hour hand and the minute hand of a clock move in relation to each other
continuously and at any given point of time, they make an angle between 0° and
180° with each other.
the time shown by the clock is known, the angle between the hands can be
calculated. Similarly, if the angle between two hands is known, the time shown
by the clock can be found out.
we say angle between the hands, we normally refer to the acute/obtuse angles
(upto 180°) between the two hands and not the reflex angle (> 180°).
solving the problems on clocks, the following points will be helpful.
hand covers 360° in 1 hour, i.e., in 60 minutes. Hence, MINUTE HAND COVERS 6°
hand covers 360° in 12 hours. Hence, HOUR HAND COVERS 30° PER HOUR.i.e., 1/2°
following additional points also should be remembered. In a period of 12 hours,
the hands make an angle of.
- 0° with each other (i.e., they coincide with each other),
- 180° with each other (i.e., they lie on the same straight
line), 11 times.
- 90° or any other angle with each other,22 times. .
We can also solve the problems on clocks using the method
of "Relative Velocity".
In 1 minute, Minute hand covers 6° and Hour hand covers
Relative Velocity = 6 – 1/2 = 5 ½,° per minute. Alternately, in 1 hour, the
minute hand covers 60 minute divisions whereas the hour hand covers 5 minute
Relative Speed = 60 – 5 =
55 minutes per hour.
adopting the approach of actual angles covered is by far the simplest and does
not create any confusion.
Points to Note :
- Any angle is made 22 times in a period of 12 hours.
- In a
period of 12 hours, there are 11 coincidences of the two hands, when the two
hands are in a straight line facing opposite directions.
- The time gap between any two coincidences is 12/11 hours
or 653/1 minutes.
- If the hands of a clock (which do not show the correct
time) coincide every p minutes, then
> 655/11, then the watch is going slow or
losing time. If p < 655/11, then the watch
is going fast or gaining time.
calculate the angle '9' between the hands of a clock, we use the following
formula (where m = minutes and h = hours)
Worked out examples :
is the angle between the minute hand and the hour hand of a clock at 4 hours 30
Where 9 = angle
m = minutes
h = hours
Here, m = 30 and h = 4
0=165 – 1209 = 45°
The angle between the two hands is 45°. Choice (c).
what time between 4 and 5 O'clock will the minute hand and the hour hand make
an angle of 30° with each other?
– 30h, here = 30°> h = 4, m = ?
11m = 60h + 29
min past 4.
11m = 60h –
min past 4 O clock
Therefore, the angle between the two hands is 30° when
the time is 4 hours min
and 4 hours yrmin.
what time between 4 and 5 O'clock will the minute hand and the hour hand
coincide with each other?
When the two hands coincide with each other the angle between
them is 0°.
h = 4 m =
min past 4.
The two hands of the clock coincide at 4 hours min.
what time between 4 and 5 O'clock will the minute hand and the hour hand are
perpendicular to each other?
the two hands of the clock are perpendicular to each other then the angle
between them is 90°.
rmin past 4.
min past 4.
the two hands of the clock are perpendicular to each other at 4 hours prmin and at 4 hours min.
time between 4 and 5 O'clock will the minute hand and the hour hand are on a
straight line but facing opposite directions?
the two hands are on a straight line but facing opposite directions then the
angle between them is 180°.
– m – 30h, where 9 = 180°, h
11m = 60h + 2m =
past 4 O’ clock.
two hands of a clock are on the same straight line but facing opposite
directions at 4 hours min.
At what time between 5 and 6 O'clock, will the hands of a
clock be at an angle of 62°?
(a) 5 hours 172/11 minutes
(b) 5 hours 386/11 minutes
(c) 5 hours 16 minutes
(d) Both (b) and (c)
= 11/2m – 30h
11/2m = +
11 m = 2(62 + 30 x 5)
min past 5 O'clock the two
hands of the clock are 62° apart.
= 30h – 11/2m 11/2m = 30h –
m = 2(30x5 – 62) m = 176/11 = At 16 minutes
past 5 O'clock, the two hands of the clock are 62° apart
clock is set to show the correct time at 10 a.m. The clock uniformly loses 12
min in a day. What will be the actual time when the clock shows 5 p.m. on the
next day? .
4 : 25 p.m. .
4 : 45 p.m.
(c) 5 : 15 p.m. .
(d) 4 : 50 p.m.
from 10 : 00 a.m. a day to 5 : 00 p.m. the next day = 31 hours 23 hours 48
minutes of this clock = 24 hours of the correct clock.
of this clock = 24 hours of
the correct clock.
31 hours of this clock
= 31 hrs 15 min (Approx.).
The correct time is 31 hours 15 minutes after 10 : 00
= 5 hours 15 minutes. .
At what time between 6 O'clock and 7 O'clock, are the
hands of a clock together?
(a) 6hrs. 087/11min.
(b) 6hrs. 328/11 min.
(c) 6hrs. 368/11min.
(d) 6hrs. 489/11 min.
30 x 6
minute hand of a clock overtakes the hour – hand at intervals of 66 minutes of
the correct time. How much in a day does the clock gain or lose?.
(a) 10113/131 minutes
(b) 11115/121 minutes
correct clock, the hands of a clock coincide every minutes. But in this case
they are together again after 66 minutes, hence clock loses time.
Loss in 66 min minutes.
Loss in 24 hours =
= 1440/121 = H109/121
The clock loses 11109/121 minutes
in 24 hours. Choice (c).
minute hand of a clock overtakes the hour – hand al intervals of 62 minutes of
a correct time. How much in a day does the clock gain or lose?.
(a) 8080/141 minutes
(b) 85090/311 minutes
(c) 8070/341 minutes
(d) 8060/341 minutes
correct clock, the hands of a clock coincide every 657 minutes. But in this
case both the hands are together again after 62 minutes, hence the clock gains
time. Gain in 62 minutes = (655/11 – 62) = 3 5/11 min
gain. Gain in 24 hours = 38/11 × 60 × 24 / 62.
So the clock gains 80%, minutes in 24 hours.