MATHEMATICS

Part - 1

Standard
VI

Government of Kerala
Department of General Education

Prepared by
State Council of Educational Research and Training (SCERT) Kerala
2025
 THE NATIONAL ANTHEM
Jana-gana-mana adhinayaka, jaya he
Bharatha-bhagya-vidhata
Punjab-Sindh-Gujarat-Maratha
Dravida-Utkala-Banga
Vindhya-Himachala-Yamuna-Ganga
Uchchala-Jaladhi-taranga
Tava subha name jage,
Tava subha asisa mage,
Gahe tava jaya gatha
Jana-gana-mangala-dayaka jaya he
Bharatha-bhagya-vidhata
Jaya he, jaya he, jaya he,
Jaya jaya jaya, jaya he.

PLEDGE
India is my country. All Indians are my brothers and
sisters.
I love my country, and I am proud of its rich and varied
heritage. I shall always strive to be worthy of it.
I shall give my parents, teachers and all
elders, respect and treat everyone with courtesy.
To my country and my people, I pledge my devotion. In
their well-being and prosperity alone, lies my happiness.

Mathematics
6
State Council of Educational Research and Training (SCERT)
Poojappura, Thiruvananthapuram 695012, Kerala
Website : www.scertkerala.gov.in
e-mail : scertkerala@gmail.com, Phone : 0471 - 2341883
Typesetting and Layout : SCERT
First Edition : 2025
Printed at : KBPS, Kakkanad, Kochi-30
© Department of General Education, Government of Kerala
Dear children,

We have already learned about different dimensions of

Mathematics. We, also, have gained a clear understanding
of counting numbers, various operations, basic concepts
of geometic shapes, number relations, fractions and
decimal forms in previous classes. The content of this
book will help us to apply the concept we have learned
and create new ideas. These lessons will guide us to
increase our Mathematical skills, solve problems logically
and understand Mathematics based on observations and
conclusions. Let's confidently enter into the fascinating
and curious world of Mathematics.
With best wishes,

Dr. Jayaprakash R.K.

Director,
SCERT, Kerala
 Textbook Development Team
Advisor Chairperson
Dr. E. Krishnan C. Venugopal
Head (Retd.), Assistant Professor (Retd.),
Department of Mathematics, IASE, Thrissur
University College, Thiruvananthapuram

Experts
Dr. Saleemudheen K. Dr. Shikhi M.
Principal (Full Additional Charge), Professor,
DIET Malappuram College of Engineering Trivandrum (CET),
Thiruvananthapuram
Members

Kunhabdhulla M. Thulaseedharan Pillai K.G. Suresh Babu T.

Headmaster (Retd.), Headmaster (Retd.), LPST (Retd.),
Muyippoth MUP School, GLPS Anchal, Kollam UPGS Punukkonnoor, Kollam
Kozhikkode
Fathima K. Nisha C. Sukanya P.
UPST, Lecturer, UPST, Ramakrishna Mission
GVHSS Nellikuthu, DIET, Malappuram Higher Secondary School,
Malappuram Meenchanda, Kozhikode
Satheesh U.K. Sandeep Mattada Ullas U.G. (Artist),
PD Teacher, UPST, AHSS, Paral, HST, CBHSS Vallikunnu,
GUPS Pathappiriyam, Mambattumoola, Malappuram Malappuram
Malappuram
Anoop K.P.
HST Drawing, MSAPTHSS Kakkovu,
Vazhur, Malappuram

Academic Coordinator
Dr. Abhilash Babu P.
Research Officer (Evaluation) &
Coordinator (State Assessment Cell), SCERT

State Council of Educational Research and Training (SCERT)

Vidhyabhavan, Poojappura, Thiruvananthapuram 695 012
 CONTENTS
1 Angles 7

2 One Fraction Many Forms 19

3 Volume 33

4 Arithmetic of Parts 47

5 Decimal Forms 67

6 Multiples and Factors 83

The icon below is used in this

textbook for convenience

Let's do the problems

 1
Angles

Angles
When lines join
Remember how we drew various shapes joining lines straight up and slanted?
(The section Line math of the lesson Lines and Circles in the class 5 textbook).

Each of these figures has four corners; and at each corner, two lines meet.
Let's look at these different pairs of lines in the first figure:

What about the pairs in the second figure?

A figure formed by two lines meeting at a point is called an angle.

Thus each of the three figures seen first has four angles. Now look at the
bottom-left corners of these:

7
Standard VI - Mathematics

The lines in the middle figure are a bit more spread out than the lines in the right figure,
isn't it?
In the left figure, the lines are spread out even more; and the top line is straight up.
So, we can say that the angle on the right is the smallest, the middle one larger and the
left one the largest.
See these pictures:

Which is the larger angle?

The lines of the angle on the right are more spread out; and so it is the larger angle.
What about these pictures?

Difficult to say, isn't it?

To compare these, we must be able
to measure angles exactly. There is an
instrument in your geometry box just for
this.

8
 Angles

It is called a protractor.

See the several lines on it.

At the end of each line, there are two numbers,

one below the other.
Look at the bottom numbers:

Starting with the bottom line marked 0, there are other lines upward; and as they move
up, the angles between them and the bottom line become larger and larger. The numbers
at their ends show the sizes of these angles.
The number at the end of the first line is 10, right? We say that the angle between this
line and the bottom line is 10 degrees and write 10°. And the angle between the bottom
line and the line marked 40 is 40°:

In other words, an angle of 40° is 4 angles of 10° stacked up:

We can draw an angle like this also:

9
Standard VI - Mathematics

It is to draw and measure angles to the left, that the protractor

has another round of numbers on top.

Now let's see how we measure angles using a protractor. For example, see these angles:

Place the protractor at the corner of each angle, as shown below:

Don't you see that the angle on the left is 50° and the angle on the right is 60°? We mark
them as shown below:

10
 Angles

Now let's look at the angles we couldn't compare earlier:

വലതുുകോ�ോൺ 60° ആണെ�ന്നുു കാാണാംം. ഇടതുുകോ�ോണോ�ോ ?

We can see that the angle on the right is 60°. What about the angle on the left?
The slanted line of this angle passes between 50 and 60 on the protractor, doesn't it?
Do you see the small lines in the protractor which divide the gaps between the multiples
of 10 into ten equal parts? Each of them shows a difference of 1°.
In our picture, the slanted line goes through the fifth (slightly larger) line between 50
and 60.
This means the angle is 55°.

Now draw a line, place the square corner of the set square at one end and draw a line
(The section Line math of the lesson Lines and Circles in the class 5 textbook).

11
Standard VI - Mathematics

Measure this angle using the protractor:

So, the angle at a square corner is 90°. This angle is also called a right angle.
In pictures, we mark a right angle like this:

A line making an angle of 90° with another line is said to be perpendicular to the first
line. For example in the picture below, both the green lines are perpendicular to the blue
line:

In a rectangle, any pair of sides which meet are perpendicular

to each other:

12
 Angles

Drawing angles
Next let's see how we can draw an angle of specified size
using a protractor.
For example, how do we draw the angle shown in the figure?
First draw a horizontal line:

Then place the protractor at its left end as shown below and mark a point at the number
40 in it.

Now remove the protractor and join this point and the left endpoint of the first line to
get a 40° angle:

(1) Measure and mark each angle below:

13
Standard VI - Mathematics

(2) Draw the pictures below in the notebook:

Circle division
A protractor is a semicircle, that is half a circle, right?

14
 Angles

The small lines in it at the very top divide the semicircle into 180 equal parts. That is, if
these lines are extended to the centre of the semicircle, we get 180 angles of the same
size.
We have already mentioned that the size of each of these angles is 1°.
Thus if in a semicircle we draw 181 equally spaced radii, then the angle between two
nearby radii would be 1°.
This is usually stated in terms of a full circle, instead of a semicircle:
If in a circle 360 equally spaced radii are drawn, then the angle between any two
nearby radii is 1°.
The sides of a 1° angle are very close to each other, right? To see this angle, we will have
to extend the sides quite a bit:

What we have said above can be put the other way round: in a circle, we can draw 360
radii, 1° apart; and they divide the circle into 360 equal parts.
What if we draw radii 10° apart?
We get 36 radii which divide the circle into
36 equal parts:

15
Standard VI - Mathematics

And radii 60° apart?

Since 360 ÷ 60 = 6, we get 6 radii and 6 equal parts of the circle:

If we join the ends of these radii, we get a six-sided figure:

Remember drawing this figure using three circles in class 5? (The section Circle math
of the lesson Lines and Circles).
So, drawing these pictures from the class 5 textbook is now a little easier:

16
 Angles

What if we want to draw a five-sided figure?

