FRACTIONS
FRACTIONS
Bombay Cambridge Gurukul
MATHEMATICS
Choose the level
Standard III
Standard IV
Standard V
What are fractions?
How to read fractions?
More about fractions…
III Parts of a collection
Revision
More about fractions…
…numerator and denominator
Back
Equivalent fractions
Types of fractions
Fraction as division
IV Mixed numbers
Comparison of fractions
Addition of like fractions
Subtraction of like fractions
Back
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
Standard III
What
are
fractions?
Look at the figure given below.
It is a whole figure.
1
We can divide it
1
2 2 into 2 equal parts
by drawing a line.
Shade only one part of the figure.
Each part is called one half of the whole.
We write it as 1
2
Look at the figure given below.
It is a whole figure.
1 1
We can divide it
2 2 into 2 equal parts
by drawing a line.
Shade only one part of the figure.
Each part is called one half of the whole.
We write it as 1
2 Back
How to read
fractions?
How to read a fraction?
1 is read as 1 upon 2 or 1 by 2.
2
3 is read as 3 upon 7 or 3 by 7.
7
2 is read as 2 upon 5 or 2 by 5.
5
7 is read as 7 upon 9 or 7 by 9.
9
Back
More about
fractions…
The following figures are divided into
two equal parts.
1 1 1 1
2
whole
2 2 whole 2
When a whole is divided into two equal parts,
each part is called half of the whole.
One half is written as 1
2
Two halves make a whole.
Each figure is divided into two parts.
Are both parts equal?
Yes Yes No Yes
No No Yes No
Which of the following figures are
divided into two equal parts?
The following figures are divided into
three equal parts.
1 1
1 1 1 3 3
3 3 3
1
3
When a whole is divided into three equal parts,
each part is called one third of the whole.
One third is written as 1
3
Each figure is divided into three parts.
Are all the three parts equal ?
No Yes Yes No
Yes No No Yes
Which of the following figures are
divided into three equal parts?
The following figures are divided into
four equal parts.
1 1
1 1 4 4
4 4
1 1
1 1
4 4
4 4
When a whole is divided into four equal parts,
each part is called one fourth of the whole.
One fourth is written as 1
4
Each figure is divided into four parts.
Are all the four parts equal?
No No Yes Yes
Yes Yes No No
Which of the following figures are
divided into four equal parts?
Draw a line or lines to divide
each of the
following shapes into:
two equal parts
threeequal parts
four equal parts
Shade onefourth(1/3) each shape
Shade third (1/4) of each shape
Shade onehalf (1/2) of of each shape
1 1
1 1 14 14
1 1
14 1 4
1 3 3
1 1 2 2
31
2 3 23
1 11 1
4 4 4 4
3
Look at the figure given below:
It has 3 equal parts.
2 parts are shaded.
The fraction for the shaded part is 2
3
It is read as two third.
It has 4 equal parts.
3 parts are shaded.
The fraction for the shaded part is 3
4
It is read as three fourth.
Match the following
1 One fourth
2
1 One third
4
3 One half
4
1 Two third
3
2 Three fourth
3
Back
Parts
of a
collection
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.
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.
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.
Total number of insects shown below is 12.
How many insects are there in 1 of the collection?
How many insects are there in 1 of the collection?
there in the
4
3
2
12 4
12 3
2 =
= 6
4
3
Encircle one third( 11 1 )of each collection.
)of each collection.
fourth( )of each collection.
half(
23 4
6 4
3
One half of of 12 is 3
One fourthof 12 is 4
third 12 is 6
Colour one third of collection.
Colour one half of thethe collection.
Colour one fourth of the collection.
Back
Revision
How many equal parts is each rod divided into?
2 equal parts
3 equal parts
4 equal parts
5 equal parts
What fraction do the colored portions in
each of the following show?
2 2
5 3
3 1
4 4
Match the following fractions to the figures.
5 2 1 6 2 4
9 8 5 7 6 6
WHOLE 1
1
HALF
2
QUARTER 1
(ONE FOURTH) 4
THREE QUARTERS 3
(THREE FOURTH) 4
1
ONE THIRD 3
TWO THIRD 2
3
Back
More about
fractions…
…numerator and
denominator
PARTS OF A WHOLE ARE CALLED
FRACTIONS.
e.g.
