FRACTIONS

1
FRACTIONS
FRACTIONS
FRACTIONS
Bombay Cambridge Gurukul
MATHEMATICS
2
Choose the level
Standard III
Standard IV
Standard V
3
What are fractions?
How to read fractions?
More about fractions
III
Parts of a collection
Revision
More about fractions
numerator and denominator
Back
4
Equivalent fractions
Types of fractions
Fraction as division
IV
Mixed numbers
Comparison of fractions
Addition of like fractions
Subtraction of like fractions
Back
5
Reduced form of fractions
Factors and Multiples
Addition of unlike fractions, mixed numbers
V
Subtraction of unlike fractions, mixed numbers
Multiplication of fractions
Reciprocal of a fraction
Division of fractions
Back
6
Standard III
7
What are fractions?
8
Look at the figure given below.
It is a whole figure.
We can divide itinto 2 equal partsby drawing a
line.
1 2
1 2
Shade only one part of the figure.
Each part is called one half of the whole.
1 2
We write it as
9
Look at the figure given below.
It is a whole figure.
We can divide itinto 2 equal partsby drawing a
line.
1 2
1 2
Shade only one part of the figure.
Each part is called one half of the whole.
1 2
We write it as
Back
10
How to read fractions?
11
How to read a fraction?
1 2
1 by 2.
is read as
1 upon 2 or
3 7
is read as
3 upon 7 or
3 by 7.
2 5
is read as
2 upon 5 or
2 by 5.
7 9
is read as
7 upon 9 or
7 by 9.
Back
12
More about fractions
13
The following figures are divided into two equal
parts.
1 2
1 2
1 2
1 2
whole
whole
When a whole is divided into two equal parts,
each part is called half of the whole.
1 2
One half is written as
Two halves make a whole.
14
Each figure is divided into two parts.
Are both parts equal?
Yes
Yes
Yes
No
No
No
No
Yes
15
Which of the following figures are divided into
two equal parts?
16
The following figures are divided into three
equal parts.
When a whole is divided into three equal parts,
each part is called one third of the whole.
1 3
One third is written as
17
Each figure is divided into three parts.
Are all the three parts equal ?
No
No
Yes
Yes
Yes
Yes
No
No
18
Which of the following figures are divided into
three equal parts?
19
The following figures are divided into four
equal parts.
1 4
1 4
1 4
1 4
1 4
1 4
1 4
1 4
When a whole is divided into four equal parts,
each part is called one fourth of the whole.
1 4
One fourth is written as
20
Each figure is divided into four parts.
Are all the four parts equal?
No
No
Yes
Yes
No
Yes
No
Yes
21
Which of the following figures are divided into
four equal parts?
22
Draw a line or lines to divide each of the
following shapes into
two equal parts
four equal parts
three equal parts
23
Shade half (1/2) of each shape
Shade one third (1/3) of each shape
Shade one fourth (1/4) of each shape
1 4
1 4
1 3
1 3
1 4
1 4
1 2
1 2
1 3
1 3
1 3
1 2
1 2
1 4
1 4
1 4
1 4
1 3
24
Look at the figure given below
It has 3 equal parts.
2 parts are shaded.
2 3
The fraction for the shaded part is
It is read as two third.
It has 4 equal parts.
3 parts are shaded.
3 4
The fraction for the shaded part is
It is read as three fourth.
25
Match the following
1 2
One fourth
1 4
One third
3 4
One half
1 3
Two third
2 3
Three fourth
Back
26
Parts of a collection
27
The box given below has 12 stars.
They can be divided into 2 equal parts.
6
6
Each part has 6 stars.
To find the number of objects in one half of a
collection, we divide the total number of
objects by 2.
28
The box given below has 12 stars.
They can be divided into 3 equal parts.
4
4
4
Each part has 4 stars.
To find the number of objects in one third of a
collection, we divide the total number of
objects by 3.
29
The box given below has 12 stars.
They can be divided into 4 equal parts.
3
3
3
3
Each part has 3 stars.
To find the number of objects in one fourth of a
collection, we divide the total number of
objects by 4.
30
Encircle one half(
Total number of insects shown below is 12.
1 2
)of each collection.
4
6
3
One half of 12 is 6
One fourth of 12 is 3
One third of 12 is 4
31
Colour one half of the collection.
Colour one fourth of the collection.
Colour one third of the collection.
Back
32
Revision
33
How many equal parts is each rod divided into?
2 equal parts
3 equal parts
4 equal parts
5 equal parts
34
What fraction do the colored portions in each of
the following show?
2 5
2 3
3 4
1 4
35
Match the following fractions to the figures.
1 5
6 7
2 8
1 5
6 7
2 8
5 9
5 9
2 6
4 6
2 6
4 6
36
1
WHOLE
1 2
HALF
QUARTER (ONE FOURTH)
1 4
3 4
THREE QUARTERS (THREE FOURTH)
1 3
ONE THIRD
2 3
TWO THIRD
Back
37
More about fractions
numerator and denominator
38
PARTS OF A WHOLE ARE CALLED
FRACTIONS.
e.g.
NUMERATOR
Parts considered
1 2
Total number of equal parts
DENOMINATOR
NUMERATOR

