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EQUIVALENT FRACTIONS - EXPLANATION, EQUIVALENT OR NOT, SIMPLEST/REQUIRED FORM.
Please study Illustrated explanation of fractions
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2⁄4
3⁄6
4⁄8
5⁄10
6⁄12
Clearly, the black portions (green portions)
of these figures are equal.
i.e. 1⁄2 = 2⁄4 = 3⁄6 = 4⁄8 = 5⁄10 = 6⁄12 = ........
These fractions are called equivalent fractions.
Two or more fractions representing the same part
of a whole are called equivalent fractions.
Note that, the above equivalent fractions are
1⁄2 = (1 x 2)⁄(2 x 2) = (1 x 3)⁄(2 x 3)= (1 x 4)⁄(2 x 4)
= (1 x 5)⁄(2 x 5) = (1 x 6)⁄(2 x 6) = ....
This shows that
Multiplying the numerator and denominator
of a fraction by the same non-zero number
does not change the value of the fraction.
Similarly
Dividing the numerator and denominator
of a fraction by the same non-zero number
does not change the value of the fraction.
To get a fraction equivalent to a given fraction,
we multiply or divide the numerator and denominator
of the given fraction by the same non-zero number.
Solved Example 1 of Equivalent Fractions : Explanation
Write five fractions equivalent
to each of the following :
(i) 2⁄3 (ii) 7⁄9
(i)Solution:
2⁄3 = (2 x 2)⁄(3 x 2) = (2 x 3)⁄(3 x 3) = (2 x 4)⁄(3 x 4)
= (2 x 5)⁄(3 x 5) = (2 x 6)⁄(3 x 6)
i.e. 2⁄3 = 4⁄6 = 6⁄9 = 8⁄12 = 10⁄15 = 12⁄18
Thus the five fractions equivalent to 2⁄3 are
4⁄6, 6⁄9, 8⁄12, 10⁄15, 12⁄18. Ans.
(ii)Solution:
7⁄9 = (7 x 2)⁄(9 x 2) = (7 x 3)⁄(9 x 3) = (7 x 4)⁄(9 x 4)
= (7 x 5)⁄(9 x 5) = (7 x 6)⁄(9 x 6)
i.e. 2⁄3 = 14⁄18 = 21⁄27 = 28⁄36 = 35⁄45 = 42⁄54
Thus the five fractions equivalent to 7⁄9 are
14⁄18, 21⁄27, 28⁄36, 35⁄45, 42⁄54. Ans.
Solved Example 2 of Equivalent Fractions :
Required Numerator/Denominator
Write an equivalent fraction of
(i) 3⁄7 with numerator of 27
(ii) 72⁄99 with numerator of 16
(iii) 4⁄9 with denominator of 81
(iv) 256⁄144 with denominator of 9
(i) Solution :
The given fraction is 3⁄7.
The numerator is 3.
To make it 27, from the knowledge of
Multiplication Tables,
we know, we have to multiply it by 9.
So, to get the equivalent fraction of 3⁄7
with numerator of 27, we have to multiply the
numerator and denominator with 9.
So, 3⁄7 = (3 x 9)⁄(7 x 9) = 27⁄63.
Thus, the equivalent fraction of 3⁄7
with numerator of 27 is 27⁄63. Ans.
(ii) Solution :
The given fraction is 72⁄99.
The numerator is 72.
To make it 16, we first make it 8
(since 72 is not a multiple of 16).
To make it 8, from the knowledge of
Division,
we know, we have to divide it by 9.
So, to get the equivalent fraction of 72⁄99
with numerator of 8, we have to divide the
numerator and denominator with 9.
So, 72⁄99 = (72 ÷ 9)⁄(99 ÷ 9) = 8⁄11.
To make the numerator 16, multiply
numerator and denominator with 2.
∴8⁄11 = (8 x 2)⁄(11 x 2) = 16⁄22
Here we first divided (Nr and Dr) with 9
and then multiplied (Nr and Dr) with 2.
Thus the equivalent fraction of 72⁄99
with numerator of 16 is 16⁄22. Ans.
(iii) Solution :
The given fraction is 4⁄9.
The denominator is 9.
To make it 81, from the knowledge of
Multiplication Tables,
we know, we have to multiply it by 9.
So, to get the equivalent fraction of 4⁄9
with denominator of 81, we have to multiply the
numerator and denominator with 9.
So, 4⁄9 = (4 x 9)⁄(9 x 9) = 36⁄81.
Thus the equivalent fraction of 4⁄9
with denominator of 81 is 36⁄81. Ans.
(iv) Solution :
The given fraction is 256⁄144.
The denominator is 144.
To make it 9, we use the knowledge of
Long Division.
Dividend
Divisor 9 ) 144 ( 16 Quotient
09
---
54
54
---
0 Remainder
---
So, 144÷9 = 16 or 144÷16 = 9
Thus we have to divide 144 by 16 to get 9..
So, to get the equivalent fraction of 256⁄144
with denominator of 9, we have to divide the
numerator and denominator with 16.
Dividend
Divisor 16 ) 256 ( 16 Quotient
16
-----
96
96
-----
0 Remainder
-----
Thus 256 ÷ 16 = 16
So, 256⁄144 = (256 ÷ 16)⁄(144 ÷ 16) = 16⁄9.
Thus, the equivalent fraction of 256⁄144
with denominator of 9 is 16⁄9. Ans.
To test whether two given fractions
are equivalent or not
Let a⁄b and c⁄dbe two given fractions.
