NCERT Textbook: Fractions & Decimals

 Page 1


MATHEMATICS 20
2.1  MULTIPLICATION OF FRACTIONS
Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length
and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its
area would be 7 × 4 = 28 cm
2
.
What will be the area of the rectangle if its length and breadth are 
7
1
2
 cm and
3
1
2
 cm respectively? Y ou will say it will be 
7
1
2
 × 
3
1
2
 = 
15
2
 × 
7
2
 cm
2
. The numbers 
15
2
and 
7
2
 are fractions. T o calculate the area of the given rectangle, we need to know how to
multiply fractions. W e shall learn that now .
2.1.1  Multiplication of a Fraction by a Whole Number
Observe the pictures at the left (Fig 2.1). Each shaded part is 
1
4
part of a circle. How much will the two shaded parts represent together?
They will represent 
1 1
4 4
+
 = 
1
2×
4
.
Combining the two shaded parts, we get  Fig 2.2 . What part of a circle does the
shaded part in Fig 2.2 represent? It represents 
2
4
 part of a circle .
Fig 2.1
Fig 2.2
Chapter  2
Fractions and
Decimals
2024-25
Page 2


MATHEMATICS 20
2.1  MULTIPLICATION OF FRACTIONS
Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length
and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its
area would be 7 × 4 = 28 cm
2
.
What will be the area of the rectangle if its length and breadth are 
7
1
2
 cm and
3
1
2
 cm respectively? Y ou will say it will be 
7
1
2
 × 
3
1
2
 = 
15
2
 × 
7
2
 cm
2
. The numbers 
15
2
and 
7
2
 are fractions. T o calculate the area of the given rectangle, we need to know how to
multiply fractions. W e shall learn that now .
2.1.1  Multiplication of a Fraction by a Whole Number
Observe the pictures at the left (Fig 2.1). Each shaded part is 
1
4
part of a circle. How much will the two shaded parts represent together?
They will represent 
1 1
4 4
+
 = 
1
2×
4
.
Combining the two shaded parts, we get  Fig 2.2 . What part of a circle does the
shaded part in Fig 2.2 represent? It represents 
2
4
 part of a circle .
Fig 2.1
Fig 2.2
Chapter  2
Fractions and
Decimals
2024-25
FRACTIONS AND DECIMALS 21
The shaded portions in Fig 2.1 taken together are the same as the shaded portion in
Fig 2.2, i.e., we get Fig 2.3.
Fig 2.3
or
1
2×
4
 =
2
4
 .
Can you now tell what this picture will represent? (Fig 2.4)
             Fig 2.4
And this? (Fig 2.5)
Fig 2.5
Let us now find 
1
3×
2
.
We have
1
3×
2
 =
1 1 1 3
2 2 2 2
+ + =
We also have                       
1 1 1 1+1+1 3×1 3
+ + = = =
2 2 2 2 2 2
So
1
3×
2
 =
3×1
2
 = 
3
2
Similarly
2
×5
3
 =
2×5
3
 = ?
Can you tell
2
3×
7
 = ?
3
4 × ?
5
=
The fractions that we considered till now, i.e., 
1 2 2 3
, , ,
2 3 7 5
 and 
3
5
 were proper fractions.
=
=
=
2024-25
Page 3


MATHEMATICS 20
2.1  MULTIPLICATION OF FRACTIONS
Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length
and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its
area would be 7 × 4 = 28 cm
2
.
What will be the area of the rectangle if its length and breadth are 
7
1
2
 cm and
3
1
2
 cm respectively? Y ou will say it will be 
7
1
2
 × 
3
1
2
 = 
15
2
 × 
7
2
 cm
2
. The numbers 
15
2
and 
7
2
 are fractions. T o calculate the area of the given rectangle, we need to know how to
multiply fractions. W e shall learn that now .
2.1.1  Multiplication of a Fraction by a Whole Number
Observe the pictures at the left (Fig 2.1). Each shaded part is 
1
4
part of a circle. How much will the two shaded parts represent together?
They will represent 
1 1
4 4
+
 = 
1
2×
4
.
Combining the two shaded parts, we get  Fig 2.2 . What part of a circle does the
shaded part in Fig 2.2 represent? It represents 
2
4
 part of a circle .
Fig 2.1
Fig 2.2
Chapter  2
Fractions and
Decimals
2024-25
FRACTIONS AND DECIMALS 21
The shaded portions in Fig 2.1 taken together are the same as the shaded portion in
Fig 2.2, i.e., we get Fig 2.3.
Fig 2.3
or
1
2×
4
 =
2
4
 .