FRACTION
DENOMINATOR
39
Remember Letteru is in the word numerator
and the word up .
3 8
3
8
Remember Letter d starts the word
denominator and the word down .
40
Write the numerator and denominator for each of
the following fractions.
Fraction
Numerator
Denominator
2
2 3
3
3 4
3
4
1
1 5
5
5 7
5
7
41
Write the fraction for the numerator and
denominator given below.
Numerator
Denominator
Fraction
1 5
1 5
1 5
1 5
1 5
4 7
4 7
4 7
4 7
4 7
3 4
3 4
3 4
3 4
3 4
5 8
5 8
5 8
5 8
5 8
42
Write the fraction for the shaded part.
5
Numerator
Numerator
5
(shaded parts)
(shaded parts)
8
Denominator
Denominator
10
(total parts)
(total parts)
5 8
5 10
Fraction
Fraction
43
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Back
44
Standard IV
45
Equivalent fractions
46
Is the shaded part in each pair of figures same?
47
Is the shaded part in both the figures same?
What is the fraction for the shaded part?

So, we see that
48
Is the shaded part in both the figures same?
What is the fraction for the shaded part?
So, we see that

49
Is the shaded part in both the figures same?
What is the fraction for the shaded part?
So, we see that

50
Fractions which are equal in value to each other
are called
equivalent fractions.
e.g.
51
Match the following equivalent fractions.
1 2
2 8
2 4
1 3
2 6
1
3 3
1 4
Back
52
Types of fractions
53
Fractions where the numerator is smaller than the
denominator
are called
proper fractions.
1 4
2 7
4 9
3 5
e.g.
etc.
54
Fractions where the numerator is greater than
the denominator
are called
improper fractions.
4 3
9 4
7 2
8 7
e.g.
etc.
55
Fractions which have same denominator
like fractions.
are called
3 9
4 9
2 9
5 9
e.g.
etc.
56
Fractions which have different denominators
are called
unlike fractions.
3 4
4 5
2 3
5 9
e.g.
etc.
57
Fractions which have numeral 1 as numerator
are called
unit fractions.
1 9
1 4
1 5
1 3
e.g.
etc.
Back
58
Fraction as division
59
We can write each division sum as a fraction.
4 12
4
12


3 6
3
6


1 5

1
5

7 10
7
10


60
We can write each fraction as a division sum.
1 8

1
8

6 9
6
9


4 12

4
12

2 9

2
9

Back
61
Mixed numbers
62
Mixed numbers include a whole number and a
fraction.