What angle should we take?
360° ÷ 5 = 72°.

To get the figure upright without tilting, first draw a horizontal radius and a radius
making an angle of 90° − 72° = 18° with it. Then the radii 72° apart can be drawn:

(1) In each of the pictures below, calculate what fractions of the circle are the
yellow and green parts:

17
Standard VI - Mathematics

(2) Mark each of the fractions below as part of a circle and colour the pictures.

(i) 83 (ii) 52 (iii) 94 5

(iv) 12 5
(v) 24

(3) Draw the pictures below:

18
 2
One Fraction Many Forms

One Fraction Many

Forms
Numerator and denominator
We have seen in class 5, how parts of a measure can be written as fractions:

19
Standard VI - Mathematics

We have also seen that fractions in different forms can be the same measure. For example,
2 1
4 and 2 give the same part:

20
 One Fraction Many Forms

Similarly we have seen 36 is also 12 .

Thus we have
1
= 2
= 3
2 4 6
How do we get different forms of 12 ?

• If an object or a measure is divided into two equal parts, each is 12 of the whole.

• What if each of these two equal parts is again split into two equal parts?

∗ 2 × 2 = 4 equal parts in all.

∗ To get 12 of the whole, we must take 2 of these.

1 = 2
∗ 2 4

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Standard VI - Mathematics

• What if each of the two equal parts is split into three equal parts?

∗ 2 × 3 = 6 in all

∗ To get 12 of the whole, we must take 3 of these.

1 = 3
∗ 2 6

We can continue like this as many times as we want.

For example, suppose we divide each of the two equal parts into 50 equal parts

50 × 2 = 100 equal parts in all.

To get 12 of the whole, how many of these should we take?

What another form of 12 do we get from this?

In the fraction 100

50
the lower number 100 shows the number of equal parts we have
divided the whole into ; and the upper number 50 shows how many of these equal parts
we take.

50 is called the numerator and 100 is called the denominator of the fraction 100
50
.

Now look at the many forms of 12

1 2 3 4
2 , 4 , 6 , 8 , ...

In all these, the denominators 2, 4, 6, 8, ... are multiples of 2; and the numerators 1, 2,
3, 4, ... the numbers by which 2 is multiplied.

Next let's look at 13 .

We have seen that 13 and 62 give the same part.

22
 One Fraction Many Forms

In what other ways can we write 13 ?

What if we split a circle into 9 equal parts? How many parts should we take to get 13 of
the whole?

In the same manner, if we divide 1 metre into 9 equal parts and take three of them, we
get 13 metre.

23
Standard VI - Mathematics

We can continue this also, as in the case of 12 .

• If an object or a measurement is divided into three equal parts, each is 13 of the
whole.
• What if each of these three equal parts is again split into two equal parts?
∗ 2 × 3 = 6 equal parts in all.
∗ To get 13 of the whole, we must take 2 of these.
1 = 2
∗ 3 6
• What if each of the three equal parts is split again into three equal parts?
∗ 3 × 3 = 9 in all
∗ To get 13 of the whole, we must take 3 of these.
1 = 3
∗ 3 9
What if we divide each of the 3 equal parts into 25 equal parts?
How many parts in all ?
How many of them should be taken to get 13 of the whole?
What another form of 13 do we get from this?
Now look at these forms of 13 :
1 2 3 4
3 , 6 , 9 , 12 , ...
In all these, the denominators 3, 6, 9, 12, ... are multiples of 3 ; and the numerators
1, 2, 3, 4, ... are the numbers by which 3 is multiplied.

(1) Find the fractions specified below :

(i) The form of 12 with denominator 24
(ii) The form of 12 with numerator 24
(iii) The form of 13 with denominator 24
(iv) The form of 13 with numerator 24
(v) The form of 14 with numerator 100
(2) Write three different forms of 14 .

24
 One Fraction Many Forms

(3) For each of the pair of fractions below, find three forms with the same
denominator:
(i) 12 , 13 (ii) 12 , 14 (iii) 13 , 14

(4) Does 13 have another form with denominator 10, 100 or 1000 ? Give reasons.

Let's look at another kind of fractions.

How do we find different forms of 32 ?
We get 32 by dividing an object or measure into 3 equal parts and taking 2 of these:

What if we divide each of the three equal parts again into two equal parts?

25
Standard VI - Mathematics

2 × 3 = 6 equal parts in all; and the 2 large equal parts taken earlier now become
2 × 2 = 4 small equal parts.
In other words, as the total number of equal parts is doubled, the number of parts taken
also is doubled.
2 = 4
3 6
This can also continue. What if we divide each of the first 3 equal parts into 3 equal
parts ?
• 3 × 3 = 9 equal parts in all
• The 2 equal parts taken first become 3 × 2 = 6 equal parts.
That is, as the total number of equal parts is tripled, the number of parts taken also is
tripled.
Thus in all the different forms of 32 , the denominators are all multiples of 3; and the
numerators are 2 multiplied by the number by which 3 is multiplied.
We can compute different forms of any fraction like this. In general, we can say this:

Multiplying the numerator and denominator of a fraction by the same

natural number gives a form of the same fraction.

Now can't you fill up the following table, by multiplying the numerator and
denominator by a number?

Fraction Multiplier Numerator Denominator Another form

2 4 8 12 2 = 8
3 3 12

2 6
3

3 12
4

3 12
4

26
 One Fraction Many Forms

Let's look at another kind of problem:

Can we write the fraction 10
15 in another form with a smaller numerator and
denominator?
Both 10 and 15 have 5 as a factor.
10 = 2 × 5 15 = 3 × 5
In other words, we get 10 and 15 by multiplying 2 and 3 by the same number 5. So by
the result above,
10 = 2
15 3
How do we reduce 18
24 like this?
18 and 24 are even numbers; that is, both have 2 as a factor:
18 = 9 × 2 24 = 12 × 2
So, as above,
18 = 9
24 12
Do 9 and 12 have a common factor?
3 is a factor, right? So,
9 = 3
12 4
Thus,
18
= =9 3
24 12 4
Like this, if the numerator and denominator of any fraction have a common factor, then
we can remove this to get another form with smaller numerator and denominator.
In general, we have this result :

If the numerator and denominator of a fraction have a common

factor, then dividing them by this common factor gives a form of this
fraction.

In the second example above, we first divided the numerator and denominator of 18
24 by
the common factor 2 and wrote it as 12
9
; then divided the numerator and denominator
again by 3 to reduce it further and wrote it as 34 .
We can't make the numerator and denominator still smaller, can we? Why?

24 in lowest terms.
is called the form of 18
3
4

27
Standard VI - Mathematics

In general , the form of a fraction in lowest terms is got by removing all common
factors of the numerator and denominator by division.

Write each of the following fractions in lowest terms:

(i) 32
64 (ii) 81
27
(iii) 30
45 (iv) 12
21 (v) 54
45

Fraction as division
If a 2 metres piece of rope is divided into 3 equal pieces, what would be the length of
each piece?
We have seen in class 5 that it is 32 metre (The section Fractional share of the lesson
Fractions).
What if we cut 6 metres long rope into 3 equal pieces?
This we compute as a division:
6÷3=2
We have seen in class 5 that, this too can be written as a fraction (The section Fraction
and division of the lesson Fractions).
That is,
6 ÷ 3 = 63 = 2
Any division can be written as a fraction like this. For example:
8÷2 = 4 2 = 4
8

15 ÷ 3 = 5 3 = 5
15

30 ÷ 10 = 3 10 = 3
30

Since any natural number divided by 1 gives that number itself, we can write all natural
numbers as fractions:
1÷1 = 1 1 = 11
2÷1 = 2 2 = 12
3÷1 = 3 3 = 13

This raises a question: In fractions written as divisions also, can we remove the common
factors of the numerator and denominator?

28
 One Fraction Many Forms

This too can be done, as we have seen in class 5 (the section Common factors of the
lesson Within Numbers).
For example to compute 12 ÷ 6, we first write these numbers as
12 = 6 × 2
6 = 3×2
Then we can remove the common factor 2 and do the division as
12 ÷ 6 = 6 ÷ 3 = 2
These computations can be written in the form of fractions as
=12 6 # 2
= 6 =
6 3 # 2 3 2

So, natural numbers can be written as fractions in various ways :

1 = =
1
1
2
=
2
3 =
3 ...

2 = =
2
1
4
=
2
6 =
3 ...

3 = =
3
1
6
=
2
9 =
3 ...