1 Parts considered NUMERATOR
2 Total number of DENOMINATOR
equal parts
NUMERATOR
FRACTION =
DENOMINATOR
Remember :
Letter‘u’ is in the word ‘numerator’ and
the word ‘ up’ .
So, in the fraction 3 , 3 is the numerator.
3 8
8
Remember :
Letter ‘d’ starts the word
denominator’ and the word ‘down’ .
So, in the fraction 3 , 8 is the denominator.
8
Write the numerator and denominator
for each of the following fractions.
Fraction Numerator Denominator
2
3
3
4
1
5
5
7
Write the fraction for the numerator
and denominator given below.
Numerator Denominator Fraction
1 1 1
5 5 5
4 4 4
7 7 7
3 3 3
4 4 4
5 5 5
8 8 8
Write the fraction for the shaded part.
Numerator 5 Numerator 5
(shaded parts) (shaded parts)
Denominator 10 Denominator 8
(total parts) (total parts)
Fraction 5 Fraction 5
10 8
The End
BOMBAY CAMBRIDGE GURUKUL
Back
Standard IV
Equivalent
fractions
Is the shaded part in each pair of figures same?
Yes Yes Yes
Is the shaded part in both the figures same?
Yes
What is the fraction for the shaded part?
So, we see that =
Is the shaded part in both the figures same?
Yes
What is the fraction for the shaded part?
So, we see that =
Is the shaded part in both the figures same?
Yes
What is the fraction for the shaded part?
So, we see that =
Fractions which are equal in value to each
other are called equivalent fractions.
e.g.
1 is equivalent to 2
2 4
3 is equivalent to 6
4 8
Match the following equivalent fractions.
1 2
2 8
1 2
3 4
1 2
6
1 3
4 3
Back
Types of
fractions
Fractions where the numerator is smaller
than the denominator are called
proper fractions.
e.g. 1 3 2 4 etc.
4 5 7 9
Fractions where the numerator is greater
than the denominator are called
improper fractions.
e.g. 9 7 4 8 etc.
4 2 3 7
Fractions which have same denominator
are called like fractions.
e.g. 5 3 2 4 etc.
9 9 9 9
Fractions which have different denominators
are called unlike fractions.
e.g. 5 3 2 4 etc.
9 4 3 5
Fractions which have numeral 1 as
numerator are called unit fractions.
e.g. 1 1 1 1 etc.
9 4 3 5
Back
Fraction
as
division
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
We can write each fraction
as a division sum.
1
= 1 8
8
6 6 9
9
=
4 4 12
=
12
2
= 2 9
9
Back
Mixed
numbers
Mixed numbers include a
whole number and a fraction.
+ =
(whole number) + (fraction) = (mixed number)
Converting mixed numbers to
improper fractions.
Convert 4 1 to a improper fraction.
2
Step 1 : Multiply the denominator 2 with
whole number 4. 2 4 = 8
Step 2 : Add numerator 1 to 8 1 + 8 = 9
Step 3 : Write 9 as the numerator 9
of the improper fraction.
Step 4 : Write denominator 2 as the 9
denominator of the improper fraction. 2
(mixed number) 4 1 = 9 (improper fraction)
2 2
Converting improper fractions to
mixed numbers.
Convert 7 to a mixed number.
3
Step 1 : Divide 7 by 3. Quotient: 2
Divisor : 3
Remainder : 1
Step 2 : Write the mixed number.
2 *
The quotient becomes the whole number. *
The divisor becomes the denominator. 2 *
3
The remainder becomes the numerator. 2 1
3
7 = 2 1 (mixed number)
(improper fraction)
3 3
Converting improper fractions to
mixed numbers.
Improper can be changed to Mixed
fractions numbers
Improper can be changed to Mixed
fractions numbers
Back
Comparison of
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.
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? 4
6
So, we can say that 4 2
>
6 6
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? 2
7
So, we can say that 2 6
7 7
If there are two like fractions,
then the fraction with smaller numerator
is lesser in value.
e.g. 2 8
or = .
4 > 1
5 5
3 6
6 Back
Addition of
like fractions
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.
Addition of like fractions
+ =
1 2 3
4 4 4
1 2 3
+ =
4 4 4
Addition of like fractions
+ =
2 3 5
6 6 6
2 3 5
+ =
6 6 6
Addition of like fractions
1
3
+
1
3
2
3
1 1 2
+ =
3 3 3
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.