Can you now tell what this picture will represent? (Fig 2.4)
             Fig 2.4
And this? (Fig 2.5)
Fig 2.5
Let us now find 
1
3×
2
.
We have
1
3×
2
 =
1 1 1 3
2 2 2 2
+ + =
We also have                       
1 1 1 1+1+1 3×1 3
+ + = = =
2 2 2 2 2 2
So
1
3×
2
 =
3×1
2
 = 
3
2
Similarly
2
×5
3
 =
2×5
3
 = ?
Can you tell
2
3×
7
 = ?
3
4 × ?
5
=
The fractions that we considered till now, i.e., 
1 2 2 3
, , ,
2 3 7 5
 and 
3
5
 were proper fractions.
=
=
=
2024-25
MATHEMATICS 22
For improper fractions also we have,
5
2 ×
3
 =
2 × 5
3
 = 
10
3
Try ,
8
3×
7
 = ?
7
4 ×
5
  = ?
Thus, to multiply a whole number with a proper or an improper fraction, we
multiply the whole number with the numerator of the fraction, keeping the
denominator same.
1. Find:   (a)   
2
×3
7
 (b)   
9
6
7
×
(c)  
1
3×
8
(d)  
13
× 6
11
         If the product is an improper fraction express it as a mixed fraction.
2. Represent pictorially :     
2 4
2×
5 5
=
To multiply a mixed fraction to a whole number, first convert the
mixed fraction to an improper fraction and then multiply.
Therefore,
5
3 2
7
×
 =
19
3
7
×
 = 
57
7
 = 
1
8
7
.
Similarly ,
2
2 4
5
×
 =
22
2
5
×
 = ?
Fraction as an operator ‘of’
Observe these figures (Fig 2.6)
The two squares are exactly similar.
Each shaded portion represents 
1
2
 of 1.
So, both the shaded portions together will represent 
1
2
 of 2.
Combine the 2 shaded 
1
2
 parts. It represents 1.
So, we say 
1
2
 of 2 is 1. We can also get it as 
1
2
 × 2 = 1.
Thus, 
1
2
 of 2 = 
1
2
 × 2 = 1
TRY THESE
TRY THESE
Find:  (i)  
3
5× 2
7
 (ii)  
4
1 × 6
9
Fig 2.6
2024-25
Page 4


MATHEMATICS 20
2.1  MULTIPLICATION OF FRACTIONS
Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length
and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its
area would be 7 × 4 = 28 cm
2
.
What will be the area of the rectangle if its length and breadth are 
7
1
2
 cm and
3
1
2
 cm respectively? Y ou will say it will be 
7
1
2
 × 
3
1
2
 = 
15
2
 × 
7
2
 cm
2
. The numbers 
15
2
and 
7
2
 are fractions. T o calculate the area of the given rectangle, we need to know how to
multiply fractions. W e shall learn that now .
2.1.1  Multiplication of a Fraction by a Whole Number
Observe the pictures at the left (Fig 2.1). Each shaded part is 
1
4
part of a circle. How much will the two shaded parts represent together?
They will represent 
1 1
4 4
+
 = 
1
2×
4
.
Combining the two shaded parts, we get  Fig 2.2 . What part of a circle does the
shaded part in Fig 2.2 represent? It represents 
2
4
 part of a circle .
Fig 2.1
Fig 2.2
Chapter  2
Fractions and
Decimals
2024-25
FRACTIONS AND DECIMALS 21
The shaded portions in Fig 2.1 taken together are the same as the shaded portion in
Fig 2.2, i.e., we get Fig 2.3.
Fig 2.3
or
1
2×
4
 =
2
4
 .
Can you now tell what this picture will represent? (Fig 2.4)
             Fig 2.4
And this? (Fig 2.5)
Fig 2.5
Let us now find 
1
3×
2
.
We have
1
3×
2
 =
1 1 1 3
2 2 2 2
+ + =
We also have                       
1 1 1 1+1+1 3×1 3
+ + = = =
2 2 2 2 2 2
So
1
3×
2
 =
3×1
2
 = 
3
2
Similarly
2
×5
3
 =
2×5
3
 = ?
Can you tell
2
3×
7
 = ?
3
4 × ?
5
=
The fractions that we considered till now, i.e., 
1 2 2 3
, , ,
2 3 7 5
 and 
3
5
 were proper fractions.
=
=
=
2024-25
MATHEMATICS 22
For improper fractions also we have,
5
2 ×
3
 =
2 × 5
3
 = 
10
3
Try ,
8
3×
7
 = ?
7
4 ×
5
  = ?
Thus, to multiply a whole number with a proper or an improper fraction, we
multiply the whole number with the numerator of the fraction, keeping the
denominator same.
