EQUIVALENT FRACTIONS - EXPLANATION, EQUIVALENT OR NOT, SIMPLEST/REQUIRED FORM.

Please study
Fractions before Equivalent Fractions
if you have not already done so.

There we studied about
half, quarter, three fourth
with examples and exercises.

Also study Fractions Made Easy.

There we studied about
concept of fraction in general
with examples and exercises.

Learn/Teach
Fractions through Fun Games.
For details, see near
the bottom of this page.

Here, we will see Illustrated explanation
of fractions which are equivalent,
testing fractions to know whether
they are equivalent or not,
reducing a fraction to its
lowest terms/Simplest Form,
to get a fraction of required
numerator/denominator which is equivalent.

Illustrated explanation of fractions
which are equivalent

In each of the figures given below,
the fraction represented by the
black portion (green portion)
is given along with figure.

1⁄2

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 
                 ---   

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 
                 -----   

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⁄d be 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
----

Let us find 56 x 9.

 56
  9                             
---           5
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

1. Write five fractions equivalent to each of the following :
   1. 4⁄5
   2. 10⁄13
2. Write an equivalent fraction of
   1. 4⁄13 with numerator of 20
   2. 49⁄63 with numerator of 14
   3. 3⁄8 with denominator of 72
   4. 126⁄144 with denominator of 16
3. Are the following fractions equivalent ?
   1. 91⁄117 and 7⁄9
   2. 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.

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
Prime Factorization method

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.

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 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 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 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 :

1. 78⁄117
2. 120⁄315
3. 207⁄299

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Answers to Exercise 1 of Equivalent Fractions : Equivalent or Not, Required Numerator/Denominator

1.  
   1. 8⁄10, 12⁄15, 16⁄20, 20⁄25, 24⁄30
   2. 20⁄26, 30⁄39, 40⁄52, 50⁄65, 60⁄78
2. 1. 20⁄65
   2. 14⁄18
   3. 27⁄72
   4. 14⁄16
3. 1. yes
   2. no

Answers to Exercise 2 of Equivalent Fractions : Simplest Form

1. 2⁄3
2. 8⁄23
3. 9⁄13