(fraction)

(whole number)
(mixed number)

63
Converting mixed numbers to improper fractions.
Step 1 Multiply the denominator 2 with whole
number 4.
Step 2 Add numerator 1 to 8
Step 3 Write 9 as the numerator
of the improper fraction.
Step 4 Write denominator 2 as the denominator
of the improper fraction.
64
Converting improper fractions to mixed numbers.
Step 1 Divide 7 by 3.
Step 2 Write the mixed number.
The quotient becomes the whole number.
The divisor becomes the denominator.
The remainder becomes the numerator.
65
Converting improper fractions to mixed numbers.
Improper fractions
Mixed numbers
Mixed numbers
Improper fractions
Back
66
Comparison of fractions
67
like fractions
How to compare like fractions ?
Look at the figures shown below.
Each figure is divided into 4 equal parts.
(A)
(B)
Which figure has more shaded parts?
The first figure (A) has more shaded parts.
68
like fractions
How to compare like fractions ?
Look at the figures shown below.
Write the fraction for both figures.
2 6
4 6
Which fractions has more shaded area?
69
like fractions
How to compare like fractions ?
Look at the figures shown below.
Write the fraction for both figures.
2 7
6 7
Which fraction has less shaded area?
70
like fractions
How to compare like fractions ?
If there are two like fractions, then the
fraction with greater numerator is greater in
value.
3 7
e.g.
4 7
gt
If there are two like fractions, then the
fraction with smaller numerator is lesser in
value.
e.g.
8 9
2 9
lt
71
Compare the following using lt , gt or .
4 5
1 5
gt
3 7
6 7
lt
2 9
2 9

4 6
3 6
gt
Back
72
Addition of like fractions
73
Addition of like fractions
In the circle given below only one part out of
five is shaded.
Two more parts of the circle are shaded.
The circle has three shaded parts.
74
Addition of like fractions


1 4
2 4
3 4
1 4
2 4
3 4


75
Addition of like fractions


2 6
3 6
5 6
2 6
3 6
5 6


76
Addition of like fractions
1 3

1 3
2 3
1 3
1 3
2 3


77
Addition of like fractions
When two or more like fractions are added, then
only the numerators are added together.
The denominators are not added together.
78
Addition of like fractions
4

4


The answer should be written in the reduced form
of fractions.
79
Addition of like fractions
2 5
2 5
4 5
2 2 5



1 7
2 7
3 7
1 2 7



5 8
2 8
7 8
5 2 8



3 9
3 9
6 9
3 3 9



Back
80
Subtraction of like fractions
81
Subtraction of like fractions
In the figure given below, three parts out of
five parts are shaded.
Two parts are taken away.
One part out of five is left.
82
Subtraction of like fractions
In the figure given below, three parts out of
four are shaded.
Two parts are taken away.
3 4
2 4
1 4
-

One part out of four is left.
83
Subtraction of like fractions
When two like fractions are subtracted, then
the smaller numerator is subtracted from the
bigger numerator.
The denominators are not subtracted.
84
Subtraction of like fractions
2
4 12
6 12
2 12
1 6
-


2
2
2
14 16
2 16
12 16
6 8
3 4
-



2
2
The answer should be written in the reduced form
of fractions.
85
Subtraction of like fractions
3 6
2 6
1 6
3 - 2 6

-

5 7
2 7
3 7
5 - 2 7

-

6 8
1 8
5 8
6 - 1 8

-

7 9
5 9
2 9
7 - 5 9

-

86
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Created by
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of Research
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Back
87
Standard V
88
Reduced form of fractions
89
Reduced form of fractions
A fraction is said to be in the reduced form if
its numerator and denominator cannot be divided
by a common number.
90
Look at the fraction given below.
6 8
We can divide the numerator and denominator both
by 2.
2
6 8
2 2
6 8
3 4

So,


2
6 divided by 2 is 3.
8 divided by 2 is 4.
Now we can not divide 3 and 4 both by any number.
6 8
3 4
is the reduced form of
So, we can say that
91
Reduce the given fraction to its lowest form.
We can divide both, the numerator and the
denominator by 3.
3
3 9
1 3

