1. Bombay Cambridge Gurukul
FRACTIONS
FRACTIONS
FRACTIONS
Bombay Cambridge Gurukul
MATHEMATICS

2. Standard III Standard IV Standard V
Choose the level
Standard III
Standard IV
Standard V

3. III What are fractions? How to read fractions? More about fractions…
Parts of a collection
Revision
More about fractions…
…numerator and denominator
Back

4. IV Equivalent fractions Types of fractions Fraction as division
Mixed numbers
Comparison of fractions
Addition of like fractions
Subtraction of like fractions
Back

5. V 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 it into 2 equal parts by 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 it into 2 equal parts by 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 is read as 1 upon 2 or 1 by 2. 2 37
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 No
Yes No No
Yes No

15.       Which of the following figures are
divided into two equal parts?      

16. The following figures are divided into three equal parts.1 3 1 3
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
Yes
Yes No
Yes No No
Yes

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
Yes
Yes No No

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:
three equal parts
two equal parts
four 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 3 1 4

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.
The fraction for the shaded part is 3 4
It is read as three fourth.

25. Match the following 1 One fourth 2 1 One third 4 3 One half 4 1
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. 4 6 3 Total number of insects shown below is 12. 12 3 ¸ 4 12 4 ¸ 3 12
How many insects are there in 1 4
of the collection?
How many insects are there in 1 2
of the collection?
How many insects are there in 1 3
of the collection? = 12 3 4 = 12 4 3 = 12 2 6
Encircle one fourth( 1 4
)of each collection.
Encircle one third( 1 3
)of each collection. 1 2
)of each collection.
Encircle one half( 4 6 3
One half of 12 is 6
One third of 12 is 4
One fourth of 12 is 3

31. Colour one half of the collection.
Colour one fourth of the collection.
Colour one half of the collection.
Colour one third of the collection.
Back

32. Revision

33. How many equal parts is each rod divided into?

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.5 9 5 9 2 8 2 8 1 5 1 5 6 7 6 7 2 6 2 6 4 6 4 6

36. 1 WHOLE 1 HALF 2 QUARTER 1 (ONE FOURTH) 4 3 THREE QUARTERS 4
(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. 1 2
Parts considered
NUMERATOR
Total number of
equal parts
DENOMINATOR
NUMERATOR
FRACTION =
DENOMINATOR

39. 3 3 8 8 So, in the fraction , 3 is the numerator. 3 8
Remember :
Letter‘u’ is in the word ‘numerator’ and the word ‘ up’ .
So, in the fraction , 3 is the numerator. 3 8 3 8 3 8
Remember :
Letter ‘d’ starts the word
denominator’ and the word ‘down’ .
So, in the fraction , 8 is the denominator. 3 8

40. Write the numerator and denominator
for each of the following fractions.
Fraction
Numerator
Denominator 2 2 3 3 3 3 4 4 1 1 5 5 5 7 5 7

41. Write the fraction for the numerator and denominator given below.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.
Numerator 5
Numerator 5
(shaded parts)
(shaded parts)
Denominator 10
Denominator 8
(total parts)
(total parts) 5 10 5 8
Fraction
Fraction

43. The End Created by Department of Research BOMBAY CAMBRIDGE GURUKUL
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44. Standard IV

45. Equivalent
fractions

46. Is the shaded part in each pair of figures same?
Yes
Yes
Yes

47. What is the fraction for the shaded part?
Is the shaded part in both the figures same?
Yes
What is the fraction for the shaded part?
So, we see that =

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

49. What is the fraction for the shaded part?
Is the shaded part in both the figures same?
Yes
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.
is equivalent to 1 2 4
is equivalent to 3 4 6 8

51. Match the following equivalent fractions.1 2 2 8 2 4 1 3 2 6 1 1 4 3
Back

52. Types of
fractions

53. Fractions where the numerator is smaller than the denominator
are called
proper fractions. 1 4 3 5 2 7 4 9
e.g.
etc.

54. Fractions where the numerator is greater than the denominator
are called
improper fractions. 9 4 7 2 4 3 8 7
e.g.
etc.

