business mathematics

Clocks problems

Clocks problems

Simple Problems:

Type1:
Find the angle between the hour hand and the minute hand
of a clock when the time is 3.25

solution : In this type of problems the formulae is as follows
30*[hrs-(min/5)]+(min/2)
In the above problem the given data is time is 3.25. that is
applied in the
formulae
30*[3-(25/5)]+(25/2)30*(15-25)/5+25/2
= 30*(-10/5)+25/2
= -300/5+25/2
= -600+(25/2)=-475/10=-47.5
i.e 47 1/20
therefore the required angle is 47 1/20.

Note:The -sign must be neglected.
Another shortcut for type1 is :
The formulae is
6*x-(hrs*60+X)/2
Here x is the given minutes,
so in the given problem the minutes is 25 minutes,
that is applied in the given formulae
6*25-(3*60+25)/2
150-205/2
(300-205)/2=95/2
=47 1/20.
therefore the required angle is 47 1/20.

Type2:
At what time between 2 and 3 o' clock will be the hands of a
clock be together?

Solution : In this type of problems the formulae is
5*x*(12/11)
Here x is replaced by the first interval of given time.
here i.e 2. In the above problem the given data is between
2 and 3 o' clock
5*2*12/11 =10*12/11=120/11=10 10/11min.
Therefore the hands will coincide at 10 10/11 min.past2.

Another shortcut for type2 is:
Here the clocks be together but not opposite
to each other so the angle is 0 degrees. so the formulae is
6*x-(2*60+x)/2=06*x-(120+x)/2=012x-120-x=0
11x=120
x=120/11=10 10/11
therefore the hands will be coincide at 10 10/11 min.past2.




Medium Problems

Type3:
At what time between 4 and 5 o'clock will the hands of a clock
be at rightangle?

Solution : In this type of problems the formulae is
(5*x + or -15)*(12/11)
Here x is replaced by the first interval of given time
here i.e 4

Case 1 : (5*x + 15)*(12/11)
(5*4 +15)*(12/11)
(20+15)*(12/11)
35*12/11=420/11=38 2/11 min.
Therefore they are right angles at 38 2/11 min .past4

Case 2 : (5*x-15)*(12/11)
(5*4-15)*(12/11)
(20-15)*(12/11)
5*12/11=60/11 min=5 5/11min
Therefore they are right angles at 5 5/11 min.past4.

Another shortcut for type 3 is:
Here the given angle is right angle i.e 900.

Case 1 : The formulae is 6*x-(hrs*60+x)/2=Given angle
6*x-(4*60+x)/2=90
6*x-(240+x)/2=90
12x-240-x=180
11x=180+240
11x=420
x=420/11= 38 2/11 min

Therefore they are at right angles at 38 2/11 min. past4.

Case 2 : The formulae is (hrs*60+x)/2-(6*x)=Given angle
(4*60+x)/2-(6*x)=90
(240+x)/2-(6*x)=90
240+x-12x=180
-11x+240=180
240-180=11x
x=60/11= 5 5/11 min

Therefore they art right angles at 5 5/11 min past4.

Type 4:
Find at what time between 8 and 9 o'clock will the hands of a
clock be in the same straight line but not together ?

Solution : In this type of problems the formulae is

(5*x-30)*12/11
x is replaced by the first interval of given time Here i.e 8
(5*8-30)*12/11
(40-30)*12/11
10*12/11=120/11 min=10 10/11 min.
Therefore the hands will be in the same straight line but not
together at 10 10/11 min.past 8.

Another shortcut for type 4 is:
Here the hands of a clock be in the same
straight line but not together the angle is 180 degrees.
The formulae is (hrs*60+x)/2-(6*x)=Given angle
(8*60+x)/2-6*x=180
(480+x)/2-(6*x)=180
480+x-12*x=360
11x=480-360
x=120/11=10 10/11 min.
therefore the hands will be in the same straight line but not
together at 10 10/11 min. past8.


Type 5:
At what time between 5 and 6 o’ clock are the hands of a
3 minutes apart ?

Solution : In this type of problems the formuae is
(5*x+ or - t)*12/11
Here x is replaced by the first interval of given time here xis 5.
t is spaces apart

Case 1 : (5*x+t)*12/11
(5*5+3)*12/11
28*12/11 = 336/11=31 5/11 min
therefore the hands will be 3 min .apart at 31 5/11 min.past5.

Case 2 : (5*x-t)*12/11
(5*5-3)*12/11
(25-3)*12/11=24 min
therefore the hands wi be 3 in apart at 24 min past 5.

