The 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.

If 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.

When 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°).

For solving the problems on clocks, the following points will be helpful.

- Minute hand covers 360° in 1 hour, i.e., in 60 minutes. Hence, MINUTE HAND COVERS 6° PER MINUTE.
- Hour hand covers 360° in 12 hours. Hence, HOUR HAND COVERS 30° PER HOUR.i.e., 1/2° per minute.
- The 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), 11 times.
- 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. .

**Note : **

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 1/2°.

Therefore, 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 divisions.

Relative Speed = 60 – 5 = 55 minutes per hour.

However, 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 65
^{3}/_{1}minutes. - If the hands of a clock (which do not show the correct time) coincide every p minutes, then

If p
> 65^{5}/_{11},_{ }then the watch is going slow or
losing time. If p < 65^{5}/_{11},_{ }then the watch
is going fast or gaining time.

To calculate the angle '9' between the hands of a clock, we use the following formula (where m = minutes and h = hours)

(i)

(ii)

What is the angle between the minute hand and the hour hand of a clock at 4 hours 30 minutes?

(a) 15°

(b) 30°

(c) 45°

(d) 60°

We have

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).

At 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.

At 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°.

.

Here = 0°,

h = 4 m = 30h

min past 4.

The two hands of the clock coincide at 4 hours min.

At what time between 4 and 5 O'clock will the minute hand and the hour hand are perpendicular to each other?

When the two hands of the clock are perpendicular to each other then the angle between them is 90°.

rmin past 4.

min past 4.

Therefore, the two hands of the clock are perpendicular to each other at 4 hours prmin and at 4 hours min.

At what time between 4 and 5 O'clock will the minute hand and the hour hand are on a straight line but facing opposite directions?

When 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 = 4.

11m = 60h + 2m =

past 4 O’ clock.

The 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 17^{2}/^{11} minutes

(b) 5 hours 38^{6}/^{11} minutes

(c) 5 hours 16 minutes

(d) Both (b) and (c)

= 11/2m – 30h

11/2m = + 30h

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

Choice (d)

A 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? .

(a) 4 : 25 p.m. .

(b) 4 : 45 p.m.

(c) 5 : 15 p.m. .

(d) 4 : 50 p.m.

Time 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 a.m.

= 5 hours 15 minutes. .

Choice (c)

At what time between 6 O'clock and 7 O'clock, are the hands of a clock together?

(a) 6hrs. 08^{7}/_{11}min.

(b) 6hrs. 328/^{11} min.

(c) 6hrs. 36^{8}/_{11}min.

(d) 6hrs. 489/_{11} min.

30 x 6

Choice (b)

The 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) 10^{113}/_{131} minutes

(b) 11^{115}/_{121} minutes

(c)
11^{109}/_{121} minutes

(d)
10^{104}/_{121} minutes

In a 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 = H^{109}/_{121}

The clock loses 11^{109}/_{121} minutes
in 24 hours. Choice (c).

The 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) 80^{80}/_{141} minutes

(b) 850^{90}/_{311} minutes

(c) 80^{70}/_{341} minutes

(d) 80^{60}/_{341} minutes

In a
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 = (65^{5}/_{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.

Choice (a)