Title: FRACTIONS
1FRACTIONS
FRACTIONS
FRACTIONS
Bombay Cambridge Gurukul
MATHEMATICS
2Choose the level
Standard III
Standard IV
Standard V
3What are fractions?
How to read fractions?
More about fractions
III
Parts of a collection
Revision
More about fractions
numerator and denominator
Back
4Equivalent fractions
Types of fractions
Fraction as division
IV
Mixed numbers
Comparison of fractions
Addition of like fractions
Subtraction of like fractions
Back
5Reduced 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
6Standard III
7What are fractions?
8Look 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
9Look 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
10How to read fractions?
11How 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
12More about fractions
13The 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.
14Each figure is divided into two parts.
Are both parts equal?
Yes
Yes
Yes
No
No
No
No
Yes
15Which of the following figures are divided into
two equal parts?
16The 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
17Each figure is divided into three parts.
Are all the three parts equal ?
No
No
Yes
Yes
Yes
Yes
No
No
18Which of the following figures are divided into
three equal parts?
19The 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
20Each figure is divided into four parts.
Are all the four parts equal?
No
No
Yes
Yes
No
Yes
No
Yes
21Which of the following figures are divided into
four equal parts?
22Draw a line or lines to divide each of the
following shapes into
two equal parts
four equal parts
three equal parts
23Shade 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
24Look 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.
25Match the following
1 2
One fourth
1 4
One third
3 4
One half
1 3
Two third
2 3
Three fourth
Back
26Parts of a collection
27The 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.
28The 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.
29The 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.
30Encircle 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
31Colour one half of the collection.
Colour one fourth of the collection.
Colour one third of the collection.
Back
32Revision
33How many equal parts is each rod divided into?
2 equal parts
3 equal parts
4 equal parts
5 equal parts
34What fraction do the colored portions in each of
the following show?
2 5
2 3
3 4
1 4
35Match 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
361
WHOLE
1 2
HALF
QUARTER (ONE FOURTH)
1 4
3 4
THREE QUARTERS (THREE FOURTH)
1 3
ONE THIRD
2 3
TWO THIRD
Back
37More about fractions
numerator and denominator
38PARTS OF A WHOLE ARE CALLED
FRACTIONS.
e.g.
NUMERATOR
Parts considered
1 2
Total number of equal parts
DENOMINATOR
NUMERATOR
FRACTION
DENOMINATOR
39Remember 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 .
40Write 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
41Write 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
42Write 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
43The End
Created by
Department
of Research
BOMBAY CAMBRIDGE GURUKUL
Back
44Standard IV
45Equivalent fractions
46Is the shaded part in each pair of figures same?
47Is the shaded part in both the figures same?
What is the fraction for the shaded part?
So, we see that
48Is the shaded part in both the figures same?
What is the fraction for the shaded part?
So, we see that
49Is the shaded part in both the figures same?
What is the fraction for the shaded part?
So, we see that
50Fractions which are equal in value to each other
are called
equivalent fractions.
e.g.
51Match the following equivalent fractions.
1 2
2 8
2 4
1 3
2 6
1
3 3
1 4
Back
52Types of fractions
53Fractions where the numerator is smaller than the
denominator
are called
proper fractions.
1 4
2 7
4 9
3 5
e.g.
etc.
54Fractions where the numerator is greater than
the denominator
are called
improper fractions.
4 3
9 4
7 2
8 7
e.g.
etc.
55Fractions which have same denominator
like fractions.
are called
3 9
4 9
2 9
5 9
e.g.
etc.
56Fractions which have different denominators
are called
unlike fractions.
3 4
4 5
2 3
5 9
e.g.
etc.
57Fractions which have numeral 1 as numerator
are called
unit fractions.
1 9
1 4
1 5
1 3
e.g.
etc.
Back
58Fraction as division
59We can write each division sum as a fraction.
4 12
4
12
3 6
3
6
1 5
1
5
7 10
7
10
60We 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
61Mixed numbers
62Mixed numbers include a whole number and a
fraction.
(fraction)
(whole number)
(mixed number)
63Converting 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.
64Converting 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.
65Converting improper fractions to mixed numbers.
Improper fractions
Mixed numbers
Mixed numbers
Improper fractions
Back
66Comparison of fractions
67like 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.
68like 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?
69like 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?
70like 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
71Compare 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
72Addition of like fractions
73Addition 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.
74Addition of like fractions
1 4
2 4
3 4
1 4
2 4
3 4
75Addition of like fractions
2 6
3 6
5 6
2 6
3 6
5 6
76Addition of like fractions
1 3
1 3
2 3
1 3
1 3
2 3
77Addition of like fractions
When two or more like fractions are added, then
only the numerators are added together.
The denominators are not added together.
78Addition of like fractions
4
4
The answer should be written in the reduced form
of fractions.
79Addition 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
80Subtraction of like fractions
81Subtraction 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.
82Subtraction 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.
83Subtraction of like fractions
When two like fractions are subtracted, then
the smaller numerator is subtracted from the
bigger numerator.
The denominators are not subtracted.
84Subtraction 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.
85Subtraction 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

86The End
Created by
Department
of Research
BOMBAY CAMBRIDGE GURUKUL
Back
87Standard V
88Reduced form of fractions
89Reduced 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.
90Look 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
91Reduce 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
92Circle 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
93Factors and Multiples
94A 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?
95List 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
96Which 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
97Which 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
98Common 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
99Common 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
100Addition of unlike fractions, mixed numbers
101When we add two unlike fractions (with different
denominators), we need to find the least common
multiple ( L.C.M.) of the two denominators.
102Addition 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
103L.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.
104Divide 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
105Addition 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.
106Addition 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
107Subtraction of unlike fractions, mixed numbers
108When we subtract two unlike fractions (with
different denominators), we need to find the
least common multiple ( L.C.M.) of the two
denominators.
109Subtraction 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
110L.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.
111Divide 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
112Subtraction 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.
113Subtraction 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
114Multiplication of fractions
115How 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,
116How 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,
117How 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
118Reciprocal of a fraction
119How to write a reciprocal fraction ?
The numerator becomes the denominator.
And the denominator becomes the numerator.
Fraction
Reciprocal fraction
7 9
7
9
1207 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
121Division of fractions
122How 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,
123How 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
124How 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,
125The End
Created by
Department
of Research
BOMBAY CAMBRIDGE GURUKUL