1. Find:   (a)   
2
×3
7
 (b)   
9
6
7
×
(c)  
1
3×
8
(d)  
13
× 6
11
         If the product is an improper fraction express it as a mixed fraction.
2. Represent pictorially :     
2 4
2×
5 5
=
To multiply a mixed fraction to a whole number, first convert the
mixed fraction to an improper fraction and then multiply.
Therefore,
5
3 2
7
×
 =
19
3
7
×
 = 
57
7
 = 
1
8
7
.
Similarly ,
2
2 4
5
×
 =
22
2
5
×
 = ?
Fraction as an operator ‘of’
Observe these figures (Fig 2.6)
The two squares are exactly similar.
Each shaded portion represents 
1
2
 of 1.
So, both the shaded portions together will represent 
1
2
 of 2.
Combine the 2 shaded 
1
2
 parts. It represents 1.
So, we say 
1
2
 of 2 is 1. We can also get it as 
1
2
 × 2 = 1.
Thus, 
1
2
 of 2 = 
1
2
 × 2 = 1
TRY THESE
TRY THESE
Find:  (i)  
3
5× 2
7
 (ii)  
4
1 × 6
9
Fig 2.6
2024-25
FRACTIONS AND DECIMALS 23
Also, look at these similar squares (Fig 2.7).
Each shaded portion represents 
1
2
 of 1.
So, the three shaded portions represent 
1
2
 of 3.
Combine the 3 shaded parts.
It represents 1
1
2
 i.e., 
3
2
.
So, 
1
2
 of 3 is 
3
2
. Also, 
1
2
 × 3 = 
3
2
.
Thus, 
1
2
 of 3 = 
1
2
 × 3 = 
3
2
.
So we see that ‘of’ represents multiplication.
Farida has 20 marbles. Reshma has 
1
th
5
of the number of marbles what
Farida has. How many marbles Reshma has? As, ‘of ’ indicates multiplication,
so, Reshma has 
1
× 20
5
 = 4 marbles.
Similarly, we have 
1
2
of 16 is  
1
×16
2
 = 
16
2
 =  8.
Can you tell, what is (i) 
1
2
of 10?,  (ii) 
1
4
of 16?,  (iii) 
2
5
 of 25?
EXAMPLE 1 In a class of 40 students 
1
5
 of the total number of studetns like to study
English, 
2
5
 of the total number like to study Mathematics and the remaining
students like to study Science.
(i) How many students like to study English?
(ii) How many students like to study Mathematics?
(iii) What fraction of the total number of students like to study Science?
SOLUTION Total number of students in the class = 40.
(i) Of these 
1
5
 of the total number of students like to study English.
Fig 2.7
TRY THESE
2024-25
Page 5


MATHEMATICS 20
2.1  MULTIPLICATION OF FRACTIONS
Y ou know how to find the area of a rectangle. It is equal to length × breadth. If the length
and breadth of a rectangle are 7 cm and 4 cm respectively, then what will be its area? Its
area would be 7 × 4 = 28 cm
2
.
What will be the area of the rectangle if its length and breadth are 
7
1
2
 cm and
3
1
2
 cm respectively? Y ou will say it will be 
7
1
2
 × 
3
1
2
 = 
15
2
 × 
7
2
 cm
2
. The numbers 
15
2
and 
7
2
 are fractions. T o calculate the area of the given rectangle, we need to know how to
multiply fractions. W e shall learn that now .
2.1.1  Multiplication of a Fraction by a Whole Number
Observe the pictures at the left (Fig 2.1). Each shaded part is 
1
4
part of a circle. How much will the two shaded parts represent together?
They will represent 
1 1
4 4
+
 = 
1
2×
4
.
Combining the two shaded parts, we get  Fig 2.2 . What part of a circle does the
shaded part in Fig 2.2 represent? It represents 
2
4
 part of a circle .
Fig 2.1
Fig 2.2
Chapter  2
Fractions and
Decimals
2024-25
FRACTIONS AND DECIMALS 21
The shaded portions in Fig 2.1 taken together are the same as the shaded portion in
Fig 2.2, i.e., we get Fig 2.3.
Fig 2.3
or
1
2×
4
 =
2
4
 .
Can you now tell what this picture will represent? (Fig 2.4)
             Fig 2.4
And this? (Fig 2.5)
Fig 2.5
Let us now find 
1
3×
2
.
We have
1
3×
2
 =
1 1 1 3
2 2 2 2
+ + =
We also have                       
1 1 1 1+1+1 3×1 3
+ + = = =
2 2 2 2 2 2
So
1
3×
2
 =
3×1
2
 = 
3
2
Similarly
2
×5
3
 =
2×5
3
 = ?
