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Equivalent Fractions

The fractions having the same value are called equivalent fractions. Their numerator and denominator can be different but, they represent the same part of a whole.

Consider the following:

(i)

Equivalent Fractions

12


(ii)

Equivalent Fractions

24


(iii)

Equivalent Fractions

48


(iv)

816


(v)

510


(vi)

Equivalent Fractions

1020


(viii)

612


(viii)

Equivalent Fractions

36


We can see the shade portion with respect to the whole shape in the figures from (i) to (viii)

In; (i) Shaded to whole shape is half to whole             i.e., 1/2

(ii) Shaded to whole shape is 2 to fourth                   i.e., 2/4 = 1/2

(iii) Shaded to whole shape is 4 to eighth                   i.e., 4/8 = 1/2

(iv) Shaded to whole shape is 8 to sixteenth              i.e., 8/16 = 1/2

(v) Shaded to whole shape is 5 to tenth                    i.e., 5/10 = 1/2

(vi) Shaded to whole shape is 10 to 20th                   i.e., 10/20 = 1/2

(vii) Shaded to whole shape is 6 to 12th                    i.e., 6/12 = 1/2

(viii) Shaded to whole shape is 3 to 6th                     i.e., 3/6 = 1/2

Thus, 12, 24, 48, 816, 510, 1020, 612, 36, etc., each fraction represents half portion of the shape, which are all equal. All have different numerator and denominator but they all have the same value because they represent the same shaded area i.e., half of the rectangle.

So, 12, 24, 48, 816, 510, 1020, 612, 36 are equivalent fractions.

We can express it as, 12 = 24 = 48 = 816 = 510 = 1020 = 612 = 36 = 12.

The fractions having different numerators and denominators but representing equal value or magnitude are called equivalent fractions.

Note:

The fraction 12 and 24 and 48 show the same amount of shaded or colored parts. 12 and 24 and 48 are equivalent fractions.

Equivalent fractions are fractions that have different forms but the same value.


Again, observe the following figures:

Equivalent Fractions

We see that the shaded regions in all the figures are the same.

So, the fractions represented by these shaded portions are equal.

Hence, 12, 24, 36, 48 are equivalent fractions.

12 = 24 = 36 = 48.

Also 12 = 1×22×2 = 1×32×3 = 1×42×4 and so on.


Again observe the following figures:

Equivalent Fractions Example

We see that the shaded regions in all the circles are the same.

So, the fractions represented by these shaded portions are equal.

Hence, 13, 26, 39, 412 are equivalent fractions.

Or, 13 = 26 = 39 = 412 are equivalent fractions.

Also 13 = 1×23×2 = 1×33×3 = 1×43×4

Similarly, we can conclude that 15 = 1×25×2 = 1×35×3 = 1×45×4 = 1×55×5 and so on.

Or 15 = 210 = 315 = 420 = 525.


Hence, we can say that,

When we multiply the numerator and the denominator of a fraction by the same non- zero number, we get its equivalent fraction.


Again observe the following equivalent fractions 27, 414, 621

414 = 4÷214÷2 = 27,

621 = 6÷321÷3 = 27

So,

We see that when the numerator and the denominator of a fraction are divided by the same non-zero number, we get its equivalent fraction.


Building Equivalent Fractions:

1. Change 2/5 to an equivalent fraction with denominator 15.

Equivalent Fractions





Note:

Multiply numerator and denominator by the same number to get the required denominator.


2. Change 912 to an equivalent fraction with denominator 4.

Equivalent Fraction 1





Note:

To find an equivalent fraction with smaller denominator, you can divide the numerator and denominator with the same number.



3. We can build equivalent fraction with multiples of numerator and denominator.

Write the next three equivalent fractions.

next three equivalent fractions
Equivalent Fractions

Note:

Equivalent fractions have the same value.

Equivalent fraction can be built to very large numbers.

Equivalent fraction can be reduced to the lowest term.


Look at the following:

Equivalent Fractions

Look at the figures above.

In A the three different fractions 13, 26

and 412 represent the same shaded part.

In B the three different fractions 14, 28 and 416 represent the same shaded part.

That is,

13 = 26 = 412;

14 = 28 = 416


Fractions which represent the same part of a whole thing or which indicates the same number of things in a group are called ‘Equivalent Fractions’.

