Shortcut -Clock
Angle between the hour hand and the minute hand.
Angle=$[30H+\dfrac{M}{2}]-6M$
H - Hour value
M - Minute value
Question:
What is the angle between minute hand and hour hand at 05:40?
Answer: H=5
M = 40
Angle = [30(5) + (40/2)] ~ 6(40)
= 170 - 240
= 70`
Note:
If the value is above 180`, subtract it from 360 to get refract angle.
Shortcut -Clock
Time at which the two hands overlap.
Time =$\dfrac{12}{11}5H$
H - Least hour between ‘H’ and ‘H+1’ hour
Question:
At what time between 3 o’ clock and 4 o’ clock the two hands of the clock will overlap?
Answer: Time = [12/11][5(3)]
= 180/11 minutes past 3 ‘o’ clock
= 16.36 minutes past 3 ‘o’ clock
= 16 minutes 22 seconds past 3
= 03 : 16 : 22
Question:
At what time between 10 o’ clock and 11 o’ clock the two hands of the clock will overlap?
Answer: Time = [12/11][5(10)]
= 600/11 minutes past 10 ‘o’ clock
= 54.54 minutes past 10 ‘o’ clock
= 54 minutes 32 seconds past 10
= 10 : 54 : 32
Shortcut -Clock
Time at which the two hands face opposite directions.
Case 1: H < 6
Time=$\dfrac{12}{11}[5H+30]$
Question:
At what time between 1 o’ clock and 2 o’ clock the two hands will be facing opposite direction?
Answer: Time = 12/11[5(1) + (180/6)]
= 12/11[5(1) + 30]
= 420/11 minutes past 1 ‘o’ clock
= 38.18 minutes past 1 ‘o’ clock
= 38 minutes 11 seconds past 1
= 01 : 38 : 11
Case 2: H > 6
Time=$\dfrac{12}{11}[5H-30]$
Question:
At what time between 9 o’ clock and 10 o’ clock the two hands will be facing opposite
Answer: Time = 12/11[5(9) - (180/6)]
= 12/11[5(9) - 30]
= 180/11 minutes past 9 ‘o’ clock
= 16.36 minutes past 9 ‘o’ clock
= 16 minutes 22 seconds past 9
= 09 : 16 : 22
Shortcut -Clock
Time at which the two hands will be certain angle apart.
$Time=\dfrac{12}{11}[5H\pm \dfrac{\theta}{6}]$
There are two different time at which the hands will be certain degrees apart.
$T_{1}$ and $T_{2}$.
Question:
At what time between 4 o’ clock and 5 o’ clock the two hands of the clock will be 60 degrees apart?
Answer: $T_{1}= 12/11[5(4) + (60/6)]$
= 360/11 minutes past 4 ‘o’ clock
= 32.72 minutes past 4 ‘o’ clock
= 32 minutes 44 seconds past 4
= 04 : 32 : 44
$T_{2} = 12/11[5(4) – (60/6)]$
= 120/11 minutes past 4 ‘o’ clock
= 10.9 minutes past 4 ‘o’ clock
= 10 minutes 54 seconds past 4
= 04 : 10 : 54
Shortcut -Clock
Time at which the two hands will be certain minute spaces apart.
$Time=\dfrac{12}{11}[5H\pm M]$
There are two different time at which the hands will be certain degrees apart.
$T_{1}$and $T_{2}$.
Question:
At what time between 7 o’ clock and 8 o’ clock the two hands of the clock will be 15 minutes apart?
Answer: $T_{1} $ = 12/11[5(7) + 15]
= 600/11 minutes past 7 ‘o’ clock
= 54.54 minutes past 7 ‘o’ clock
= 54 minutes 33 seconds past 7
= 07 : 54 : 33
$T_{2} $ = 12/11[5(7) – 15]
= 240/11 minutes past 7 ‘o’ clock
= 21.81 minutes past 7 ‘o’ clock
= 21 minutes 48 seconds past 7
= 07 : 21 : 48