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Class 8: Fractions (Lecture Notes)
Fractions:
𝑎
𝑎
𝑏
𝑏
The number of the form , where a and b are natural numbers are known as fractions. In a fraction ,
a is known as numerator and b is known as denominator. Note:
i)
ii)
Fraction is a part of whole.
The numerator and denominator of a fraction are called its terms.
Example:
7
10
is a fraction where the numerator is 7 and the denominator is 10.
Types of Fractions:
1. Decimal Fractions: A fraction whose denominator is any of the numbers 10, 100, 1000 etc., is
called a decimal fraction.
Examples:
7
71
,
77
,
, etc. are all decimal fractions
10 100 1000
2. Vulgar Fraction: A fraction whose denominator is a natural number other than the numbers 10,
100, 1000, etc., is called a Vulgar Fraction.
7 71 771
Examples: ,
,
, etc. are all Vulgar fractions
9 87 876
3. Proper Fraction: A fraction whose numerator is less than its denominator is called a proper
fractions.
7
71
Examples: ,
771
,
, etc. are all Proper fractions
19 827 876
4. Improper Fraction: A fraction whose numerator is greater than or equal to its denominator, is
called an improper fraction.
Examples:
71 1171 2771
19
,
827
,
876
, etc. are all Improper fractions
5. Mixed Number (or Mixed Fraction): When an improper fraction is written as a combination of
whole number and a proper fraction, it is called a mixed number (or mixed fraction).
Examples: 2
7
10
,3
71
,4
100
77
1000
etc. are all Mixed Number (or Mixed Fraction)
How to Convert a Mixed Fraction into an Improper Fraction?
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The numerator is obtained by multiplying the whole number part with the denominator of the
fractional part and then adding the numerator of the fractional part to the product. The
denominator of the fractional part forms the denominator of the improper fraction.
Example: 9
2
3
9×3+2
=
3
=
29
3
How to convert improper fraction into a Mixed Fraction?
Divide the numerator of the given improper fraction by the denominator. The quotient obtained
forms the whole number part, the remainder obtained forms the numerator of the fractional part
and the divisor forms the denominator of the fractional part.
Example:
29
3
=9
2
3
6. Simple Fraction: A fraction each of whose term is a whole numbers is called a simple fraction.
2 9
7
Example: , ,
, etc. are all simple fractions
3 5 70
7. Complex Fraction: A fraction at least one of whose terms is itself a fraction is called a complex
fraction.
Example:
2
9
,
3
(4)
(23)
5
,
7
70 ,
(87)
etc. are all complex fractions
8. Equivalent Fractions: Fractions having the same value are called equivalent fractions.
2 4 10
Example: , ,
, etc. are all equivalent fractions
3 6 20
9. Like Fractions: Fractions having the same denominators are called like fractions.
2 3 5
Example: , , , etc. are all decimal fractions
9 9 9
10. Unlike Fractions: Fractions having different denominators are called unlike fractions.
2
a. Example: ,
3
,
5
, etc. are all Unlike fractions
9 19 29
How to convert Unlike Fractions into Like Fractions?
1. Find the L.C.M of the denominators of all the given unlike Fractions.
2. Convert each of the fractions into an equivalent fraction having denominator equal to the L.C.M
of the denominator.
3. All the fractions would have the same denominator now and would be like fractions.
Example: Convert the fractions
3 11
,
,
8
8 12 15
into like fractions.
First take the LCM of 8, 12, and 15.
2
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2
2
3
2
5
8
4
2
2
1
1
12
6
3
1
1
1
15
15
15
5
5
1
LCM = 2 x 2 x 3 x 2 x 5 = 120
Now convert the fractions to 120 as denominators:
3
3 × 15
45
=
=
8
120
120
11
11 × 10 110
=
=
12
120
120
8
8 ×8
64
=
=
15
120
120
Fractions in Lowest Terms or in Simplest Form
A fraction is said to be in lowest or simplest form if the H.C.F of the numerator and
denominator is 1.
The fraction can be reduced to its lowest form if you divide both the numerator and the
denominator by their H.C.F.
Example: Reduce
126
162
to its lowest form.
H.C.F of 126 and 162 = 18
Therefore
126
162
=
126 ÷18
162 ÷18
=
7
9
Comparison of Fractions
1. Comparison of Fractions with Like Denominators and Unlike Numerators
If the fraction has the same denominator, then the fraction with larger Numerator is the
larger fraction. Example
5
7
>
3
7
2. Comparison of Fractions with Like Numerators by unlike Denominator
If the two fractions have the same numerator but different Denominator, then the fraction
with the smaller Denominator is the larger fraction. Example
7
24
>
7
36
3. Comparison of Fractions with Unlike Numerators and Unlike Denominators
a. Method 1
i. Step 1: First take the LCM of the denominators of the fractions
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ii. Step 2: Then convert the given unlike fractions into equivalent fractions with the
LCM as the common denominator
iii. You can compare the like fractions so obtained
5 11 13
,
Example: Arrange the fractions
,
9 18 24
17
, 𝑎𝑛𝑑
36
in an ascending order
LCM of 9, 18, 24 and 36 is 72.