Again, we have seen in class 5 that divisions which involve remainders can also be
written as fractions (the section New fractions of the lesson Fractions).
For example, if 3 cakes are divided equally between 2 children, each gets a full cake and
half a cake. That is 1  12  1 12 cake. This also we can write as a fraction:
3 ÷ 2 = 32 = 1 12
In such fractions also, can we remove common factors of the numerator and
denominator?
For example, see this problem :
If 6 cakes are divided equally among 4 children, how much would each get?

29
Standard VI - Mathematics

After giving one cake to each , there will be 2 cakes left:

If each of these is cut into half, there would be 4 pieces, each of them 12 of a cake:

If these pieces are also distributed one to each, then what each gets altogether is 1 12
cake:

30
 One Fraction Many Forms

This can be done in another way:

To divide 6 cakes between 4 children, we can split the kids into 2 sets of 2 and give
3 cakes to each set:

In each set, when one cake is given to each of the 2 kids, one cake will be left:

If the remaining one cake in each group is cut into half and a piece given to each of the
two children, then everyone gets half a cake more.

What did we see here?

6
= 3
= 1
4 2 12

31
Standard VI - Mathematics

Let's look at another problem:

If 33 litres of milk is divided equally between 12 persons, how much would each
get?
Here we first split 33 and 12 like this:
33 = 11 × 3 12 = 4 × 3
So we can think like this, as in the previous problem:
Split the 12 persons into 3 groups of 4 and give 11 litres to each group.
When the 4 persons in each group divide the 11 litres equally among themselves, each
person gets 11
4 litres.
That is,
33 = 11
12 4
Now how do we write 11
4 as a whole number and a fraction?
First we divide 11 by 4 and write as multiple and remainder:
11 = (2 × 4) + 3
Dividing the remainder 3 also by 4, we can write 3 ÷ 4 = 34 .
So,
33
= 11
= 3
12 4 24
Thus each gets 2 34 litres.
In general, if we remove common factors of the numerator and denominator of
a fraction, then we get another form of the same fraction.
That is, in writing divisions with remainders as fractions, we can remove common
factors of the numerator and denominator.

(1) 20 litres of water is used to fill 8 identical bottles. How much litres of water is
there in each bottle?
(2) A rope of length 140 centimetres is cut into 16 equal pieces. What is the length
of each piece?
(3) If 215 kilograms of rice is divided equally among 15 people, how much kilogram
of rice would each get?

32
 3
Volume

Volume
Large and small
Athira has collected many things and has arranged them into different lots.

Look at the two things from the first lot.

Which is bigger?
How did you find out?
Now look at the two things from the second lot:

33
Standard VI - Mathematics

How do we find out which is bigger?

To find out the bigger of the two sticks, we need only measure their lengths.
What about two rectangles?
We have to calculate their areas, right?

Rectangular blocks
Look at two wooden blocks from Athira’s collection.
Which is larger?

How did you decide?

Now look at these two.
Which is larger?

Let’s see how we can decide.

34
 Volume

Size of a rectangular block

Look at these rectangular blocks:

They are all made by stacking smaller blocks of the

same size.
Which of them is the largest?
We need only count the little blocks in each, right?
Can you find how many little blocks make up each of
the blocks below?
Is there a quick way to find the number of little blocks How many small cubes are used
in each, without actually counting all? to make this large cube? If one
small block is removed from each
corner of the large block,
how many
would be left?

Which of these is the largest?

And the smallest?

35
Standard VI - Mathematics

Look at these blocks:

How many small blocks are there in each?

Do they have the same size?
To compare sizes by just counting, what kind of little blocks should be used in both?

Size as number
Look at this picture:

What is the area of this rectangle?

How many small squares of side 1 centimetre are
in it? All sides of the
large cube are painted.
4 × 3 = 12 How many small
The area of a square of side 1 centimetre is 1 cubes would have
square centimetre; the area of the whole rectangle is no paint at all?
12 square centimetres.

36
 Volume

Now look at the rectangular block:

It is made by stacking cubes of side 1 centimetre.

How many?
So, the size of this block is equal to 30 such cubes.
Size measured like this is called volume in mathematics.
We say that a cube of side 1 centimetre has a volume of 1 cubic centimetre.
30 such cubes make the large block in the picture.
Its volume is 30 cubic centimetres.

All blocks shown below are made up of cubes of side 1 centimetre.

Calculate the volume of each:

37
Standard VI - Mathematics

Volume calculation
See this rectangular block:

1 cm

5 cm
m
3c

How do we calculate its volume?

For that, we must find out how many cubes of side 1 centimetre we need to make it.

1 cm
5 cm
m
3c

38
 Volume

So, its volume is 15 cubic centimetres.

What about this block?

2 cm

5 cm
m
3c

This can be made by stacking one over another, the two blocks seen first:

1 cm

1 cm
5 cm
m
3c

So, to make it, how many cubes of side 1 centimetre do we need?

Thus the volume of this block is 30 cubic centimetres.

39
Standard VI - Mathematics

Like this, calculate the volume of each of the rectangular blocks shown below and write
it beside each:

3 cm
1 cm

7 cm m
4c 6 cm
m
3c
5 cm

5 cm

5 cm m
5c 5 cm 4c
m

So, now you know how to calculate the volume of a rectangular block, don't you?

The volume of a rectangular block is the product of its length, width

and height.

(1) The length, width and height of a brick are 21 centimetres, 15 centimetres and
7 centimetres. What is its volume?
(2) An iron cube is of side 8 centimetres. What is its volume? 1 cubic centimetre of
iron weighs 8 grams. What is the weight of the this cube?

40
 Volume

Volume and length Area and volume

A wooden block of length 9 centimetres and width
What is the area of a rectangle of
4 centimetres has a volume of 180 cubic centimetres.
What is its height? length 8 centimetres and width
2 centimetres? What about
Volume is the product of length, width and height.
the volume of a rectangular
So in this problem, the product of 9 and 4 multiplied block of length 8 centimetres,
by the height is 180.
width 2 centimetres and height
That is, 36 multiplied by the height gives 180. 1 centimetre?
So to find out the height, we need only divide 180
by 36.

The table shows the measurement of some rectangular blocks. Calculate

the missing measures.
Length Width Height Volume
1 3 cm 8 cm 7 cm ... cc
2 6 cm 4 cm 5 cm ... cc
3 6 cm 4 cm ... cm 48 cc
4 8 cm ... cm 2 cm 48 cc
5 ... cm 2 cm 2 cm 48 cc
6 ... cm 2 cm 4 cm 80 cc
7 14 cm ... cm 5 cm 210 cc

New shapes
We can make shapes other than rectangular block, by stacking cubes. For example, see
this:

It is made by stacking cubes of side 1 centimetre. Can you calculate its volume?

41
Standard VI - Mathematics

How many cubes are there at the very bottom?

And in the step just above it?
Thus we can count the number of cubes in each step.
How many cubes in all?
What is the volume of the stairs?
Now look at this figure:
What is the volume of
a rectangular block of
length 4 centimetre, width
3 centimetre and height
1 centimetre?
If the length, width and
height are doubled, what
happens
to the volume?

It is made by stacking square blocks. The bottom block is of side 9 centimetres. As we

move up, the sides decrease by 2 centimetres at each step.
All blocks are of height 1 centimetre. What is the volume of this figure?
Just calculate the volume of each square block and add. Try it.

Calculate the volumes of the shapes shown below. All lengths are in
centimetres.
12 6
4 3
20 4 8
6
4 4 4
16
8 6 4
2
12 4 6
3
11
4
3

42
 Volume

Large measures
What is the volume of a cube of side 1 metre?
1 metre means 100 centimetres.
So, we must calculate the volume of a cube of side 100 centimetres.
How much is it?
We say that the volume of a cube of side 1 metre is 1 cubic metre.
So,
1 cubic metre = 1000000 cubic centimetres.
Volume of large objects are often said as cubic metres.

(1) A truck is loaded with sand, 4 metre long, 2 metre wide and 1 metre high. The
price of 1 cubic metre of sand is 1000 rupees. What is the price of this truck
load?
(2) What is the volume in cubic centimetres of a platform 6 metres long, 1 metre
wide and 50 centimetres high?
(3) What is the volume of a piece of wood which is 5 metres long, 1 metre wide and
25 centimetres high? The price of 1 cubic metre of wood is 60000 rupees. What
is the price of this piece of wood?
Capacity
Look at this hollow box:
It is made with thick wooden planks. Because of the
thickness, its inner length, width and height are less than
the outer measurements.
The inner length, width and height are 40 centimetres,
20 centimetres and 10 centimetres.
So, a rectangular block of these measurement can exactly
fit into the space within this box.