When we multiply the numerator and denominator of a fraction by the same number (other than zero) we get an equivalent fraction with a higher numerator and denominator.

For example;

1×26×2 = 212

1×36×3 = 318

1×46×4 = 424

16 = 212 = 318 = 424


When we divide the numerator and denominator of a fraction by the same number (other than zero) we get an equivalent fraction with a lower numerator and denominator.

For example:

1. 36÷248÷2 = 1824

18÷324÷3 = 68

6÷28÷2 = 34

34 = 68 = 1824 = 3648


2. 46 = .......18

Find the relation between the denominators.

6 × ....... = 18 (since 6 < 18)

6 × 3 = 18

Therefore, 4×36×3 = 1218

Therefore, 46 = 1218


3. 910 = 27.......

Find the relation between the numerators.

6 × ....... = 27 ? (since 9 < 27)

9 × 3 = 27

Therefore, 9×310×3 = 2730

Therefore, 910 = 2730


4. 2432 = .......8

32 ÷ 8 = 4 (since 32 >8)

24÷432÷4 = 68

Therefore, 2432 = 68



Checking for Equivalence of Fractions:

To check whether the given fractions are equivalent or not, we cross-multiply the two fractions. If the products come out to be equal, the fractions are equivalent otherwise they are not.

Cross Product Rule:

If the cross products of two fractions are equal, then they are equivalent fractions.


Note: 

Cross multiplication means multiplying the numerator of a fraction with the denominator of the other fraction and multiplying the denominator of the other fraction with the numerator of the first fraction.


For example:

1. Is 23 equivalent to 46?



2 × 6 = 12

4 × 3 = 12

Therefore, 23 is equivalent to 46


2. Is 24 equivalent to 35?

Equivalent Fractions

2 × 5 = 10

3 × 4 = 12

Therefore, 24 is not equivalent to 35


3. Check whether 27 and 828 are equivalent fractions or not.

1st numerator × 2nd  denominator = 2 × 28 = 56

1st denominator × 2nd numerator = 7 × 8 = 56

Since the products are equal, 27 and 828 are equivalent fractions.


4. Check whether 36 and 515 are equivalent fractions or not.

1st numerator × 2nd  denominator = 3 × 15 = 45

1st denominator × 2nd numerator = 6 × 5 = 30

Since the products are not equal, 36 and 515 are not equivalent fractions.



Solved Examples on Equivalent Fractions:

1. Write the next three equivalent fractions of 23.

Solution:

Multiply the numerator and the denominator of the given fraction:

by 2, 3, 4 and 5.

23 = 2×23×2 = 46;

23 = 2×33×3 = 69;

23 = 2×43×4 = 812

So, 46 , 69 and 812 are equivalent fractions of 23


2. Find an equivalent fraction of 2/5 with numerator 8.

Solution:

Let 25 = 8.....

To get 8 in the numerator, we multiply 2 by 4, and also 5 by 4.

So, 25 = 2×45×4 = 820

Hence 25 and 820 are equivalent fractions.


3. Find an equivalent fraction of 4256 with numerator 8.
Solution:

Let 4256 = .....8

To get 8 in the denominator, we divide 56 by 7 and also 42 by 7.

So 42÷756÷7 = 68

Hence 4256 and 68 are equivalent fractions.



Activity on Equivalent Fractions:

Objective: Understanding equivalent fractions by paper folding.

Materials Required: White sheets of paper, pencil colours.

Procedure/Demonstration: Divide students in groups of 3.

They will be given A-4 sheets or smaller sheets (but all the same size) and asked by the teacher to fold and create the following:

12, 24, 48


Each sheet will represent one of the above. They will of course be folded successively by the students. There will be some experimentation in this.
Once they have folded and unfolded, they will need to use colour to shade
12, 24, 48

Equivalent Fractions

Now, in each group, every bit of fraction will be cut out.
So, each group will have 14 pieces.
Then on direction from the teacher, students will find how many of the smaller pieces cover fully the piece with written on it. They will then paste those together on a separate sheet and present this finding thus.

Equivalent Fractions

So, 12 = 24 = 48.


