are gained
in (180 x 5/54 x 5) hrs. =
83
hrs 20 min. = 3 days 11 hrs 20
min.
Watch is correct
3 days 11 hrs 20 min. after 8 a.m. of Sunday.
It will be
correct at 20 min. past 7 p.m. on Wednesday.
Ex 8. A clock is
set right at 6 a.m. The clock loses 16 minutes in 24 hours. What will be the
true time when the clock indicates 10 p.m. on 4th day?
Sol. Time from 5 a.m.
on a day to 10 p.m. on 4th day =
89
hours.
Now 23 hrs 44 min. of this clock = 24 hours of correct clock.
356/15 hrs of this clock = 24 hours of
correct clock.
89 hrs of this clock = (24 x 31556 x
89) hrs of correct clock.
= 90 hrs of correct
clock.
So, the correct time is 11 p.m.
Ex. 9. A clock is set right at 8
a.m. The clock gains 10 minutes in 24 hours will be the true time when the
clock indicates 1 p.m. on the following day?
Sol. Time from 8 a.m.
on a day 1 p.m. on the following day = 29 hours.
24 hours 10 min. of this clock = 24 hours of the
correct clock.
145 /6 hrs of this clock = 24 hrs of the
correct clock
29 hrs of this
clock =
(24
x 6/145 x 29) hrs of the correct
clock
= 28 hrs 48 min. of correct clock
The correct time
is 28 hrs 48 min. after 8 a.m.
This is 48 min.
past 12.