43
Standard VI - Mathematics

The volume of this rectangular block is the volume within the box.
This volume is called the capacity of the box.
Thus the capacity of this box is; Litre and
cubic metre
40 × 20 × 10 = 8000 cubic centimetres.
1 litre is 1000 cubic
centimetres and
So, what is the capacity of a box whose inner length, width and 1 cubic metres is
height are 50 centimetres, 25 centimetres and 20 centimetres? 1000000 cubic
centimetres. So,
Liquid measures 1 cubic metre is
1000 litres.
What is the capacity of a cubical vessel of inner side
10 centimetres?
10 × 10 × 10 = 1000 cubic centimetres
1 litre is the amount of water that fills this vessel.
1 litre = 1000 cubic centimetres

We can look at this in another way. If a cube of side 10 centimetres is completely

immersed in a vessel filled with water, then the amount of water that overflows would
be 1 litre.

So, how many litres of water do a vessel of length In the water

20 centimetres, width 15 centimetres and height A vessel is filled with
10 centimetres contain? water. If a cube of side 1
centimetre is immersed
Let’s look at another problem: into it, how many cubic
centimetre of water would
A rectangular tank of length 4 metres and height overflow? What if 20 such
cubes are immersed?
2 2 metres can contain 15000 litres of water. What
1

is the width of the tank?

If we find the product of length, width and height in
metres, we get the volume in cubic metres.
Here the volume is given as 15000 litres.

44
 Volume

That is, 15 cubic metres.

The product of length and height is 4 × 2 12 = 10

So, width multiplied by 10 is 15.

From this, we can calculate the width as

1 2 metres.
15 = 1
10

Now suppose this tank contains 6000 litres of water. What is the height of the water?

The amount of water is 6 cubic metres. Raising water

So, the product of the length and width of A swimming pool is 25 metres long, 10
metres wide and 2 metres deep. It is half
the tank and the height of the water, all in filled. How many litres of water does it
metres is 6. contain now?
25 × 10 × 1 = 250 cubic metres
Product of length and width is 4 × 1 12 = 6 = 250000 litres
Suppose the water level is increased by
So, height is 6 ÷ 6 = 1 metre. 1 centimetre. How many more litres of
water does it contain now?

(1) The inner sides of a cubical box are of length 4 centimetres. What is its capacity?
How many cubes of side 2 centimetres can be stacked inside it?

(2) The inner sides of a rectangular tank are 70 centimetres, 80 centimetres and 90
centimetres. How many litres of water can it contain?

(3) The length and width of a rectangular box are 90 centimetres and
40 centimetres. It contains 180 litres of water. How high is the water level?

(4) The inner length, width and height of a tank are 80 centimetres, 60 centimetres
and 50 centimetres, and it contains water 15 centimetres high. How much more
water is needed to fill it?

(5) The panchayat decided to make a rectangular pond. The length, width and depth
were decided to be 20 metres, 15 metres and 2 metres. How many litres of water
is needed to fill this pond to a height of one and a half metres?

45
Standard VI - Mathematics

(6) The inner length and width of an aquarium are 60 centimetres and 30
centimetres. It is half filled with water. When a stone is immersed in it, the
water level rose by 10 centimetres. What is the volume of the stone?

(7) A rectangular iron block has length 20 centimetres, width 10 centimetres and
height 5 centimetres. It is melted and recast into a cube. What is the length of
a side of this cube?

(8) A tank 2 metres long and 1 metre wide is to contain 10000 litres of water. What
should be the height of the tank?

(9) From the four corners of a square piece of paper of side 12 centimetres, small
squares of side 1 centimetre are cut off. The edges of this are bent up and
joined to form a container of height 1 centimetre. What is the capacity of
this container? If squares of side 2 centimetres are cut off, what would be the
capacity?

46
 4
Arithmetic of Parts

Arithmetic of Parts
Joining parts
If a circle is divided into four equal parts and two of these are joined together,
we get half a circle:

Thus a quarter circle joined to a quarter circle makes half a circle.

That is, two quarters make a half.
We can write it like this:
1  1  1
4 4 2
Now draw a circle and divide it into six equal parts. Colour one part :

Now, colour one more part:

47
Standard VI - Mathematics

Now 62 of the circle is coloured. 62 is another form of 13 , isn't it?

We can write this also as a sum:

1  1  2  1
6 6 6 3
Suppose a circle is divided into eight equal parts and two of them are joined together .
Can you mentally calculate what fraction of the circle we have now?
2 of 8 equal parts is 82 ; also we can see that
=2 1# 2
= 1
8 4 # 2 4
So, we have
1  1  2  1
8 8 8 4
We can also show this by drawing circles and colouring parts:

If we join 18 of the circle and 83 of the circle, what fraction of the circle would we get?

We took 1 + 3 = 4 parts of 8 equal parts; that is 84 . We can reduce the numerator and
denominator of this:
1  3  4  1
8 8 8 2
Draw pictures of circle with coloured parts to show this.

48
 Arithmetic of Parts

Take a long white ribbon and mark 9 equal parts:

Colour 2 parts:

Then colour 4 more:

Now 2 + 4 = 6 parts are coloured.

We can say this in another way: first we coloured 92 of the ribbon; and then coloured 94
of it; altogether 69 of the ribbon.

We can write this as a sum of fractions:

2  4  6
9 9 9

In this, we can reduce 69 to the lowest terms:

=6 2 # 3
= 2
9 3 # 3 3

Thus we have,
2  4  6  2
9 9 9 3

Now look at this picture:

What fraction of the ribbon is coloured red?

And the fraction coloured green?
What fraction is coloured altogether?
What sum of fractions do we get from this?

49
Standard VI - Mathematics

1  5  6  3
8 8 8 4

In each of the pictures below, write the parts of each colour and the total coloured parts
as fractions. Write the sum of fractions got from each picture in lowest terms.

50
 Arithmetic of Parts

Addition of fractions
If a circle is cut into four equal pieces and two of them joined together, we get half a
circle:

What if one more piece is joined to this?

We get three quarters of a circle. Thus half and a quarter make three quarters:

1  1  3
2 4 4

Now see this picture:

A circle is divided into six equal pieces, one of which is coloured red and two of them
coloured yellow. Total number of the pieces coloured is 1 + 2 = 3. How do we write this
as a sum of fractions?
1  2  3
6 6 6

51
Standard VI - Mathematics

In this 62 and 36 can be reduced to their lowest terms:

2 = 1 3 = 1
6 3 6 2
So, we can write this sum as

1  1  1
6 3 2

What sum do we get from the picture below?

1  3  4
6 6 6
If we write 36 = 12 and 64 = 32 , in lowest terms, then this sum can be written like this:

1  1  2
6 2 3

What about this picture?

The sum we get is

2  3  5
6 6 6

52
 Arithmetic of Parts

Reducing 62 and 36 to lowest terms, this becomes:

1  1  5
3 2 6
Now see these pictures:

1
4 of a circle and 83 of another circle of the same size are cut out and these pieces are
put together. What fraction of the full circle is this?
If the pieces are all alike, we need only count their numbers. What
if we see the one-fourth of the circle as two one-eighths joined
together?
3
8 is 3 such parts.
So 2 + 3 = 5 of 8 equal parts of the circle, which make 85 .
Thus we get
1 3235
4 8 8 8 8
Let's look at another problem.
Two ribbons of lengths 10
3
metre and 52 metre are joined end to end. What is the
total length?
The first ribbon is 3 of 10 equal parts of a 1 metre long ribbon:

53
Standard VI - Mathematics

The second ribbon is 2 of 5 equal parts of a 1 metre long ribbon:

Both are parts of 1 metre, but when we put them together, they are not equal parts. So
how do we write it as a fraction of a metre?
We can look at 52 metre as 4 of 10 equal parts of a metre:

Thus the first ribbon is 3 of 10 equal parts of a metre and the second is 4 of 10 equal
parts of a metre; altogether 7 equal parts:

Thus the total length is 10

7
metre.
As a sum of fractions:
3  2  3  4  7
10 5 10 10 10
What if we join 12 metre and 5 metre?
2

What did we see in all these problems?

To find the sum of two fractional measures put together, we must see them as sets of
the same equal pieces.
In other words, we must convert both to forms with the same denominator.

54
 Arithmetic of Parts

For example, let's compute:

1 +2
4 5

First we must change them into forms with the same denominator.

In all the different forms of 14 , the denominator is a multiple of 4.

In all the different forms of 52 , the denominator is a multiple of 5.

So the same denominator we want for both must be a multiple of 4 and 5.

4 × 5 = 20 is a multiple of both 4 and 5 , isn't it?