55. Fractions which have same denominator
are called
like fractions. 5 9 3 9 2 9 4 9
e.g.
etc.

56. Fractions which have different denominators
are called
unlike fractions. 5 9 3 4 2 3 4 5
e.g.
etc.

57. Fractions which have numeral 1 as numerator are called unit fractions.9 1 4 1 3 1 5
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= 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.+ =
(whole number) +
(fraction) =
(mixed number)

63. Converting mixed numbers to improper fractions.1 2 4
to a improper fraction.
Step 1 : Multiply the denominator 2 with
whole number 4. = 2 4 8 = + 1 8 9
Step 2 : Add numerator 1 to 8
Step 3 : Write 9 as the numerator
of the improper fraction. 9
Step 4 : Write denominator 2 as the
denominator of the improper fraction. 9 2 1 2 4 = 9
(improper fraction)
(mixed number)

64. Converting improper fractions to mixed numbers.7 3
to a mixed number.
Step 1 : Divide 7 by 3.
Divisor : 3
Quotient: 2
Remainder : 1
Step 2 : Write the mixed number. * 2
The quotient becomes the whole number. * 3 2
The divisor becomes the denominator.
The remainder becomes the numerator. 1 3 2 7 3 1 2 =
(improper fraction)
(mixed number)

65. Converting improper fractions to mixed numbers.
can be changed to
Improper
fractions
can be changed to
Mixed numbers
Back

66. Comparison of
fractions

67. How to compare like fractions ? 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. How to compare like fractions ? like fractions
Look at the figures shown below.
Write the fraction for both figures. 2 6 4 6 4 6
Which fractions has more shaded area?
So, we can say that 4 6 2
>

69. How to compare like fractions ? like fractions
Look at the figures shown below.
Write the fraction for both figures. 2 7 6 7 2 7
Which fraction has less shaded area?
So, we can say that 2 7 6
<

70. How to compare like fractions ? like fractions
If there are two like fractions,
then the fraction with greater numerator
is greater in value.
e.g. 4 7 3 7
>
If there are two like fractions,
then the fraction with smaller numerator
is lesser in value.
e.g. 2 9 8 9
<

71. Compare the following using
< , > or = . 4 5 1 5
> 3 7 6 7
< 2 9 2 9 = 4 6 3 6
>
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 fractions1 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 ¸ The answer should be written4 + 4 +
The answer should be written
in the reduced form of fractions.

79. Addition of like fractions2 5 2 5 4 5
2 + 2 5 + = = 1 7 2 7 3 7
1 + 2 7 + = = 5 8 2 8
5 + 2 8 7 8 + = = 3 9 3 9
3 + 3 9 6 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 6 12 4 12 2 12 1 6 - = = 2 14 16 2 16 12 16 2 2 6 8 3 4 - = = = 2 2
The answer should be written
in the reduced form of fractions.

85. Subtraction of like fractions3 6 2 6
3 - 2 6 1 6 - = = 5 7 2 7
5 - 2 7 3 7 - = = 6 8 1 8
6 - 1 8 5 8 - = = 7 9 5 9
7 - 5 9 2 9 - = =

86. The End Created by Department of Research BOMBAY CAMBRIDGE GURUKUL
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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. 6 8 6 8 2
So, 2 3 4 = = 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. 3 4 6 8
So, we can say that
is the reduced form of

91. Reduce the given fraction to its lowest form.3 9
We can divide both, the numerator
and the denominator by 3. 3 3 9 1 3 = 3
The reduced form of 3 9 1 is
We can divide both, the numerator
and the denominator by 2. 10 12 10 12 2 5 6 = 2
The reduced form of 10 12 5 6 is

92. Circle the fractions which
are in the reduced form. 5 6 3 3 9 2 8 2 1 4 3 2 4 12 4 5 7 3 5 9 18 9 4 9 6 14 2 3 8 4 9 8 12 2 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
Is 3 a factor of 20? No
Is 20 a multiple of 3? No

95. List the numbers that divide 15 exactly.3 5 15 15
So, we can say that factors of 15 are
1, 3, 5 and 15.
List the numbers that divide 12 exactly. 1 1 2 3 4 6 12 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.