Typicalproblems

problems:
A watch which gains uniformly ,is 5 min,slow at 8 o'clock in
the morning on sunday and it is 5 min.48 sec.fast at 8 p.m on
following sunday. when was it correct?

Solution :
Time from 8 am on sunday to 8 p.m on following sunday = 7 days
12 hours = 180 hours
the watch gains (5+(5 4/5))min .or 54/5 min. in 180 hours
Now 54/5 minare gained in 180 hours.
Therefore 5 minutes are gained in(180*5/54*5)hours=83 hours20 min.
=3 days11hrs20min.
therefore watch is correct at 3 days 11 hours 20 minutes after 8 a.m
of sunday
therefore it wil be correct at 20 min.past 7 p.m on wednesday

CLOCKS practice problems:

For clocks concept click: http://business-maths.blogspot.com/2008/09/clocks.html

For clocks solved problems click: http://business-maths.blogspot.com/2008/11/clocks-problems.html

Practice problems:

1.Find the angle between the hands of a clock when the time
is 3:20
a)20 b)30 c)40 d)50

2.At what time between 4 and 5'o clock will the 2 hands of
a clock beat an angle of 60 degrees with each other?
a)4:10 10/11 b)4:32 8/11 c)both a and b d)none of these

3.Find the time at which the hands of a clock are at right angles
between 7 and 8'o clock?
a)7:21 9/11 b)7:54 6/11 c)both a & b d)none of these

4.Find the time at which the hands of a clock are exactly opposite
directions between 8 and 9'o clock?
a)8:10 10/11 b)8:02 c)8:3/11 d)none of these

5.At what time between 2 and 3'o clock are the hands of a clock
together?
a)2:10 b)2:10 10/11 c)2:10 10/12 d)none

6.In 24 hours,how many times do the two hands of a clock coincide?
a)20 times b)21 times c)22 times d)23 times

7.If the hands of a clock coincide every 65 minutes,how much time
does the clock gain or lose per day?
a)10 100/146 b)10 100/145 c)10 100/156 d)10 100/143

8.In a country R,clocks are manufactured in a special way.The
total area covered by the hour hand in 2 days is 2/3rd of the
total area covered by the minute hand in one day.What is the
ratio of the length of the minute hand to the length of the
hour hand?
a)1:2 b)2:1 c)4:1 d)1:4

9.There are two clocks,which are set to correct time on sunday
at 12 noon.The first clock gains 2 1/2 minutes every hour while
the second clock loses 1 1/2 minutes every hour.When will they
be 2 hours apart?
a)Monday 9 p.m b)Tuesday 12 midnight
c)Monday 6 p.m d)Tuesday 6 a.m

10.A watch which is set correctly at 12 noon on sunday shows
6:20 a.m on Monday when the correct time is 6:00 a.m.If the
correct time is 6:00 p.m. on tuesday,what time does the
clock show?
a)6:45 p.m b)7:00 p.m c)7:30 p.m d)8:00 p.m

11.At what time between 1 and 2'o clock are the two hands
coincident?
a)1:12 5/11 b)1:07 5/11 c)1:05 5/11 d)1:08 5/11

12.Ajay started a test at a certain time between 5 and 6 and
ended between 6 and 7.He observed that the minutes and hours
hands when he ended the test had inter-changed their positions
with those when he started.How much time did he take for
the test?
a)52 3/11 b)55 5/13 c)47 4/9 d)45 min

13.A clock is started at noon.By 10 minutes past 5,the hour
hand has turned through
a)145 deg b)150 deg c)155 deg d)160 deg

14.An accurate clock shows 8'o clock in the morning.Through
how many degrees will the hour hand rotate when the clock
show 2'o clock in the afternoon?
a)144 deg b)150 deg c)168 deg d)180 deg

15.At 3:40 the hour hand and the minute hand of a clock
form an angle of
a)120 b)125 c)130 d)135

16.The angle between the minute hand and the hour hand of a
clock when the time is
8:30 is
a)80 b)75 c)60 d)105

17.The angle between the minute hand and the hour hand of a
clock when the time is
4:20 is
a)0 b)10 c)5 d)20

18.At what angle the hands of a clock are inclined at 15
minutes past 5?
a)58 1/2 b)64 c)67 1/2 d)72 1/2

19.The reflex angle between the hands of a clock at 10:25 is
a)180 b)192 1/2 c)195 d)197 1/2

20.How many times do the hands of a clock coincide in a day?
a)20 b)21 c)22 d)24

21.How many times do the hands of a clock are straight?
a)22 b)24 c)44 d)48

22.How many times are theh hands of a clock at right angle
in a day?
a)22 b)24 c)44 d)48