=1 5=#1 5
4 5 # 4 20

=2 4=#2 8
5 4 # 5 20
Now we can calculate the sum:
1  2  5  8  5  8  13
4 5 20 20 20 20
Let's calculate
5+2
8 3
like this.
To write these as forms with the same denominator, what number do we take as the
denominator?

=5 3=# 5 15
8 3 # 8 24

=2 8=# 2 16
3 8 # 3 24

Now we can add these:

5  2  15  16  15  16  31
8 3 24 24 24 24

In this, we can write 31

24 as
31  24  7  24  7   7  7
24 24 24 24 1 24 1 24

55
Standard VI - Mathematics

(1) In each pair of pictures below, find the fraction of the circle we get by cutting
up the coloured pieces of both circles and putting them together:

(2) Calculate the sums given below:

(i) 14 + 18 (ii) 34 + 16 (iii) 13 + 52 (iv) 12 + 52 (v) 32 + 15

56
 Arithmetic of Parts

(3) There are two taps to fill a tank with water. If the first tap alone is opened, the
tank would fill up in 10 minutes. If the second tap alone is opened, it would take
15 minutes to fill up the tank.
(i) If the first tap alone is opened, what fraction of the tank would be filled in
one minute?
(ii) If the second tap alone is opened, what fraction of the tank would be filled
in one minute?
(iii) If both the taps are opened, what fraction of the tank would be filled in one
minute?
(iv) If both the taps are opened, how much time would it take for the tank to
be filled up?

Some other sums

See these sums :
1 + 1
2 2 = 1
1 + 2
3 3 = 1
1 + 3
4 4 = 1
1 + 4
5 5 = 1
2 + 3
5 5 = 1
What do we do in all these?
Of the two fractions added, one is some parts of one divided into equal parts taken
together; and the other is the remaining parts taken together. When both these are taken
together, we get all the parts, right? In other words, we get the whole.

For each fraction given below, can you mentally calculate the fraction to be
added to make it 1?
(i) 72 (ii) 74 (iii) 83 (iv) 10
3

57
Standard VI - Mathematics

Now, look at this problem:

A jar contains three - quarter litre of milk,
and half a litre more is poured into it.
How much does it contain now ?
Suppose the half litre is poured into it in two
steps, a quarter litre each time. Then the first
quarter makes one litre (three quarters and a
quarter) in the jar. How much when the second
quarter is also added? One and a quarter litres.

Let's write this as a sum of fractions :

= 34 + 14 + 14 3 + 1
4 2
= 1 + 14
= 1 14
What about doing this sum as before, by equalizing the denominators ?
3  1  3  2  5
4 2 4 4 4
Then we can split 54 like this :
5  4  1  4  1   1  1
4 4 4 4 1 4 14

So, we can do it either way.

What about adding three - quarter litre itself to three - quarter litre ?
Three quarters and a quarter make one; half more is to be added. Altogether one and a
half litres.
= 34 + 14 + 24 3 + 3
4 4
= 1 + 12
= 1 12
We can also add like this :
3363 # 2 32  121  1  1
4 4 4 2 # 2 2 2 2 2 1 2 12

58
 Arithmetic of Parts

Let's look at another problem :

Draw two circles of the same size and
colour half of one and two thirds of the
other. If we cut out these pieces, can we
put them together ?
What if we cut them like this ?

We can put them together as one circle and a

remaining small part.
Let's write this down as a sum of fractions :
1 + 2
2 3 = 36 + 64
= 36 + 36 + 16
= 1 + 16
= 1 16
Another problem:
Anoop and his father went to
buy material for shirts. One and
a half metres for Anoop and two
and a quarter metres for father.
If they buy the same kind of
material for both, how much in
all?

59
Standard VI - Mathematics

We can do it like this:

One and two make three; half and quarter make three quarters. So, three and three
quarter metres in all.
That is,
= c1 + 12 m + c 2 + 14 m 12 + 24
1 1

= ^1 + 2h + c 12 + 14 m
= 3 + 34
= 3 34

(1) Calculate the sums below :

(i) 56 + 13 (ii) 87 + 14 (iii) 56 + 14 (iv) 85 + 34 (v) 2 13 + 3 12
(2) One jar contains one and a half litres of milk and another contains two and a
three quarters litres of milk. How much milk in both the jars together?
(3) Two strings of lengths one and a half metres are joined end to end. What is the
total length?
(4) Ahirath bought one and a half kilograms of beans and three quarter kilograms
of yam. What is the total weight?

Removing parts
From a ribbon three quarters of a metre long , a quarter metre piece is cut off; What is
the length of the remaining piece?
3
4 metre means 3 of 4 equal pieces into which one metre is divided:

Each of these equal parts is 14 metre long. So, cutting off 14 metre means removing one
of these equal parts.

60
 Arithmetic of Parts

What remains is 2 of the 4 equal parts; that is

2 = 1
4 2
Thus the remaining piece is half a metre long.
We can write the calculation like this:
3  1  1
4 4 2
We can do this subtraction just as we did addition of fractions:
3  1  3  1  2  1
4 4 4 4 2
If a half metre piece is cut off from a three-quarter metre long ribbon, then the
remaining piece is a quarter metre long:

We write this computation as

3  1  1
4 2 4
As in the case of sums, we can do this by equalising denominators:
3  1  3  2  3  2  1
4 2 4 4 4 4
What if one-third of a metre is removed from half a metre?
We must compute 12 − 13 .
We can do this by equalising denominators:
1  1  3  2  3  2  1
2 3 6 6 6 6
Thus what remains is 16 metre.

(i) 12 − 18 (ii) 34 − 18 (iii) 13 − 15 (iv) 52 − 13 (v) 32 − 15

61
Standard VI - Mathematics

We have seen some pairs of fractions which add up to 1. We can rewrite them as
subtractions:
Sum Difference
1  1  1 1
2 2 1 1  2  2

1  2  1 2 2 1
3 3 1 1  3  3 1  3  3

1  3  1 3 3 1
4 4 1 1  4  4 1  4  4

1  4  1 4 4 1
5 5 1 1  5  5 1  5  5

2  3  2 3 3 2
5 5 1 1  5  5 1  5  5

Now look at this problem:

From one litre of milk, a quarter litre was used up. How much milk is left?
A quarter and three quarters make one; so, what remains is three-quarter litre of milk.
We write it like this:
1  4  c4  4m  4  4
1 3 1 1 3

We can also do it like this:

1 4 1 4  1  3
1  4  4  4  4 4
Now see this picture:

What fraction of the circle is coloured?

What fraction remains to be coloured?
The fraction to be coloured can be calculated like this:
5 3
1  8  8

62
 Arithmetic of Parts

This we can also do as

5 8 5 8  5  3
1  8  8  8  8 8
Another problem:
From a two and a half kilograms yam,
one and a quarter kilograms piece is cut
off. What is the weight of the remaining
piece?
We can do it in head like this: One kilogram
cut off from two kilograms leaves one
kilogram and a quarter kilogram cut off from
half a kilogram leaves a quarter kilogram. So
what remains is one and a quarter kilograms.
We can write it out like this:
2 2  1 4  c 2  2 m  c1  4 m  ^2  1 h  c 2  4 m  1  4  1 4
1 1 1 1 1 1 1 1

What if we changed the cloth problem done earlier like this?

One and a half metres of cloth was bought for Anoop and two and a quarter
metres for his father. How much more was bought for father?
Here, we cannot subtract half a metre from a quarter metre. So, let's think of another
way.
Half a metre added to one and a half metres makes two metres; another quarter of a
metre added makes two and a quarter metres. Total added is half and a quarter, which is
three quarters. So, three quarters of a metre more. That is,
3
24  12  4
1 1

Let's write down our reasoning:

• 1
2  12  2
1

• 1 1 1 1 1
24  2  4  12  2  4

• 24  12  4
1 1 3

• 24  12  4
1 1 3

63
Standard VI - Mathematics

(1) Natasha drew a circle and coloured 12

5
of it. What fraction of the circle
remains to be coloured?
(2) A bucket can hold 10 litres of water and it contains 3 34 litres. How much more
is needed to fill it?
(3) From a string, one and three quarters of a metre long, a piece half a metre long
is cut off. What is the length of the remaining piece?
(4) A panchayat constructed a new road, 14 34 kilometres long last year. This year, a
road 16 14 kilometres long was constructed. How much more was constructed
this year?
(5) Ashadev bought 20 metres of string. He cut off a piece 5 34 metres long first,
and then a piece 6 12 metres long later. What is the length of string left?
(6) The milk society got 75 14 litres in the morning and 55 12 litres in the evening.
Of this, 85 34 litres of milk is sold. How much milk is left?

Large and small

Which is larger, 52 or 53 ?
How did you decide that 53 is larger?
2
5 is 2 parts of 5 equal parts taken together. To get 53 we must take 3 such parts together.

