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

Your Ad Here



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

                                                                                                


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 ab and cdbe two given fractions.
If two cross products are equal, i.e. ad = bc,
then ab and cd 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                             
---           5
504
---

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

  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
For Answers, see at the bottom of the page.

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.

Get The Best Grades With the Least Amount of Effort

Here is a collection of proven tips,
tools and techniques to turn you into
a super-achiever - even if you've never
thought of yourself as a "gifted" student.

The secrets will help you absorb, digest
and remember large chunks of information
quickly and easily so you get the best grades
with the least amount of effort.

If you apply what you read from the above
collection, you can achieve best grades without
giving up your fun, such as TV, surfing the net,
playing video games or going out with friends!

Know more about the

Speed Study System.



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.

Great Deals on School & Homeschool Curriculum Books

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 = 11438513171= 3821719= 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.

Great deals on School & Homeschool Curriculum Books and Software

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


Progressive Learning of Math : Equivalent Fractions

Recently, I have found a series of math curricula
(Both Hard Copy and Digital Copy) developed by a Lady Teacher
who taught everyone from Pre-K students to doctoral students
and who is a Ph.D. in Mathematics Education.

This series is very different and advantageous
over many of the traditional books available.
These give students tools that other books do not.
Other books just give practice.
These teach students “tricks” and new ways to think.

These build a student’s new knowledge of concepts
from their existing knowledge.
These provide many pages of practice that gradually
increases in difficulty and provide constant review.

These also provide teachers and parents with lessons
on how to work with the child on the concepts.

The series is low to reasonably priced and include

Elementary Math curriculum

and

Algebra Curriculum.



Learning/Teaching Math Can Be Fun

Here is a collection of kids math
games and fun math activities for
the class room or for the home, to
make math exciting and easy to learn.

They help you

* To save you time and money to be
spent on resources, games and books.

* To become a wonderful, fun teacher
or parent who knows how to make math
fun, interesting and effective.

* To cater for all different ability
levels and cater for different
learning styles.

* To see your kids math skills soar
and their grades in math going
up and up.

This Collection of Fun Math Games
are electronic books (e-books)
that are downloaded to your computer
in a flash. You can start printing
games right away. You get to print
only what you want and as many
copies as you need.

For more information or to have
some FREE samples or to order
click HERE.

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
    1. 20⁄65
    2. 14⁄18
    3. 27⁄72
    4. 14⁄16
    1. yes
    2. no

Answers to Exercise 2 of Equivalent Fractions : Simplest Form

  1. 2⁄3
  2. 8⁄23
  3. 9⁄13