23.How many times in a day,are the hands of a clock in
straight line but opposite in direction?
a)20 b)22 c)24 d)48

24.How much does a watch lose per day,if its hands coincide
every 60 minutes?
a)32 8/11 b)36 5/11 c)90 min d)96 min

25.At what time,in minutes , between 3'o clock and 4'o clock ,
both the needles will coincide each other?
a)5 1/11 b)12 4/11 c)13 4/11 d)16 4/11

26.At what time between 9 and 10'o clock will the hands of a
watch be together?
a)45 min past 9 b)50 min past 9
c)49 1/11past9 d)48 2/11 past 9

27.At what time between 7 and 8'o clock will the hands of a
clock be in the same straight line but,not together?
a)5 min.past 7 b)5 2/11 past 7
c)5 3/11 past 7 d)5 5/11 past 7

28.At what time between 4 and 5'o clock will the handle of
a watch point to opposite directions?
a)45 min past 4 b)40 min past 4
c)50 4/11 past 4 d)54 6/11 min past4

29.At what time between 5:30 and 6 will the hands of a
clock be at right angles?
a)43 5/11 b)43 7/11 past 5
c)cannot be determined d)none

30.A watch which gains uniformly in 2 min low at noon on monday
and is 4 min.48sec fast at 2 pp.m.on the following monday.
when was it correct?
a)2 p.m on tuesday
b)2 p.m on wednesday
c)3 p.m on thursday
d)1 p.m on friday

31.A watch which gains 5 seconds in 3 minutes was set right at
7 a.m.In the afternoon of the same day,when the watch
indicated quarter past 4'o clock,the true time is
a)59 7/12 past 3 b)4 p.m c)5 p.m d)7 p.m

32.A clock is set right at 8 a.m.The clock gains 10 minutes
in 24 hours.What will be the true time whne the clock
indicates 1 p.m on the following day?
a)48 min past 12 b)48 min c)47 min past 4 d)none

33.A clock is set right at 5 a.m.The clock loses 16 min
in 24 hours.What will be the true time when the clock
indicates 10 p.m on 4th day?
a)12 p.m b)1 p.m c)1 a.m d)11 p.m

34.The minute hand of a clock overtakes the hour hand at
intervals of 65 min of the correct time.How much a day
does the clock gain or lose?
a)10 10/43 in 24 hrs b)10 10/23 in 24 hrs
c)10 10/34 in 24 hrs d)none

35.Find the angle between the hour hand and the minute hand of
a clock when the time is 3:25
a)42 b)45 c)47 1/2 d)73

36.At what time between 2 and 3'o clock will the hands of a
clock be together?
a)10 10/11 min past 2 b)10 10/11 past 3
c)10 10/11 past4 d)none

37.At what time between 4 and 5'o clock will the hands of a
clock be at right angle?
a) 5 5/11 min past 4 b)38 2/11 min past 4
c)both a&b d)none

38.At what time between 4 and 5'o clock will the hands of a
clock be at right angle?
a) 5 5/11 min past 4 b)38 2/11 min past 4
c)both a&b d)none

39.Find at what time between 8 and 9'o clock will the hands
of a clock be in the same straight line but not together
a)10 10/11 min past 8 b)10 10/11 past7
c)10 10/11past2 d)none

40.At what time between 5 and 6'o clock are the hands of a
clock 3 minutes apart?
a)31 5/11 min past 5
b)31 5/18 min past 5
c)31 5/19 min past 5
d)31 5/16 min past 5

General Concepts:

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.

Important points:

a) In every 60 minutes, the minute hand gains 55 minutes on the
hour hand

b)In every hour, both the hands coincide once ,i.e 0 degrees.

c)the hands are in the same straight line when they are coincident
or opposite to each other. i.e 0 degrees or 180 degrees.

d)when the two hands are at right angles, they are 15 minute spaces
apart,i.e 90 degrees.

e)when the hands are in the opposite directions,they are 30 minute
spaces apart,i.e 180 degrees.

f)Angle traced by hour hand in 12hrs = 360 degrees.

g)Angle traced by minute hand in 60 min = 360 degrees. If a watch
or a clock indicated 8.15,when the correct time is 8, it 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 slow.

h)60 min --> 360 degrees
1 min --> 60

i)the hands of a clock coincide in a day or 24 hours is 22 times,
in 12hours 11minutes.

j)the hands of clock are straight in a day is 44 times .

k)the hands of a clock at right angle in a day is 44 times .

l)the hands of a clock in straight line but opposite in direction is
22 times per day