Thus the number 52 is less than the number 53 . We write it like this:
2 3
5 1 5

64
 Arithmetic of Parts

On the other hand, 53 is greater than 52 .

This we write in shorthand as
3 2
5 2 5
Similarly , which of 83 and 85 is smaller and which is larger?
How do we write these in shorthand?
In general, we can say this

Of two fractions with the same denominator, the one with larger
numerator is the larger and the one with smaller numerator is the smaller.

In other words, by increasing the numerator alone makes a fraction larger. For example:
1 2 3 4
5151515
On the other hand, what happens if the denominator alone is increased?
For example, let's take 34 and 53
Each of 4 equal parts is larger than each of 5 equal parts:

So, 3 of the first kind of pieces taken together is larger than 3 of the second kind of
pieces:

65
Standard VI - Mathematics

This means 34 2 53
What can we say in general?

Of two fractions with the same numerator, the one with larger
denominator is smaller and the one with smaller denominator is larger

We can say it like this: if the denominator alone of a fraction is increased, the fraction
becomes smaller. For example,
5 5 5 5
6272829
To compare two fractions with the same denominator, we need only compare the
numerators; to compare two fractions with the same numerator, we need only compare
the denominators.
What if both the numerator and denominator are different?
For example, which of 12 and 32 is the larger?
We can convert any two fractions into forms with the same denominators, right?
=1 3=2 4
2 6 3 6
Which of 36 and 64 is the larger?
So, which of 12 and 32 is the larger?
1 2
2 1 3

(1) Find the larger and smaller of each pair of fractions below and write this using
the < or > symbol:
(i) 52 , 53 (ii) 52 , 32 (iii) 52 , 34 (iv) 37 , 92 (v) 72 , 83 (vi) 94 , 83
(2) Arrange each triple of fractions below from the smallest to the largest and write
it using the < symbol :
(i) 52 , 34 , 53 (ii) 37 , 92 , 72 (iii) 12 , 13 , 32

66
 5
Decimal Forms

Decimal Forms
Decimal places
We have seen in class 5 how various
measures can be written as fractions
and as their decimal forms (The lesson
Measure Math).
For example, the length of a pencil can
be said in different ways:
• 5 centimetres 7 millimetres
• 7 centimetres
5 10

• 5.7 centimetres
We can write other measures also like this:
7 litres = 5.7 litres
5 10
7 kilograms = 5.7 kilograms
5 10

We can drop all references to measures and simply say that 5.7 is the decimal
7.
form of the number 5 10
7 = 5.7
5 10
29
Similarly, 4.29 is the decimal form of 4 100
29 = 4.29
4 100

We must note another thing here. We write natural numbers using ones, tens,
hundreds and so on. For example:
247 = 2 hundreds + 4 tens + 7 ones

67
Standard VI - Mathematics

What about 247.3 ?

First we split it as the sum of a whole number and a fraction, as
3 = 247 + 3
247.3 = 247 10 10
3 here can be written as
The 10

3  1  1  1
10 10 10 10
That is, 3 tenths. So, we can write 247.3 in terms of hundreds, tens, ones and tenths:
247.3 = 2 hundreds + 4 tens + 7 ones + 3 tenths
How do we split 247.39 like this?
First, we write it as
39 = 247 + 39
247.39 = 247 100 100
39 as
Then, we can split 100
39  30  9  30  9  3  9
100 100 100 100 10 100
3 here is 3 tenths; and 9 is 9 hundredths. So,
The 10 100
247.39 = 2 hundreds + 4 tens + 7 ones + 3 tenths + 9 hundredths

In general,

In the decimal form of a number, the dot separates the whole number
part and the fractional part. Digits to the left of the dot show the
multiples of ones, tens, hundreds and so on; the digits to the right show
the multiples of tenths, hundredths, thousandths and so on.
For example, the two numbers used in the above examples can be split according to
place value like this:

68
 Decimal Forms

Split the numbers below according to place value:

(i) 4.5 (ii) 4.57 (iii) 4.572 (iv) 45.72 (v) 457.2

Let's look at the decimal form of some measures again. For example, what is the decimal
form of 23 metres and 40 centimetres?
40 metres = 23.40 metres
23 metres 40 centimetres = 23 100
Taking only the numbers, we have
40 = 23.40
23 100
40 here as
We can write the 100
40 = 4
100 10
So, we get
23=40 = 4
100 23 10 23.4
This means
23.40 = 23.4
We can see this using place values also:

Thus we can write 23 metres and 40 centimetres in two different ways:

23 metres 40 centimetres = 23.40 metres
23 metres 40 centimetres = 23.4 metres
What about 23 metres and 4 centimetres?
4 of a metre; that is 4 metre.
4 centimetres is 100 100
4 metres
23 metres 4 centimetres = 23 100

69
Standard VI - Mathematics

4 according to place value:

We can split 23 100
4 = 2 tens + 3 ones + 4 hundredths
23 100

4 ?
So what is the decimal form of 23 100
4 = 23.04
23 100

In our problem of lengths,

23 metres 4 centimetres = 23.04 metres

What about 23 metres and 4 millimetres?

4 metres. So
4 millimetres means 1000
4 metres
23 metres 4 millimetres = 23 1000
4 according to place value?
How do we split 23 1000

4 as
So, we can write the decimal form of 23 1000
4 = 23.004
23 1000

In terms of lengths

23 metres 4 millimetres = 23.004 metres

70
 Decimal Forms

Now try to write 4 kilograms and 55 grams as kilograms in decimal form.

Convert the measures below into the measures specified, using fractions and
decimal forms.

Measure Fractional form Decimal form

4 metres 30 centimetres 30 metres

4 100 4.3 metres

4 metres 3 centimetres ... metres ... metres

4 metres 3 millimetres ... metres ... metres

3 kilograms 200 grams ... kilograms ... kilograms

3 kilograms 250 grams ... kilograms ... kilograms

3 kilograms 25 grams ... kilograms ... kilograms

30 kilograms 250 grams ... kilograms ... kilograms

10 litres 750 millilitres ... litres ... litres

10 litres 75 millilitres ... litres ... litres

10 litres 705 millilitres ... litres ... litres

100 litres 750 millilitres ... litres ... litres

71
Standard VI - Mathematics

Decimals and fractions

We have seen how measures of various kinds, given as decimals, can be converted to
fractions.

For example,
3 centimetres
7.3 centimetres = 7 10

We can also write it as millimetres

1 centimetre means 10 millimetres; so

7 centimetres = 70 millimetres
3 centimetres?
What about 10
3 1
10 is three 10
1 centimetres is 1 millimetre;
And 10

so, 3
10 centimetres = 3 millimetres
Now we can write
3 centimetres
7.3 centimetres = 7 10

= 73 millimetres
1 centimetres
We can change this again. 73 millimetres mean 73 of 10
73 centimetres
73 millimetres = 10

In terms of just numbers without using measures, this means

73
7.3 = 10
Similarly, how do we write 7.31 metres as a fraction?
First, we write it as a whole number and a fraction, as seen in class 5:
31 metres
7.31 metres = 7 100

72
 Decimal Forms

1 metre is 100 centimetres, so that

7 metres = 700 centimetres
1 metres means 1 centimetre,
On the other hand, since 100
31
100 metres = 31 centimetres
Thus
31 metres
7.31 metres = 7 100
= 731 centimetres
If 731 centimetres is converted back to metres,
731 metres
731 centimetres = 100
Thus,
731 metres
7.31 metres = 100
In terms of numbers alone
731
7.31 = 100
In this way, how do we write 7.319 litres as a fraction?
319 litres
7.319 litres = 7 1000
In this
7 litres = 7000 millilitres
319
1000 litres = 319 millilitres
From this we get
7.319 litres = 7319 millilitres
Converting back to litres
7319 litres
7319 millilitres = 1000
Thus
7319 litres
7.319 litres = 1000

73
Standard VI - Mathematics

In terms of numbers
7319
7.319 = 1000
In all the examples above, we converted numbers in decimal form to fractions:
73
7.3 = 10
731
7.31 = 100
7319
7.319 = 1000
What change do you see in the denominators of the fractions, as the number of digits
after the decimal point increases?
Can you say the fractional form of the decimal 12.03?
What is the numerator of the fraction?
And the denominator?
In 12.03, how many digits are there after the decimal point?
12.03 = 1203
100
On the other hand, what is the decimal form of 1203
1000 ?
Looking at the denominator, can you say how many digits after the decimal point does
it have?
1203 = 1.203
1000

(1) The decimal form of some numbers are given below. Write each of them as a
fraction with denominator 10, 100 or 1000.
(i) 3.7 (ii) 3.07 (iii) 30.7 (iv) 3.72 (v) 37.2 (vi) 3.072 (vii) 30.72
(2) Write the decimal form of the fractions given below.
(i) 10
51
(ii) 513
10
513 (iv) 513 (v) 5130
(iii) 100 1000 1000

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 Decimal Forms

Addition and subtraction

A line 4.3 centimetres long was drawn and then extended by another 2.5 centimetres:

How many centimetres long is the line now?