Hence we can write
5 11 13
,
,
9 18 24
17
, 𝑎𝑛𝑑
Hence the ascending order is
36
as
34 39 40
,
,
72 72 72
40 44 39
,
,
72 72 72
, 𝑎𝑛𝑑
, 𝑎𝑛𝑑
34
72
44
72
b. Method 2
i. Short Cut Method (Cross Multiplication Method)
ii. If
𝑎
𝑏
i.
ii.
iii.
𝑐
𝑎𝑛𝑑
𝑑
𝑎
𝑏
𝑎
𝑏
𝑎
𝑏
>
=
<
are two fractions to be compared then
𝑐
𝑑
𝑐
𝑑
𝑐
𝑑
if ad > bc
if ad = bc
if ad < bc
To Insert Fractions between Fractions
If
𝑎
𝑏
𝑎𝑛𝑑
𝑐
𝑑
are two fractions, then If
𝑎+𝑐
𝑏+𝑑
lies between
𝑎
𝑏
𝑎𝑛𝑑
𝑐
𝑑
Example
If
5
6
𝑎𝑛𝑑
7
8
are two fractions, then If
5+7
6+8
=
12
14
=
6
7
lies between
5
6
𝑎𝑛𝑑
7
8
Operations on Fractions
1. Addition of Fractions
a. Addition of Like Fractions
Sum of like Fractions =
Sum of the Numerators
Common Dinominators
b. Addition of unlike Fractions
i. Step 1: Find the LCM of Denominators of unlike fractions
ii. Step 2: Convert unlike fractions into equivalent Fractions with their LCM as the
common Denominators
iii. Step 3: Add the like fractions so obtained
Example: Add
5
6
7
5
8
5
7
6
6
8
𝑎𝑛𝑑 . LCM of 6 and 8 is 24. So
Therefore the sum of +
=
20
24
+
21
24
=
41
𝑎𝑛𝑑
7
8
can be written as
20
24
𝑎𝑛𝑑
21
.
24
24
4
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2. Subtraction of Fractions
a. The same rules are followed as above. Instead of + operator, - operator comes into
play.
Example: Subtract
5
6
7
5
8
6
𝑓𝑟𝑜𝑚 . LCM of 6 and 8 is 24. So
5
Therefore the subtract of −
6
7
8
=
20
24
−
21
24
=
−1
𝑎𝑛𝑑
7
8
can be written as
20
24
𝑎𝑛𝑑
21
24
.
24
3. Multiplication of Fractions
Product of Fractions =
Product of the Numerators
Product of Dinominators
The answer obtained is reduced to the lowest form by dividing the numerator and denominator
by their HCF.
Easier way of multiplying fractions is that if there are common factors in one of the numerators
and one of the denominators, then we can cancel that before multiplying.
Example: Multiply
3
14
×
7
18
=
3 ×7
14×18
=
1
12
4. Division of Fractions
In order to divide two fractions, we just multiply the dividend by the reciprocal of the Divisor.
Reciprocal of a Non-Zero Fraction
Let
𝑎
𝑏
be a non-zero fraction. Then a and b are natural numbers.
The Fraction
𝑏
𝑎
is called the reciprocal of
𝑎
𝑏
Simplification of Expressions Involving Fractions
1. Use of BODMAS Rule (remember the word BODMAS)
a. We simplify the expressions by applying the operations strictly in the order
i. Brackets
ii. Of
iii. Division
iv. Multiplication
v. Addition
vi. Subtraction
b. Removal of Brackets: Follow this order
i. Bar or Vinculum (---)
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ii. Parenthesis ( )
iii. Curly Brackets { }
iv. Square Brackets [ ]
Example 1: Simplify the following
1 1 1
1 2
1
7
2
1
7 + ÷ 𝑜𝑓 − × 2 ÷ 1 𝑜𝑓 (1 − 1 )
2 2 2
4 5
3
8
5
3
=
15 1 1
1 2 7 15
7 4
+ ÷ 𝑜𝑓 − × ÷
𝑜𝑓 ( − )
2
2 2
4 5 3 8
5 3
=
15 1 1
1 2 7 15
21 − 20
+ ÷ 𝑜𝑓 − × ÷
𝑜𝑓 (
)
2
2 2
4 5 3 8
5
=
15 1 1
1 2 7 15
1
+ ÷ 𝑜𝑓 − × ÷
𝑜𝑓 ( )
2
2 2
4 5 3 8
5
=
15 1 1 2 7 1
+ ÷ − × ÷
2
2 8 5 3 8
=
15 1 8 2 7 8
+ × − × ×
2
2 1 5 3 1
=
15
112
+ 4 −
2
15
=
225 − 120 − 224
30
=
121
30
= 4
1
30
Example 2: Simplify
[3
1
1 1 1
1 1
÷ {1 − (2 − ( − ))}]
4
4 2 2
4 6
=[
13
5 1 5
1 1
÷ { − ( − ( − ))}]
4
4 2 2
4 6
=[
13
5 1 5 3−2
÷{ − ( −
)}]
4
4 2 2
12
=[
13
5 1 5
1
÷{ − ( −
)}]
4
4 2 2 12
=[
13
5 1 30 − 1
÷{ − (
)}]
4
4 2 12
=[
13
5 29
÷{ −
}]
4
4 24
=[
13
30 − 29
÷{
}]
4
24
=[
13
1
÷ ]
4
24
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=[
13
4
×
24
1
] = 78
7
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