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

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

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.

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.

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

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.

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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 = 114^{38}⁄513^{171}= 38^{2}⁄171^{9}= 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