We must add 4.3 centimetres and 2.5 centimetres.

We can do this in several ways.

First, we can convert these to centimetres and millimetres.
4.3 centimetres = 4 centimetres 3 millimetres
2.5 centimetres = 2 centimetres 5 millimetres
And add the centimetres and millimetres separately.
4 centimetres + 2 centimetres = 6 centimetres
3 millimetres + 5 millimetres = 8 millimetres
The length of the line is
6 centimetres 8 millimetres
Then we can convert back to centimetres.
68 centimetres
6 centimetres 8 millimetres = 10
= 6.8 centimetres
Or, first we can write the lengths in millimetres.
4.3 centimetres = 43 millimetres
2.5 centimetres = 25 millimetres

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Standard VI - Mathematics

Now we need only add 43 and 25; and this too can be done in several ways.
For example,
43 + 25 = 40 + 20 + 3 + 5 = 68
Thus the length of the line is 68 millimetres.
This we can write as centimetres in decimal form.
68 millimetres = 6 centimetres 8 millimetres

= 6.8 centimetres
Another method is to remove the measures and write the numbers as fractions.
43
4.3 = 10
25
2.5 = 10
And these fractions we can add like this:
43  25  43  25  68
10 10 10 10
This fraction can be written in decimal form as
68
10 = 6.8
Now, we can add the measures and say the length of the line is 6.8 centimetres.
What if we want to add 4.3 centimetres and 2.8 centimetres?
We can add by changing the lengths to millimetres:
4.3 centimetres = 43 millimetres
2.8 centimetres = 28 millimetres
Now, we need only add 43 and 28. It can be done like this:
43 + 28 = 40 + 20 + 3 + 8 = 60 + 11 = 71
Thus the length of this line is 71 millimetres.
This we can write in centimetres as a decimal:
71 millimetres = 7 centimetres 1 millimetres

= 7.1 centimetres

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 Decimal Forms

We can do it by converting just the numbers to

fractions. Place values
and addition
43
4.3 = 10 We can add numbers in
28 decimal form without
2.8 = 10 converting to fractions,
using place values. For
Now, add the fractions:
example, see how calculate
43  28  43  28  71 4.3 + 2.8
10 10 10 10
Finally, convert the fraction back to the decimal
form.
71
10 = 7.1
The length of the line is 7.1 centimetres.
Another problem:
A jar contains 3.5 litres of oil and 6.25 litres more is poured into it. How much oil
does the jar contain now?
First, let's convert just the numbers to fractions:
35
3.5 = 10
625
6.25 = 100
How do we add these?
35 also as a fraction with denominator 100:
We can write 10
35 35 # 10
10 = 10 # 10
= 100350

Now we can add like this:

35 + 625 350 + 625
10 100 = 100 100
+ 625
= 350100

= 100 975

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Standard VI - Mathematics

We now convert this fraction back to the decimal form:

975
100 = 9.75
Thus the jar contains 9.75 litres of oil now.
Now look at this problem:
A person bought 2.5 kilograms of rice and 3.125 kilograms of vegetables. What is
the total weight?
This also we do with fractions
25
2.5 = 10
3125
3.125 = 1000
25 to a form
To add these, we change 10
with denominator 1000.
25 25 # 100
10 = 10 # 100
2500
= 1000
Now, we add the fractions:
25 + 3125 2500 + 3125
10 1000 = 1000 1000
+ 3125
= 2500 1000
One way of adding 2500 and 3125 is this:
2500 + 3125 = 2000 + 3000 + 500 + 125 = 5000 + 625 = 5625
So, we can continue our addition of fractions:
25 + 3125 2500 + 3125
10 1000 = 1000
= 1000 5625

Converting this to decimals:

5625
1000 = 5.625
Thus the total weight is 5.625 kilograms.

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 Decimal Forms

(1) Anu made an 8.5 metre long festoon and Sarah made a 7.8 metre long one to
decorate their classroom for the school anniversary. What is the total length of
the festoon they made?
(2) Amal needs 2.25 metres of cloth and Sagar, 1.85 metres for school uniform.
How many metres of cloth in all?
(3) A tin weighs 2.85 kilograms and it is filled with 12.5 kilograms of rice. What is
the total weight?
(4) Bakul walks 2.25 kilometres in the morning and 1.5 kilometres in the evening
everyday. What is the total distance she walks each day?
(5) Two small bottles contain 0.850 litre and 0.375 litre of honey. If both the
bottles are emptied into a large bottle, how much honey does it contain?

Next let's look at some instances where we have to subtract measures. See this problem:
From an 8.5 centimetres long eerkkil, a 3.2 centimetres long piece is broken off.
What is the length of the remaining piece?
Thinking in terms of numbers alone, what we need is to subtract 3.2 from 8.5.
We change the numbers to fractions.
85
8.5 = 10
32
3.2 = 10
Now we can subtract:
85 − 32 85 − 32
10 10 = 10
One way to subtract 32 from 85 is this:
85 − 32 = (80 − 30) + (5 − 2)
= 50 + 3
= 53

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Standard VI - Mathematics

So we get
85 − 32 53
10 10 = 10
Finally, we switch back to decimals:
53
Place value and subtraction
10 = 5.3 We can subtract numbers in decimal
Thus the length of the remaining piece of eerkkil form without changing them to
fractions. For example, consider
is 5.3 centimetres.
8.5 − 3.7
What if we change the problem like this? To do this, we first write like this
From an 8.5 centimetres long eerkkil, a 3.7
centimetres long piece is broken off. What is
the length of the remaining piece?

We start as before by converting the decimals to Since 7 tenths cannot be subtracted

fractions: from 5 tenths, we rewrite this as
85
8.5 = 10 shown below and proceed:

37
3.7 = 10

And then subtract

85 − 37 85 − 37
10 10 = 10
We have seen in earlier classes that subtractions like 85 − 37 can be done in several
different ways. For example,
85 − 37 = (85 − 35) − 2 = 50 − 2 = 48
85 − 37 = (87 − 37) − 2 = 50 − 2 = 48
85 − 37 = (85 − 40) + 3 = 45 + 3 = 48
Anyway, we find
85 − 37 48
10 10 = 10
Changing back to decimals,
48
10 = 4.8

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 Decimal Forms

The remaining piece of eerkkil is 4.8 centimetres long.

Look at another problem:

There is 15 kilograms of rice in a sack. 4.25 kilograms from this is put in a bag.
How much rice remains in the sack?

Thinking just in terms of numbers, what we have to do is subtract 4.25 from 15.

We first write 4.25 as a fraction:

425
4.25 = 100

How do we write 15 as a fraction with denominator 100?

Recall the section Fraction as division of the lesson One Fraction Many Forms.

15 = 15 15 # 100 1500
1 = 1 # 100 = 100
Now can't we subtract?
1500 − 425 = 1500 − 425
100 100 100

We can do 1500 − 425 in several ways:

1500 − 425 = 1000 + 500 − 425 = 1000 + 75 = 1075

1500 − 425 = 1425 − 425 +75 = 1000 + 75 = 1075

1500 − 425 = 1500 − 500 + 75 = 1000 + 75 = 1075

Thus, we have:
1500 − 425 = 1075
100 100 100
Changing back to decimals:
1075 = 10.75
100
So, there is 10.75 kilograms of rice still in the sack.

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Standard VI - Mathematics

(1) From a rod 14.7 metres long, a piece 7.75 metre long is cut off. What is the
length of the remaining piece?
(2) There was 38.7 kilograms of rice in a sack and 12.350 kilograms of this is used
up. How much rice remains in the sack?
(3) The perimeter of a rectangle is 24 centimetres and the length of one side is
6.4 centimetres. What is the length of the other side?
(4) There was 2.50 litres of oil in a bottle and 0.475 litres of this was used for
cooking . How much oil is left in the bottle?
(5) What number we must add to 14.32 to get 16.43?