If two cross products are equal, i.e. ad = bc,
then a⁄b and c⁄d are equivalent fractions,
otherwise they are not equivalent.
Solved Example 3 of Equivalent Fractions : Equivalent or Not
Are the following fractions equivalent ?
(i) 21⁄56 and 9⁄24(ii) 20⁄42 and 5⁄7
(i) Solution :
The given fractions are 21⁄56 and 9⁄24.
The cross products are 21 x 24 and 56 x 9.
Using the knowledge of
Multiplication,
let us find 21 x 24.
21 24 ---- 84 42 ---- 504 ----
Thus 21 x 24 = 504.
Let us find 56 x 9.
56 9 ---5504 ---
Thus 56 x 9 = 504.
The cross products (21 x 24 = 56 x 9 = 504) are equal.
Hence, The given fractions are 21⁄56 and 9⁄24
are equivalent fractions.
(ii) Solution :
The given fractions are 20⁄42 and 5⁄7.
The cross products are 20 x 7 and 42 x 5.
20 x 7 = 140 and 42 x 5 is more than 200.
So, the cross products are not equal.
Hence 20⁄42 and 5⁄7 are not equivalent fractions.
Exercise 1 of Equivalent Fractions : Equivalent
or Not, Required Numerator/Denominator
- Write five fractions equivalent to each of the following :
- 4⁄5
- 10⁄13
- Write an equivalent fraction of
- 4⁄13 with numerator of 20
- 49⁄63 with numerator of 14
- 3⁄8 with denominator of 72
- 126⁄144 with denominator of 16
- Are the following fractions equivalent ?
- 91⁄117 and 7⁄9
- 121⁄142 and 11⁄13
Fractions in Lowest Terms
or in Simplest Form.
If the numerator and denominator of a fraction have
no common factor except 1, then the fraction is said
to be in its lowest terms or in simplest form.
In other words, a fraction is said to be in its
lowest terms or in simplest form, if the G.C.F.
of its numerator and denominator is 1.
A fraction in simplest form is called an irreducible fraction,
otherwise it is known as a reducible fraction.
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Methods of reducing a fraction
to its lowest terms
Method 1:
Find the
Greatest Common Factor (G.C.F.)
of numerator and denominator using either
Euclidean Algorithm method
or
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Divide the numerator and denominator with this G.C.F.
to get the simplest form of the fraction.
Method 2:
Go on dividing the numerator and denominator of
the given fraction by common factor till we are left
with common factor 1 only.
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Solved Example 4 of Equivalent Fractions : Simplest Form
Reduce each one of the following
fractions to its lowest terms :
(i) 38⁄95 (ii) 92⁄207 (iii) 114⁄513
(i) Solution :
The given fraction is 38⁄95
Let us find the
G.C.F.
of the numerator (38)
and the denominator (95).
38 ) 95 ( 2
76
------
G.C.F.← 19 ) 38 ( 2
38
-------
0
-------
Thus G.C.F. of 38 and 95 = 19.
Let us divide the the numerator and the
denominator of the fraction with 19.
38⁄95 = (38 ÷ 19)⁄(95 ÷ 19) = 2⁄5.
Thus the fraction 38⁄95 reduced
to the lowest terms is 2⁄5. Ans.
(ii) Solution :
The given fraction is 92⁄207
Let us find the
G.C.F.
of the numerator (92)
and the denominator (207).
92 ) 207 ( 2
184
------
G.C.F.← 23 ) 92 ( 4
92
-------
0
-------
Thus G.C.F. of 92and 207 = 23.Let us divide the the numerator and the
denominator of the fraction with 23.
92⁄207 = (92 ÷ 23)⁄(207 ÷ 23) = 4⁄9.
Thus the fraction 92⁄207 reduced
to the lowest terms is 4⁄9. Ans.
(iii) Solution :
The given fraction is 114⁄513
Let us find the
G.C.F.
of the numerator (114)
and the denominator (513).
114 ) 513 ( 4
456
------
G.C.F.← 57 ) 114 ( 2
114
------
0
------
Thus G.C.F. of 114 and 513 = 57.
Let us divide the the numerator and the
denominator of the fraction with 57.
114⁄513 = (114 ÷ 57)⁄(513 ÷ 57) = 2⁄9.
Thus the fraction 114⁄513 reduced
to the lowest terms is 2⁄9. Ans.
Second method :
The given fraction is 114⁄513.
As the sum of the digits in both the numerator
and denominator are divisible by 3, they are
divisible by 3.
114⁄513 = 11438⁄513171= 382⁄1719= 2⁄9
First, we divided the numerator and
denominator with 3 to get 38⁄171.
Then, we divided the numerator and
denominator with 19 to get 2⁄9. Ans.
Exercise 2 of Equivalent Fractions :
Simplest Form
Reduce each one of the following fractions to its lowest terms :
- 78⁄117
- 120⁄315
- 207⁄299
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Answers to Exercise 1 of Equivalent Fractions : Equivalent or Not, Required Numerator/Denominator
-
- 8⁄10, 12⁄15, 16⁄20, 20⁄25, 24⁄30
- 20⁄26, 30⁄39, 40⁄52, 50⁄65, 60⁄78
-
- 20⁄65
- 14⁄18
- 27⁄72
- 14⁄16
-
- yes
- no
Answers to Exercise 2 of Equivalent Fractions : Simplest Form
- 2⁄3
- 8⁄23
- 9⁄13