3
We can divide both, the numerator and the
denominator by 2.
2
10 12
5 6

2
92
Circle the fractions which are in the reduced
form.
3
2
2 8
1 4
5 6
3 9
3
2
9
4
3 5
9 18
5 7
4 12
9
4
2
2
3 8
6 14
4 9
8 12
2
2
Back
93
Factors and Multiples
94
A number that divides a given number completely
(without leaving a remainder) is called its
factor.
e.g.
5 divides 20 exactly.
So, 5 is a factor of 20.
And 20 is a multiple of 5.
Is 20 exactly divisible by 3?
No
No
Is 3 a factor of 20?
No
Is 20 a multiple of 3?
95
List the numbers that divide 15 exactly.
So, we can say that factors of 15 are 1, 3, 5
and 15.
List the numbers that divide 12 exactly.
So, we can say that factors of 12 are 1, 2, 3,
4, 6 and 12.
Every number has at least 2 factors
the number itself.
1 and
96
Which of the following are factors of 16?
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 16
?
?
?
?
?
?
?
?
?
?
?
1
2
4
16
8
Try the following.
Is 4 a factor of 14 ?
No
Yes
Is 6 a factor of 24 ?
Yes
Is 4 a factor of 32 ?
Is 3 a factor of 17 ?
No
97
Which of the following are multiples of 4 ?
8, 10, 12, 14, 16, 18, 20, 22, 24, 26,
28, 30
?
?
?
?
?
?
?
?
?
?
?
?
12
16
20
28
8
24
Try the following.
No
Is 15 a multiple of 6 ?
Yes
Is 28 a multiple of 7 ?
Yes
Is 24 a multiple of 8 ?
Is 21 a multiple of 9 ?
No
98
Common factors
The factors of 24 are
1, 2, 3, 4, 6, 8, 12 and 24.
The factors of 30 are
1, 2, 3, 5, 6, 10, 15 and 30.
Common factors of 24 and 30 are
3,
1,
2,
6
Highest common factor (H.C.F.) of 24 and 30 is
99
Common multiples
The multiples of 3 are
3, 6, 9, 12, 15, 18, 21, 24
The multiples of 4 are
4, 8, 12, 16, 20, 24, 28 ...
Common multiples of 3 and 4 are
12 ,
24
Least common multiple (L.C.M.) of 3 and 4 is
Back
100
Addition of unlike fractions, mixed numbers
101
When we add two unlike fractions (with different
denominators), we need to find the least common
multiple ( L.C.M.) of the two denominators.
102
Addition of unlike fractions
1 6
1 2

To change both fractions to like fractions, we
find the L.C.M. of 2 and 6.
2, 4, 6, 8, 10, 12
Multiples of 2 are
Multiples of 6 are
6, 12, 18, 24, 30
Common multiples of 2 and 6 are
6, 12
Least common multiple (L.C.M.) of 2 and 6 is
6
103
L.C.M. of 2 and 6 is 6.
Now we can add

The denominator of both the fractions is the
same as the L.C.M.
Step 1
Step 2
Divide the common denominator with the
denominator of the first fraction.
Step 3
Multiply 3 with the numerator of the first
fraction.
Step 4
Write 3 in place of the first numerator.
104
Divide the common denominator with the
denominator of the second fraction.
Step 5
Step 6
Multiply 1 with the numerator of the second
fraction.
Step 7
Write 1 in place of the second numerator.
Step 8
Add the numerators
105
Addition of unlike fractions
The denominators are different, so, we find the
L.C.M. of 2 and 4.
L.C.M. of 2 and 4 is 4.