Addition of like fractions
4
4
+
+
The answer should be written
in the reduced form of fractions.
Addition of like fractions
2 2 2+2 4
+ = =
5 5 5 5
1 2 1+2 3
+ = =
7 7 7 7
5 2 5+2 7
+ = =
8 8 8 8
3 3 3+3 6
+ = =
9 9 9 9
Back
Subtraction of
like fractions
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.
Subtraction of like fractions
In the figure given below, three parts
out of four are shaded.
Two parts are taken away.
3 - 2 1
=
4 4 4
One part out of four is left.
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.
Subtraction of like fractions
2
6 - 4 2 = 1
=
12 12 12 2
6
2 2
14 - 2 12 = 6 = 3
=
16 16 16 2
8 2
4
The answer should be written
in the reduced form of fractions.
Subtraction of like fractions
3 - 2 = 3-2 = 1
6 6 6 6
5 - 2 = 5-2 = 3
7 7 7 7
6 - 1 = 6-1 = 5
8 8 8 8
7 - 5 = 7-5 = 2
9 9 9 9
The End
BOMBAY CAMBRIDGE GURUKUL
Back
Standard V
Reduced form
of fractions
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.
Look at the fraction given below.
6
8
We can divide the numerator and
denominator both by 2.
6 2 6 2
So, = = 3
8 2 82
4
6 divided by 2 is 3.
8 divided by 2 is 4.
Now we can not divide 3 and 4 both by
any number.
So, we can say that 3 is the reduced form of 6
4 8
Reduce the given fraction to its lowest form.
3 We can divide both, the numerator
9 and the denominator by 3.
3
3 3 = 1
9 3
The reduced form of 3 is 1
9 3
We can divide both, the numerator
10 and the denominator by 2.
2
12 10 = 5
2
12 6
The reduced form of 10 is 5
12 6
Circle the fractions which
are in the reduced form.
3 2
5 3 2 1
6 9 3
8 2
4
4 9
4 5 3 9
12 4
7 5 18 9
2 2
6 3 4 8
14 2
8 9 12 2
Back
Factors and
Multiples
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
Is 3 a factor of 20? No
Is 20 a multiple of 3? No
List the numbers that divide 15 exactly.
1 3 5 15
So, we can say that factors of 15 are
1, 3, 5 and 15.
List the numbers that divide 12 exactly.
1 2 3 4 6 12
So, we can say that factors of 12 are
1, 2, 3, 4, 6 and 12.
Every number has at least 2 factors :
1 and the number itself.
Which of the following are factors of 16?
3, 4, 5, 6, 7, 9,
1, 2, 8, 10, 16
1 2 4 8 16
Try the following….
Is 4 a factor of 14 ? No
Is 6 a factor of 24 ? Yes
Is 4 a factor of 32 ? Yes
Is 3 a factor of 17 ? No
Which of the following are multiples of 4 ?
8,
10, 12, 14, 16, 18,
20, 28,
22, 24, 26,
30
8 12 16 20 24 28
Try the following….
Is 15 a multiple of 6 ? No
Is 28 a multiple of 7 ? Yes
Is 24 a multiple of 8 ? Yes
Is 21 a multiple of 9 ? No
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 :
1, 2, 3, 6
Highest common factor (H.C.F.)
of 24 and 30 is : 6
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 : 12
Back
Addition of…
…unlike fractions,
mixed numbers
When we add two unlike fractions
(with different denominators),
we need to find the
least common multiple ( L.C.M.)
of the two denominators.
Addition of unlike fractions
1 1
+
2 6
To change both fractions to like fractions,
we find the L.C.M. of 2 and 6.
Multiples of 2 are : 2, 4, 6, 8, 10, 12…
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
1 + 1 L.C.M. of 2 and 6
Now we can add
2 6 is 6.
Step 1 : The denominator of both ________
the fractions is the same =
as the L.C.M. 6
Step 2 : Divide the common denominator with
the denominator of
the first fraction. 6 2=3
Step 3 : Multiply 3 with the numerator
3 1=3
of the first fraction.
Step 4 : Write 3 in place of the 3+
=
first numerator. 6
Step 5 : Divide the common denominator with
the denominator of
the second fraction. 6 6=1
Step 6 : Multiply 1 with the numerator of
the second fraction. 1 1=1
Step 7: Write 1 in place of the = 3+1
second numerator. 6
Step 8 : Add the numerators = 4
6
So, 1 + 1 4
=
2 6 6
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.