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 6
Multiples and Factors

Multiples
and Factors
Multiples of multiples
We have learnt about multiples in the lesson Within Numbers of the class 5
textbook.
The multiples of a natural number are the product of that number with the
natural numbers 1, 2, 3, ...
For example, the multiples of 2 are the numbers 2, 4, 6, ... got by multiplying the
natural numbers by 2.
The multiples of 3 are the numbers 3, 6, 9, ... got by multiplying the natural
numbers by 3.
What about the multiples of 4?
Let's compare the multiples of 2 and 4:
Multiples of 2: 2 4 6 8 10 12 14 16 18 20
Multiples of 4: 4 8 12 16 20 24 28 32 36 40
All multiples of 4 are among the multiples of 2, right?
Why is this so?
Since 4 = 2 × 2, all multiples of 4 can be written as multiples of 2 also:
1×4= 4 2×2 =4
2×4= 8 4×2 =8
3 × 4 = 12 6 × 2 = 12
4 × 4 = 16 8 × 2 = 16
5 × 4 = 20 10 × 2 = 20
.................... ....................

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Standard VI - Mathematics

Similarly, 6 is a multiple of both 2 and 3.

So every multiple of 6 can be written as a multiple of what all numbers?
1×6=6 3×2=6 2 × 3= 6
2 × 6 = 12 6 × 2 = 12 4 × 3 = 12
3 × 6 = 18 9 × 2 = 18 6 × 3 = 18
4 × 6 = 24 12 × 2 = 24 8 × 3 = 24
5 × 6 = 30 15 × 2 = 30 10 × 3 = 30
.................... .................... ....................
So what can we say in general?

All multiples of the multiple of a number are multiples of that

number also.

We have seen in class 5 that the statements about multiples can also be put in terms of
factors (The section Division and factors of the lesson Within Numbers).
For example, the statement
4 is a multiple of 2
can also be put as
2 is a factor of 4
Similarly, how do we write
6 is a multiple of 2 and 3
in terms of factors?
2 and 3 are two factors of 6
So, how do we write the general result above in terms of factors?

All multiples of a number are also multiples of any of its factors.

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 Multiples and Factors

(1) For each of the multiples given below, find the other numbers they are multiples
of:
(i) Multiples of 8
(ii) Multiples of 10
(iii) Multiples of 12
(2) Check whether each of the statements below is true or false. For true
statements, explain why they are so. For the false statements, give an example in
which it is not true.
(i) All multiples of 20 are multiples of 10
(ii) All multiples of 10 are multiples of 2
(iii) All multiples of 15 are multiples of 5
(iv) All multiples of 15 are multiples of 3
(v) All multiples of 5 are multiples of 15
(vi) All multiples of 3 are multiples of 15

Primary factors
A number can be written as the product of its factors in different ways. For example
let's consider the number 30:
30 = 1 × 30
30 = 2 × 15
30 = 3 × 10
30 = 5 × 6
Is there any other way?
We can change the order of the factors in each. But that's trivial.

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Standard VI - Mathematics

What if we write 30 as a product of three factors instead of two?

In 30 = 1 × 30, we can split the 30 on the right of the equation as 2 × 15 and write
30 = 1 × 2 × 15.
But then, we can simply rewrite each of the others using 1 as another factor.
Can we write 30 as a product of three factors, without using 1?
We can write
30 = 2 × 3 × 5
Is there any other way? (Apart from changing just the order of the factors).
Why can't we get smaller factors without using 1?
Does 2 have any factors other than 1 and 2?
This is so for 3 and 5 also, right?
Thus, the only factors of each of the numbers 2, 3 and 5 are 1 and the number itself
(For any number, 1 and the number itself are factors, aren't they?).
Can you think of any other number like this?
Is 6 such a number?
What about 7?
Let's write such numbers below 20:
1, 2, 3, 5, 7, 11, 13, 17, 19
Such numbers, excluding 1, are said to be prime numbers.

A natural number greater than 1, which has no factors other than 1 and
itself is called a prime number.

Numbers greater than 1, which are not primes are called composite numbers.
For example, 6 is a composite number.
When a number is written as the product of two factors and any one of them is not a
prime, then that factor can be again written as the product of two factors. This can be
continued till all factors are prime.

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 Multiples and Factors

Thus we have this general statement:

Any composite number can be written as a product of primes.

For example, let's see how 60 can be written as a product of primes:

(i) We start by writing 60 = 2 × 30.
(ii) The 30 in this can be written 2 × 15.
(iii) Finally 15 can be written 3 × 5.

This factorization can be shown as a picture shown below:

See how 84 is written as a product of primes like this:

84 = 2 × 42
= 2 × 2 × 21
= 2 × 2 × 3 × 7
From this, we see that
84 = 2 × 2 × 3 × 7
We can also draw a picture like this:

87
Standard VI - Mathematics

Can you write the numbers below as a product of primes?

(i) 24 (ii) 35 (iii) 36 (iv) 60 (v) 100

Once we write two numbers as a product of primes, it is easy to write the product of
these numbers also as a product of primes.

For example, 12 and 30 can be split like this:

12 = 2 × 2 × 3
30 = 2 × 3 × 5

Now we can split 12 × 30 as shown below :

360 = 12 × 30
= (2 × 2 × 3) × (2 × 3 × 5)
= 2 × 2 × 2 × 3 × 3 × 5

On the other hand, to split a number into a product of primes, we first split it into the
product of any two factors, then split each of these factors into a product of primes and
finally put these prime factors together.

For example, to split 90 into a product of primes, we first write:

90 = 9 × 10

Next write 9 and 10 as products of primes:

9 = 3×3
10 = 2 × 5

Now we can write 90 like this:

90 = 9 × 10
= (3 × 3) × (2 × 5)
= 2 × 3 × 3 × 5

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 Multiples and Factors

Write each of the numbers below as a product of primes.

(i) 72 (ii) 105 (iii) 144 (iv) 330 (v) 900

All factors
Once we know the prime factors of a number, we can find all its factors.
For example, the prime factors of 6 are 2 and 3; what are its other factors?
For any number, 1 and the number itself are factors.
So, 1 and 6 are also factors of 6.
Does 6 have any factors other than 1, 2, 3, 6?
Let's take 15 next. What are its prime factors?
And the other factors?
Does 15 have any factors other than 1, 3, 5, 15?
Like this, can you first write 21 as a product of two primes and then find all its factors?
In general, how do we find all factors of a number which is a product of two different
primes?

Find all the factors of the numbers below:

(i) 35 (ii) 77 (iii) 26 (iv) 51 (v) 95

Now let's look at the product of three different primes. For example,
30 = 2 × 3 × 5
Apart from 2, 3, 5, two other factors of 30 are 1 and 30, as seen for other numbers.
Does 30 have any other factors?

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Standard VI - Mathematics

We can rewrite 30 = 2 × 3 × 5 as
30 = 6 × 5
So, isn't 6 also a factor of 30?
Can you find two other factors?
So we can arrange the factors of 30 as shown below:
1
2, 3, 5 (Prime factors)
2×3 = 6
2 × 5 = 10 (Product of two prime factors)
3 × 5 = 15
Like this, can you write 42 as the product of three prime numbers and find all the
factors?

Write each of the numbers below as a product of three primes and find all its
factors:
(i) 66 (ii) 70 (iii) 105 (iv) 110 (v) 130

Prime numbers
The prime numbers continue like this: 2, 3, 5, 7, 11, ... The only even number among
them is 2. All the primes afterwards are odd numbers. Any even number greater than
2 has the factor 2, other than 1 and the number itself.

But not all odd numbers are primes; for example 9 = 3 × 3

We don't see any definite pattern for the odd primes. For example, after 3, 5, 7 which are
two apart, the next prime is not 9, but 11. Thus the difference between 7 and the next
prime 11 is 4.

Again, after the prime 31, the next prime is 37, and their difference is 6; the prime after
89 is 97, with difference 8.

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 Multiples and Factors

But even as such consecutive primes drift further apart, there are consecutive primes
like 41 and 43 or 71 and 73 in between, which are only two apart.
There is a technique to list all primes less than a specified number. Let's see how we can
use this to find all primes less than 50.
First, write all numbers up to 50 in rows and columns like this:

Next, strike off 1 from this. Then strike off all multiples of 2, except 2:

Now, 3 is the least number greater than 2 which is not stricken off; keep 3 and strike off
all multiples of 3:

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Standard VI - Mathematics

Next, find the least number greater than 3 which is not stricken off, and it is 5. Strike all
multiples of 5, except 5 itself.
If we remove the multiples of 7 other than itself also, we can see that there are no
multiples, except themselves, of the other numbers remain:

Now the numbers not struck off are

and these are the primes less than 50.

(1) Find all primes less than 100. Find the primes that differ by 2 among these.
(2) Can the product of two natural numbers be a prime?

(3) Can the sum of two prime numbers be prime?

92
Notes

93
 Notes

94