Can you tell
2
3×
7
 = ?
3
4 × ?
5
=
The fractions that we considered till now, i.e., 
1 2 2 3
, , ,
2 3 7 5
 and 
3
5
 were proper fractions.
=
=
=
2024-25
MATHEMATICS 22
For improper fractions also we have,
5
2 ×
3
 =
2 × 5
3
 = 
10
3
Try ,
8
3×
7
 = ?
7
4 ×
5
  = ?
Thus, to multiply a whole number with a proper or an improper fraction, we
multiply the whole number with the numerator of the fraction, keeping the
denominator same.
1. Find:   (a)   
2
×3
7
 (b)   
9
6
7
×
(c)  
1
3×
8
(d)  
13
× 6
11
         If the product is an improper fraction express it as a mixed fraction.
2. Represent pictorially :     
2 4
2×
5 5
=
To multiply a mixed fraction to a whole number, first convert the
mixed fraction to an improper fraction and then multiply.
Therefore,
5
3 2
7
×
 =
19
3
7
×
 = 
57
7
 = 
1
8
7
.
Similarly ,
2
2 4
5
×
 =
22
2
5
×
 = ?
Fraction as an operator ‘of’
Observe these figures (Fig 2.6)
The two squares are exactly similar.
Each shaded portion represents 
1
2
 of 1.
So, both the shaded portions together will represent 
1
2
 of 2.
Combine the 2 shaded 
1
2
 parts. It represents 1.
So, we say 
1
2
 of 2 is 1. We can also get it as 
1
2
 × 2 = 1.
Thus, 
1
2
 of 2 = 
1
2
 × 2 = 1
TRY THESE
TRY THESE
Find:  (i)  
3
5× 2
7
 (ii)  
4
1 × 6
9
Fig 2.6
2024-25
FRACTIONS AND DECIMALS 23
Also, look at these similar squares (Fig 2.7).
Each shaded portion represents 
1
2
 of 1.
So, the three shaded portions represent 
1
2
 of 3.
Combine the 3 shaded parts.
It represents 1
1
2
 i.e., 
3
2
.
So, 
1
2
 of 3 is 
3
2
. Also, 
1
2
 × 3 = 
3
2
.
Thus, 
1
2
 of 3 = 
1
2
 × 3 = 
3
2
.
So we see that ‘of’ represents multiplication.
Farida has 20 marbles. Reshma has 
1
th
5
of the number of marbles what
Farida has. How many marbles Reshma has? As, ‘of ’ indicates multiplication,
so, Reshma has 
1
× 20
5
 = 4 marbles.
Similarly, we have 
1
2
of 16 is  
1
×16
2
 = 
16
2
 =  8.
Can you tell, what is (i) 
1
2
of 10?,  (ii) 
1
4
of 16?,  (iii) 
2
5
 of 25?
EXAMPLE 1 In a class of 40 students 
1
5
 of the total number of studetns like to study
English, 
2
5
 of the total number like to study Mathematics and the remaining
students like to study Science.
(i) How many students like to study English?
(ii) How many students like to study Mathematics?
(iii) What fraction of the total number of students like to study Science?
SOLUTION Total number of students in the class = 40.
(i) Of these 
1
5
 of the total number of students like to study English.
Fig 2.7
TRY THESE
2024-25
MATHEMATICS 24
Thus, the number of students who like to study English = 
1
5
 of  40 = 
1
40
5
×
 = 8.
(ii) Try yourself.
(iii) The number of students who like English and Mathematics = 8 + 16 = 24. Thus, the
number of students who like Science = 40 – 24 = 16.
Thus, the required fraction is 
16
40
.
EXERCISE 2.1
1. Which of the drawings (a) to (d) show :
(i)
1
2
5
×
(ii)
1
2
2
×
(iii)
2
3
3
×
(iv)
1
3
4
×
(a) (b)
(c) (d)
2. Some pictures (a) to (c) are given below. T ell which of them show:
(i)
1 3
3
5 5
× =
(ii)
1 2
2
3 3
× =
(iii)
3
3
4
2
1
4
× =
(a) (b)
(c)
3. Multiply and reduce to lowest form and convert into a mixed fraction:
(i)
3
7
5
×
(ii)
1
4
3
× (iii)
6
2
7
× (iv)
2
5
9
× (v)
2
4
3
×
   (vi)
5
6
2
× (vii)
4
11
7
×
(viii)
4
20
5
× (ix)
1
13
3
× (x)
3
15
5
×
=
=
2024-25