96. Which of the following are factors of 16?
1, 2, 3, 4, 5, 6, 7, 8, 9, 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

97. Which of the following are multiples of 4 ?
8, 10, 12, 14, 16, 18,
20, 22, 24, 26, 28, 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

98. Common factors Common factors of 24 and 30 are :
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

99. Common multiples Common multiples of 3 and 4 are :
The multiples of 3 are :
3, 6, 9, 12, 15, 18, 21, 24 …
The multiples of 4 are :
4, 8, 12, 16, 20, 24,
Common multiples of 3 and 4 are :
12 ,
24 …
Least common multiple (L.C.M.)
of 3 and 4 is : 12
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 fractions1 2 1 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

103. ¸ ¸ ________ 6 ´ 1 2 6 + 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 :
________ 6 =
Step 2 :
Divide the common denominator with the denominator of
the first fraction. 6
2 = 3
Step 3 :
Multiply 3 with the numerator of the first fraction. 3
1 = 3
Step 4 :
Write 3 in place of the
first numerator. =
3 + 6

104. ¸ ´ Divide the common denominator with the denominator of
the second fraction.
Step 5 : 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
second numerator. =
3 + 1 6 = 4 6
Step 8 :
Add the numerators 1 2 6 +
So, = 4

105. The denominators are different,
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 numbers1 5 4 2 5 +
Step1:
Change the mixed number to an improper fraction. 1 5 4 4 5 + 1 21 5 = =
Step 2:
(21 + 2)
Add both the fractions. 2 5 + 21
21 + 2 5 23 5 = =
So, 1 5 4 2 + = 23
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 fractions2 3 _ 1 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

110. ¸ ________ 6 ´ 2 3 1 6 - 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 :
________ 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
first numerator. =
4 - 6

111. Subtract the numerators
Divide the common denominator with the denominator of
the second fraction.
Step 5 : 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
second numerator. =
4 - 1 6
Step 8 : = 3 6
Subtract the numerators 2 3 1 6 -
So, =

112. The denominators are different,
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 numbers2 7 3 -
Change the mixed number to an
improper fraction.
Step 1: 2 7 3 3 7 + 2 23 7 = =
Step 2:
Subtract both the fractions.
( ) 2 7 - 23
23 - 2 7 21 7 = =
So, 2 7 3 - = 21
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. 4 5 8 20 8 =
We should write the answer
in the reduced form of fractions. 20 8 2 10 4 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. 6 5 9 30 9 =
We should write the answer
in the reduced form of fractions. 30 9 3 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. 6 2 7 3 12 21 =
We should write the answer
in the reduced form of fractions. 12 21 3 4 7 = 2 3 6 7 4 7
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. REMEMBER
The reciprocal of 1 7 is 7 1
or 7
The reciprocal of a unit fraction is a whole number.
The reciprocal of 7 or 7 1 is 1 7
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 4 8
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 32 6 16 3
Reduce the fraction to its lowest form. = 2 6 8 16 3
So, 4 =

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 4 20 2 10 1 5
Reduce the fraction to its lowest form. = = 2 2 4 5 1 5
So, 4 =

124. ¸ ¸ ¸ ´ ´ How to divide a fraction by a fraction ? 4 8 2 3
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 12 16 6 8 3 4
Reduce the fraction to its lowest form. = = 2 2 4 8
So, 2 3 3 4 =

125. The End
Created by
Department
of Research
BOMBAY CAMBRIDGE GURUKUL