Then, numerators are added.
106
Addition of mixed numbers
2 5

Change the mixed number to an improper fraction.
Step1
21 5


(21 2) 5 5
Step 2
Add both the fractions.
21 2 5
23 5


So,
Back
107
Subtraction of unlike fractions, mixed numbers
108
When we subtract two unlike fractions (with
different denominators), we need to find the
least common multiple ( L.C.M.) of the two
denominators.
109
Subtraction of unlike fractions
1 6
2 3
_
To change both fractions to like fractions, we
find the L.C.M. of 3 and 6.
3, 6, 9, 12, 15, 18
Multiples of 3 are
Multiples of 6 are
6, 12, 18, 24, 30
Common multiples of 8 and 4 are
6, 12
Least common multiple (L.C.M.) of 8 and 4 is
6
110
L.C.M. of 3 and 6 is 6.
Now we can subtract
The denominator of both the fractions is the
same as the L.C.M.
Step 1
Step 2
Divide the common denominator with the
denominator of the first fraction.
Multiply 2 with the numerator of the first
fraction.
Step 3
Step 4
Write 4 in place of the first numerator.
111
Divide the common denominator with the
denominator of the second fraction.
Step 5
Step 6
Multiply 1 with the numerator of the second
fraction.
Step 7
Write 1 in place of the second numerator.
Step 8
Subtract the numerators
112
Subtraction of unlike fractions
The denominators are different, so, we find the
L.C.M. of 2 and 4.
L.C.M. of 2 and 4 is 4.
Then, numerators are subtracted.
113
Subtraction of mixed numbers
Change the mixed number to an improper fraction.
Step 1
23 7


Step 2
(23 - 2) 7 7
Subtract both the fractions.
23 - 2 7
21 7


So,
Back
114
Multiplication of fractions
115
How to multiply a fraction by a whole number ?
5 8

4
We multiply only the numerator of the fraction
with the whole number.
The denominator remains the same.
20 8

We should write the answer in the reduced form
of fractions.
2 2
20 8
10 4
2 2
5 2




5 8
4
5 2


So,
116
How to multiply a whole number by a fraction ?
6 9

5
We multiply the whole number only with the
numerator of the fraction.
The denominator remains the same.
30 9

We should write the answer in the reduced form
of fractions.
3 3
30 9
10 3


6 9
10 3

5
So,

117
How to multiply a fraction by a fraction ?
2 3
6 7

We multiply both the numerators.
And we multiply both the denominators.
12 21

We should write the answer in the reduced form
of fractions.
3 3
12 21
4 7


6 7
4 7
2 3
So,


Back
118
Reciprocal of a fraction
119
How to write a reciprocal fraction ?
The numerator becomes the denominator.
And the denominator becomes the numerator.
Fraction
Reciprocal fraction
7 9
7
9
120
7 1
1 7
The reciprocal of
or 7
is
The reciprocal of a unit fraction is a whole
number.
1 7
7 1
The reciprocal of 7 or
is
The reciprocal of a whole number is a unit
fraction.
Back
121
Division of fractions
122
How to divide a whole number by a fraction ?
6 8

4
We change the division sign to multiplication.


4
Then we write the reciprocal of the second
fraction.

8 6
4
Multiply the numerators.
32 6
2
16 3
32 6

Reduce the fraction to its lowest form.
2
6 8
16 3

4
So,

123
How to divide a fraction by a whole number ?
4 5

4
We change the division sign to multiplication.
4 5


Then we write the reciprocal of the whole number.

4 5
1 4
Multiply the numerators.
4 20
2
2
1 5
4 20
2 10


Reduce the fraction to its lowest form.
2
2
4 5
1 5

So,
4

124
How to divide a fraction by a fraction ?
2 3
4 8

We change the division sign to multiplication.
4 8


Then we write the reciprocal of the second
fraction.

4 8
3 2
Multiply the numerators and the denominators.
12 16
2
2
3 4
12 16
6 8


Reduce the fraction to its lowest form.
2
2
4 8
3 4

2 3
So,

125
The End
Created by
Department
of Research
BOMBAY CAMBRIDGE GURUKUL