Clock maths
Clocks maths problems
The angle between the minute hand and hour hand of a clock Part-I
Concepts of clockA. Every Clock has three hands:-
1. Second hand, 2. The minute hand, 3. Hour hand
All problems in every competitive exam asked about only those of three hands.
B. Formulae to find out
· Normal Angle ( i.e. The angle between the minute hand and hour hand of a clock )= 30H-11/2XM or 11/2XM-30H
(This formula is a universal formula to find out the angle formed by the hands of a clock )
· Reflex angle ( i.e. the angle between the minute hand and hour hand ) = 3600 –Normal angle.
· ( Reflex angle is always greater than 1800 .)
C. Short tricks to find out the angle between the hands of a clock-
1. If the ratio between the hour hand and the minute hand is 1:3 then angle=9M/2.
2. If the ratio between the hour hand and the minute hand is 1:4 then angle=MX2
3. If the ratio between the hour hand and the minute hand is 1:5or 1:6 then angle=M/2
4. If the ratio between the hour hand and the minute hand is 1:8 then angle=(M-H)X2
5. If the ratio between the hour hand and the minute hand is 1:10 then angle=5M/2
Where M→ Minute. H → Hour
6. Angle trace/move/create by Minute hand in 1 minute= 60 .
15 minures =900
30 mimutes=1800
60 minutes=3600
7. Angle trace /move/create by Hour hand in 60 mimutes=300
1 minute= 0.50
8. Hands of Clocks become right angle (900) 2 times every 1 hour, that is in 1st time and 2nd time ( except 3 O'Clock and 9 O'Clock, in 3 & 9 o’clock they form only 1 time 900 ). i.e. every 12 hrs right angle form 22 times and in 24 hrs right angle form 44 times.
D. Coincides of hands of a Clock
v Coincide or together or straight in the same direction:-
· Normally, in every hour both minutes hand and hours hand clocks, coincide one time but between 12 noon/ midnight and 1AM/PM they never get together or coincide.
Every 12 hours hands of a clock coincide 11 times (except 12 to 1 hour).
In a complete day, there are 24 hours and so in the 24-hour hands of
a clock coincides a total of 22 times.
v Coincide or together or straight in opposite direction:
· In every hour both minutes hand and hours hand become straight 1 time in the opposite direction but from 6 o’clock to 7 o’clock they never get straight in opposite direction.
So, in a complete day, ( in 24 hrs ) the hands of a clock become straight in opposite direction 22 times.
How to find the angle between the hands of a clock-
Q.1 What is the angle between the minute hand and hour hand of a clock at 8: 30?
a. 1000 b. 200
c. 1360 d. 750
Trick Here H=8, M=30
Angle =30H-(11/2) x M ( universal formula ) OR Angle = (11/2)x M-30H
= 30X8-(11/2)x30 =(11/2)x 30-30x8
= 240-330/2 =330/2-240
=240-165 =165-240
=75 =-750
* Both are the same in degree measurement, i.e. 750 . {i.e. + or-}
Q.2 What is the angle between the hour hand and the minute hand of a clock at 4: 30?
a. 800 b. 1360
c.100 d. 450
Trick Here H=4, M=30
Angle =30H-(11/2) x M ( universal formula)
=30x4-(11/2)x30
=120- 330/2
=120- 165
=450.
Q.3 What is the angle between the hands of a clock at 6: 18?
a. 900 b. 320
c. 810 d. 800
Trick Here H=6, M=18
Angle =30H-(11/2) x M ( universal formula)
=30x6-(11/2)x18
=180- 198/2
=180- 99
=810.
Short trick: Ratio of Hour hand and minute hand, H: M=6:18=1:3
Angle =9M/2=9x18/2=810.
Q.4 What is the angle between the hands of a clock at 3: 30?
a. 900 b. 320
c. 810 d. 750
Trick Here H=3, M=30
Angle =30H-(11/2) x M ( universal formula)
=30x3-(11/2)x30
=90- 330/2
=90- 165
=750.
Short trick: Ratio of Hour hand and minute hand, H: M=3:30=1:10
Angle =5M/2=5x30/2=750 .
Q.5 What is the angle between the hour hand and the minute hand of a clock at 9: 54?
a. 270 b. 1360
c.1000 d. 450
Trick Here H=9, M=54
Angle =30H-(11/2) x M ( universal formula)
=30x9-(11/2)x54
=270- 594/2
=270- 297
=-270.
= 270
Short trick: Ratio of Hour hand and minute hand, H: M=9:54=1:6
Q.6 What is the angle between the hour hand and the minute hand of a clock at 5: 25?
a. 270 b. 1360
c.1000 d. 450
Trick Here H=5, M=25
Angle =30H-(11/2) x M ( universal formula)
=30x5-(11/2)x25
=150- 275/2
=150- 137.5
= 12.50
Short trick: Ratio of Hour hand and minute hand, H:M=5:25=1:5
Angle =M/2=25/2=12.50 .
Reflex Angle Problems:-
Q.1 The reflex angle between the hands of a clock at 6.30 will be-
a.450 b. 150
c.1790 d. 3450
Short trick: As we know the reflex angle is always greater than 1800. So, the answer will be d.3450 because all other option is less than 1800.
Or.
By formula-
Reflex angle = 3600- Normal angle Here H=6 ,M=30
= 3600- {30H-(11/2)xM}
=3600- {30x6-(11/2)x30}
=3600- {180- 330/2}
=3600- {180- 165}
=.3600- 150
=3450
a.330 b.254
c. 260 d. None
Trick Here H=7, M=30
Reflex angle = 3600- Normal angle
=3600- {30x7-(11/2)x20}
=3600- {210- 220/2}
=3600- {210- 110}
=.3600- 1000
=2600.
Q.3 The reflex angle between the hour hand and the minute hand of a clock at 10:25 is-
a. 1800 b. 192½0
c. 1950 d. 197½0
Trick Here H=10, M=25
Reflex angle = 3600- Normal angle
=3600- {30x10-(11/2)x25}
=3600- {300- 275/2}
=3600- {300- 137.5}
=3600- 162.50
=197.50
= 197½
Q.4 What angle is made by minute hand in 21 sec.?
a. 17000 b. 290
c. 2600 d. 2.10
Short trick: Angle made by minute hand in S sec= (1/10) Xs
So, Angle made by the minute hand in 21 sec = (1/10) x21
= 21/10
= 2.10
Q.5 What angle is made by minute hand in 90 sec.?
a. 90 b. 9.010
c.900 d. 2.10
Short trick: Angle made by minute hand in S sec= (1/10) Xs
So, Angle made by the minute hand in 90 sec = (1/10) x90
= 90/10
=90
Q.6 What angle is made by the hour hand in 36 sec.?
a. 90 b. (3/10)0
c.900 d. 2.10
Short trick: Angle made by hour hand in S sec= (1/120) xS
So, Angle made by hour hand in 36 sec = (1/120) x36
= 36/120
= (3/10)0
Q.7 In a clock time shows 10: 30. Accordingly, the minute hand is at East, then in which direction is the hour hand?
a. North-West
b. South-East.
c. North-East.
