Chapter 29 - Clock Problems
900
900
The distribution of minutes and hours and the degree of angle on a clock face is
represented by the above two diagrams. A clock face or dial is divided into 12 hours and 60
minutes. There are 59 marks on the clock, with 60 spaces between them. They represent every
minute of an hour. There are also 12 hour marks that show the hours, represented by the
numerals from 1 to 12. The entire clock face is 36 . 36 divided by 60 is
and hence
every minute is represented by .
A clock has two hands. The smaller one is called the hour hand or short hand and the
larger one is called the minute hand or the long hand. (Problems involving second hand are not
asked for examinations)
In one hour, the minute hand covers 36 and the hour hand 5 x 6 = 3 . So in one hour,,
the minute hand gains 33 over the hour hand. In one hour, the minute hand gains 55 minutes
over the hour hand. In one minute, the minute hand covers
So in one minute, the minute hand gains
and the hour hand
.
over the hour hand.
The hands are in the same straight line when they are coincident or opposite to each
other. Between every two hours, except between (11 and 12) and (12 and 1), the minute hand
and the hour hand coincides only once. Between (11 and 12) and (12 and 1), they coincide only
when the time is 12. So in twelve hours the minute hand and the hour hand coincide 11 times
only and so in a day they coincide 22 times.
Clock problems can be broadly classified into four types as follows;
1. What is the angle between the hour hand and the minute hand at a given point of time.
2. After a given time, when will the hour hand and the minute hand of the clocks will be;
(a) Coincident (b) Opposite to each other (c) at right angles.
3. For a given time, what will the mirror image appear as.
4. When is the correct time shown by a malfunctioning clock.
Excel Speed Maths - Chapter 29 - Clock Problems
554
Exercise :
1.
What will be the angle between the hour hand and the minute hand of a clock at 4.40 hours?
Conventional method : 4.40 hours or
In
So in
hours or
hours.
hours, the degrees covered by the hour hand : In one hour, the hour hand covers 3 .
hours, the hour hand covers
x 30 = 14 .
In 40 minutes, the degrees covered by the minute hand : In one minute, the minute hand covers
. So in 40 minutes, the minute hand covers 6 x 40 = 24 .
So the hour hand is at 14 and the minute hand is at 24 and so the angle between the hour
hand and the minute hand of the clock = 240 - 140 =
.
This is the method being highly practised by the student community. For conceptual
understanding we may study this method but for examination will use a direct formula as this
method is time consuming and prone for mistakes. The direct formula is based on the positioning
of the Hour Hand and the Minute Hand and for this positioning purpose as per the formula, we
will be using the numerals from 1 to 12.
Direct Formula
1.
When the Hour Hand (HH) is ahead of the Minute Hand (MH) :
Angle = (HH - MH) x 30 +
2.
.
When the Minute Hand (MH) is ahead of the Hour Hand (HH) :
Angle = (MH - HH) x 30 -
.
So in this problem, we know that, when it is 4.40, the minute hand is ahead of the hour hand and
the formula is, (MH - HH) x 30 -
= (8 - 4) x 30 -
= 4 x 30 - 20 =
. {Position of
40 minutes is represented by the numeral 8 as 8 is }. The direct formula is very short and
simple and the answer can be derived even without any arithmetical operations.
2.
What will be the angle between the hour hand and the minute hand of a clock at 9.35 hours?
Hour Hand at 9 and Minute Hand at 7 { i.e. } .
Hour Hand is ahead of the Minute Hand and the angle between the Hour Hand and the Minute
Hand of the clock = (HH - MH) x 30 +
3.
= 60 + 17.5 =
.
What will be the angle between the hour hand and the minute hand of a clock at 11.45 hours?
HH=11. MH=
4.
= (9 - 7) x 30 +
= 9. Angle= (HH - MH) x 30 +
= (11 - 9) x 30 +
= 60+22.5 =
.
What will be the angle between the hour hand and the minute hand of a clock at 10.20 hours?
HH = 10. MH =
= 4. Angle = (HH - MH) x 30 +
= (10 - 4) x 30 +
= 180 + 10= 19 .
Now, 19 is more than 18 and is a reflex angle. If 19 is not given in the answer options,
then to get the answer substract 19 from 36 i.e.17 .
5.
What will be the angle between the hour hand and the minute hand of a clock at 12.30 hours?
Excel Speed Maths - Chapter 29 - Clock Problems
555