Problems on Clocks

General Concepts
The face or dail 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 hand is called the minute hand or long hand.

In 60 minutes , the minute hand gains 55 minutes on the hour hand.
In every hour , both the hands coincide once.
The hands are in the same straight line when they are coincident or opposite to each other
When the two hands are at right angles, they are 15 minute spaces apart.
When the hands are in opposite directions, they are 30 minute spaces apart.

Too Fast and Too Slow :If a watch or a clock indicates 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 too slow.

Solved Problems

1. Find the angle between the minute hand and hour hand of a clock when the time is 7.20

Sol : Angle traced by hour hand in 12 hours = 360 Degrees

Angle traced by it in 7 hrs 20 min i.e 22/3 hrs = (360/12) x (22/3)= 220⁰

Angle traced by minute hand in 60 min = 360⁰

Angle traced by it in 20 minutes = (360/60) x 20 = 120⁰

Required angle = (220⁰ - 120⁰) = 100⁰

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

Sol : At o' clock, the hour hand is at 2 and minute hand at 12, i.e they are 10 minute spaces apart.

To be together , the minute hand must gain 10 minutes over the hour hand.

Now, 55 minutes are gaines by it in 60 minutes

10 minutes will be gained in {(60/55) x 10} min =10 10/11 min

The hands will coincide at 10 10/11 min past 2

3. At what time between 4 and 5 o' clock will the hands of a clock be at right angle ?

Sol : At 4 o' clock, the minute hand will be 20 min spaces behind the hour hand.

Now , when the two hands are at right angles , they are 15 min. spaces apart.

So, they are at right angles in following two cases

Case I : When minute hand is 15 minute spaces behind the hour hand

In this case min hand will have to gain (20 -15) = 5 minute spaces

55 min. spaces will be gained by it in (60 x 5)/55 min = 5 5/11 min

They are at right angles at 5 5/1 min past 4

Case II : When the minute hand is 15 minute spaces ahead of the hour hand

They are at right angles at 38 2/11 min past 4

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