Addition of mixed numbers
4 1 + 2
5 5
Step1: Change the mixed number to an improper fraction.
4 1 = 4 5 + 1
=
21
5 5 5
Step 2: Add both the fractions. (21 + 2)
5 5
21 + 2 = 21 + 2 = 23
5 5 5 5
So, 4 1 + 2 = 23
5 5 5
Back
Subtraction of…
…unlike fractions,
mixed numbers
When we subtract two unlike fractions
(with different denominators),
we need to find the
least common multiple ( L.C.M.)
of the two denominators.
Subtraction of unlike fractions
2 _ 1
3 6
To change both fractions to like fractions,
we find the L.C.M. of 3 and 6.
Multiples of 3 are : 3, 6, 9, 12, 15, 18…
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
2 - 1 L.C.M. of 3 and 6
Now we can subtract
3 6 is 6.
Step 1 : The denominator of both ________
the fractions is the same =
as the L.C.M.
6
Step 2 : Divide the common denominator with
the denominator of
the first fraction. 6 3=2
Step 3: Multiply 2 with the numerator of the
first fraction. 2 2=4
Step 4 : Write 4 in place of the 4-
=
first numerator. 6
Step 5 : Divide the common denominator with
the denominator of
the second fraction. 6 6=1
Step 6 : Multiply 1 with the numerator of the
second fraction. 1 1=1
Step 7: Write 1 in place of the 4-1
second numerator. =
6
Step 8 : Subtract the numerators = 3
6
So, 2 - 1 3
=
3 6 6
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.
Subtraction of mixed numbers
3 2 - 2
7 7
Step 1: Change the mixed number to an
improper fraction.
3 2 = 3 7 + 2 =
23
7 7 7
Step 2: Subtract both the fractions. (23 - 2)
7 7
23 - 2 = 23 - 2 = 21
7 7 7 7
So, 3 2 - 2 = 21
7 7 7
Back
Multiplication
of fractions
How to multiply a fraction by a whole number ?
5 4
8
We multiply only the numerator of the fraction
with the whole number.
The denominator remains the same.
5 4 = 20
8 8
We should write the answer
in the reduced form of fractions.
20 2 = 10 2 = 5
8 2 4
2 2
So, 5 4 = 5
8 2
How to multiply a whole number by a fraction ?
5 6
9
We multiply the whole number
only with the numerator of the fraction.
The denominator remains the same.
5 6 = 30
9 9
We should write the answer
in the reduced form of fractions.
30 3 = 10
9 3 3
So, 5 6 = 10
9 3
How to multiply a fraction by a fraction ?
2 6
3
7
We multiply both the numerators.
And we multiply both the denominators.
2 6 12
=
3 7 21
We should write the answer
in the reduced form of fractions.
12 3 = 4
21 3 7
So, 2 6 4
3 7 = 7 Back
Reciprocal
of a fraction
How to write a reciprocal fraction ?
The numerator becomes the denominator.
And the denominator becomes the numerator.
Fraction Reciprocal fraction
7
9
9
REMEMBER
The reciprocal of 1 is 7 or 7
7 1
The reciprocal of a unit fraction is a whole number.
The reciprocal of 7 or 7 is 1
1 7
The reciprocal of a whole number is a unit fraction.
Back
Division
of fractions
How to divide a whole number by a fraction ?
4 6
8
We change the division sign to multiplication.
4
Then we write the reciprocal of the second fraction.
4 8
6
Multiply the numerators.
32 2
6 32 = 16
Reduce the fraction to its lowest form.
62 3
So, 4 6 = 16
8 3
How to divide a fraction by a whole number ?
4 4
5
We change the division sign to multiplication.
4
5
Then we write the reciprocal of the whole number.
4 1
5 4
Multiply the numerators.
4 2 2
20 4 = 2 =1
Reduce the fraction to its lowest form.
20 102 5
2
So, 4 4 = 1
5 5
How to divide a fraction by a fraction ?
4 2
8 3
We change the division sign to multiplication.
4
8
Then we write the reciprocal of the second fraction.
4 3
8 2
Multiply the numerators and the denominators.
12
2 2
16 12 = 6 = 3
Reduce the fraction to its lowest form.
16 8 2 4
2
So, 4 2 = 3
8 3 4
The End
BOMBAY CAMBRIDGE GURUKUL