Worksheet on Equivalent Fractions

I. Find 4 equivalent fractions for the given fractions by multiplying.

(i) 37

(ii) 29

(iii) 45

(vi) 711


Answers: 

I. (i) 614, 921, 1228, 1535

(ii) 418, 627, 836, 1045

(iii) 810, 1215, 1620, 2025

(vi) 1422, 2133, 2844, 3555


II. Fill the boxes to make equivalent fractions:

(i) 34 = 16

(ii) 59 = 35

(iii) 78 = 64

(vi) 7 = 6399

(v) 213 = 51

(vi) 1117 = 51


Answers:

II. (i) 12

(ii) 63

(iii) 56

(vi) 11

(v) 8

(vi) 33


III. Write two equivalent fractions for the following.

(i) 23, ........., .........

(ii) 45, ........., .........

(iii) 67, ........., .........

(iv) 15, ........., .........

(v) 38, ........., .........

(vi) 510, ........., .........


Answer:

III. (i) 46, 69

(ii) 810, 2025

(iii) 2428, 3642

(iv) 315, 1050

(v) 924, 2156

(vi) 510, 38



IV. Find the missing terms of the following fractions.

(i) 17 = 5.......

(ii) 34 = 12.......

(iii) 25100 = 1.......

(iv) 55100 = .......20

(v) 37 = .......63

(vi) 56 = .......24

(vii) 611 = 18.......

(viii) 1548 = .......16

(ix) 2540 = .......8 


Answer:

IV. (i) 35

(ii) 16

(iii) 4

(iv) 11

(v) 27

(vi) 20

(vii) 33

(viii) 5

(ix) 5


V. Find an equivalent fraction of 34 with

(i) Numerator 27

(ii) Denominator 28

(iii) Numerator 30

(iv) Denominator 12


Answer:

V. (i) 2736

(ii) 2128

(iii) 3040

(iv) 912


VI. Find an equivalent fraction of 5460 with

(i) Numerator 27

(ii) Denominator 10

(iii) Numerator 9

(iv) Denominator 20


Answer:

VI. (i) 2730

(ii) 910

(iii) 910

(iv) 1820


VII. Indicate which of the following pairs of fractions are equivalent:

(i) 35 and 915

(ii) 28 and 1040

(iii) 57 and 2542

(iv) 911 and 2734

(v) 413 and 1239


Answer:

VII: (i) Equivalent Fraction

(ii) Equivalent Fraction

(iii) Not Equivalent Fraction

(iv) Not Equivalent Fraction

(v) Equivalent Fraction


VIII. Write the next three equivalent fractions of each of the following:

(i) 29

(ii) 12

(iii) 48

(iv) 17

(v) 25

(vi) 29

(vii) 12

(viii) 48

(ix) 17

(x) 25


Answer:

VIII. (i) 418627836

(ii) 2448612

(iii) 81612241632

(iv) 214321428

(v) 410615820

(vi) 418627836

(vii) 2448612

(viii) 81612241632

(ix) 214321428

(x) 410615820


IX. Fill in box with the missing numerator to make the statement true:

(i) 13.....15

(ii) .....93054

(iii) 34.....20

(iv) .....82772

(v) 34.....24

(vi) 12.....14

(vii) 34.....32

(viii) 47.....14


Answer:

IX. (i) 5

(ii) 5

(iii) 15

(iv) 3

(v) 18

(vi) 7

(vii) 24

(viii) 8 


X. Fill in each box with the missing denominator to make the statement true:

(i) 4.....2863

(ii) 2314.....

(iii) 6724.....

(iv) 81596.....

(v) 5715.....

(vi) 276.....

(vii) 3524.....

(viii) 81116.....


Answer:

X. (i) 9

(ii) 21

(iii) 28

(iv) 180

(v) 21

(vi) 21

(vii) 40

(viii) 22


Answer:

XI. (i) Equivalent Fractions.

(ii) Non-equivalent Fractions.

(iii) Equivalent Fractions.

(iv) Non-equivalent Fractions.

(v) Non-equivalent Fractions.

(vi) Equivalent Fractions.


XII. Fill in the boxes to get equivalent fractions.

Equivalent Fractions Worksheet

Answer:

XII: 46, 812, 1015, 1421, 2233, 2030, 1015, 2030.

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