Comparing Unlike Fractions

In comparing unlike fractions, we first convert them into like fractions by using the following steps and then compare them.

Step I:

Obtain the denominators of the fractions and find their LCM (least common multiple).


Step II:

Each fractions are converted to its equivalent fraction with denominator equal to the LCM (least common multiple) obtained in Step I.


Step III:

Compare the numerators of the equivalent fractions whose denominators are same.

Solved Examples on Comparing Unlike Fractions:

1. Which is larger 34 or 512?

Solution:

Let us first find the LCM (least common multiple) of the denominators 4 and 12.

We have,

Comparing Unlike Fractions


Therefore, LCM (least common multiple) of 4 and 12 is 2 × 2 × 3 = 12.

Now we convert the given fractions to equivalent fractions with denominator 12

We have,

34 = 3×34×3 = 912

512 = 5×112×1512


Now we will observe the numerator, that is 9 > 5.

So, 912 > 512

Therefore, 34 > 512.



2. Compare 78 and 56.

Solution:

First we find the LCM of denominators .

We have,

Comparing Unlike Fractions


Therefore, LCM (least common multiple) = 2 × 2 × 2 × 3 = 24.

Now, we convert each fraction into equivalent fraction with 24 as its denominator.

We have,

78 = 7×38×32124[since, 24 ÷ 8 = 3]

56 = 5×46×42024[since 24 ÷ 6 = 4]


Now we will observe the numerator, that is 20 < 21

So, 2024 < 2124

Therefore, 56 < 78.


3. Arrange the fractions 58, 56, 74, 35 in ascending order.

Solution:

Let us first find the LCM (least common multiple) of the denominators:

We have,

Comparing Unlike Fractions


Therefore, LCM (least common multiple) = 2 × 2 × 2 × 3 × 5 = 120. Now, we convert each fraction into equivalent fraction with 120 as its denominator.

We have,

58 = 5×158×1575120[since 120 ÷ 8 = 15].

56 = 5×206×20100120[since 120 ÷ 6 = 20].

74 = 7×304×30210120[since 120 ÷ 4 = 30].

35 = 3×245×2472120[since 120 ÷ 5 = 24].


Now we will observe the numerator, that is 72 < 75 < 100 < 210.

So, 72120 < 75120 < 100120 < 210120.

Therefore, 35, 58, 56, 74.



4. Arrange the following fractions in descending order 38, 56, 24, 13, 68.

Solution:

We observe that the given fractions neither have common denominator nor common numerator.

So, first we convert them into like fractions i.e. fractions having common denominator. For this, we first find the LCM (least common multiple) of the denominators of the given fractions.

Denominators are 8, 6, 4, 3, 8.

We have,

Comparing Unlike Fractions

Therefore, LCM (least common multiple) = 2 × 2 × 2 × 3 = 24.

Now, we convert each fraction into equivalent fraction with 24 as its denominator.

Thus,

383×38×3924; [since 24 ÷ 8 = 3]

565×46×42024 ; [since 24 ÷ 6 = 4]

242×64×61224 ; [since 24 ÷ 4 = 6]

131×83×8824 ; [since 24 ÷ 3 = 8]

68 = 6×38×31824 ; [since 24 ÷ 8 = 3]


Now we will observe the numerator, that is 20 > 18 > 12 > 9 > 8.


So, 202418241224  > 924824 .

Therefore, 56682438 > 13 


Cross Multiplication Method of Comparing Two Unlike Fractions:

Let ab and cd be the two given fractions. To compare these fractions, we cross multiply: as shown:

Comparing Two Unlike Fractions

Then, we find the cross products ad and bc.

(i) ad > bc then ab > cd

(ii) ad < bc then ab < cd

(iii) If ad = bc then ab = cd


1. Compare 28 and 35

Solution:

We can compare the given fractions by cross multiplication method easily.

2 × 5 = 10 and 3 × 8 = 24

Therefore, 2 × 5 < 3 × 8

i.e., 28 < 35


We can also compare the given fractions by converting them to like fractions.

The LCM of denominators 8 and 5 is 40.

Therefore, we convert 28 and 35 into like fractions each having denominator 40.

28 = 2×58×5 = 1040 and 35 = 3×85×8 = 2440

Now 1040 and 2440 are like fractions.

Clearly, 10 < 24.

Therefore, 1040 < 2440,

i.e., 28 < 35.

Comparing Unlike Fractions

To compare two unlike fractions, we first convert them to like fractions.

1. Compare 712 and 518.

Solution:

Let us first convert both fractions to like fractions and then compare.

LCM of denominators 12 and 18 is 36.

Divide 36 by 12, we get 3. Now, multiply both numerators and denominator of the fraction 712 by 3.

7×312×3 = 2136

Divide 36 by denominator of the second fraction. i.e. 18, we get 2.

Multiply numerator and denominator of the fraction 518 by 2.

5×218×2 = 1036

Compare the two fractions 2136 and 1036. Here 21 > 10. Thus, 712 > 518.


Alternate Method:

We can compare unlike fractions by cross multiplication in the following way.

Compare two Unlike Fractions

Here, 7 × 18 > 12 × 5; 126 > 60

So, 712 > 518


2. Arrange the following fractions in ascending order 14, 32, 78, 116

Solution: LCM of denominators 4, 2, 8 and 16 = 16

Here, the given fractions can be written as

14 = 1×44×4 = 416

32 = 3×82×8 = 2416

78 = 7×28×2 = 1416

116 = 1×116×1 = 116

Ascending order is 116 < 416 < 1416 < 2416

                        or, 116 < 14 < 78 < 32


Questions and Answers on Comparing Unlike Fractions:

1. Compare the given fractions by putting the right sign <, > or =.

(i) 15 ___ 111

(ii) 76 ___ 46

(iii) 318 ___ 315

(iv) 23 ___ 34

(v) 612 ___ 816

(vi) 55 ___ 57

(vii) 56 ___ 1218

(viii) 1015 ___ 1421

(ix) 27 ___ 513


Answers:

(i) >

(ii) >

(iii) <

(iv) <

(v) =

(vi) >

(vii) >

(viii) =

(ix) <


2. Arrange the given fractions in ascending order.

(i) 36, 38, 34

(ii) 116, 14, 12

(iii) 12, 34, 58

(iv) 25, 34, 35


Answers:

(i) 38, 36, 34

(ii) 116, 14, 12

(iii) 12, 58, 34

(iv) 25, 35, 34


Word Problems on Comparing Unlike Fractions:

3. Robert ate 922 part of the pizza and Maria ate 511 part of the pizza. Who ate the greater part of the pizza? What fraction of pizza was finished by the two girls?

Solution:

Robert ate 922 part of the pizza.

Maria ate 511 part of the pizza.

Let us first convert both fractions to like fractions and then compare.

Let us find the LCM of the denominators 22 and 11. 

The LCM of 22 and 11 is 22.

922 = 9×122×1 = 922

511 = 5×211×2 = 1022

Compare the two fractions 922 and 1022.

Here 10 > 9. Thus, 511 > 922.

Maria ate the greater part of the pizza.


Now add the two fractions922511

We get the like fractions 922 and 522.

Now, 922 + 1022       

= 9+1022       

= 1922

Therefore, 1922 of the pizza was finished by the two girls?


3. Ron covered a distance of 56 km and Jon covered a distance of 34 km. Who covered the greater distance?

Answer: Ron 

4. Adrian cycled for 628 km and Steven cycled 274 km during the weekend. Who cycled more and by how much?

Answer: Adrian, 1 km 

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