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Methods to Compare Order Like, Unlike & Unit Fractions

Found in topics: Fractions

Introduction

What a fraction is and what the types of fractions are, have been explained in detail in the chapter Fraction & Types Like, Equivalent, Mixed, Proper & More.

This chapter Comparing and order of fractions describes about how two or more than two fractions are compared as bigger and smaller. After learning the comparing of fractions, you can learn how they are written in ascending order or descending order at the end of this chapter.

Methods to compare the fractions depend upon the type of fractions they are. The types of fractions can be like fractions or unlike fractions.

Like fractions are the fractions which have same denominator and unlike fractions are the fractions with different denominators.

Let’s start with learning methods to compare like fractions and unlike fractions.

Comparing like fractions

As already said above in the introduction, like fractions have the same denominator.
So, while making comparison of like fractions, the comparison is made on the basis of only the numerator leaving the denominator as it is. The numerators of all fractions are compared in the same way as the normal numbers are compared.
Greater the numerator, greater will be the fraction. Smaller the numerator, smaller the fraction will be.

Example

Example 1: Compare fractions 165 and 115.
As these two fractions have the same denominators as 5, so this is an example of like fractions.
So, in the case of like fractions, only numerators will be compared leaving denominators aside.
Here, the numerators are 16 and 11 in 165 and 115 respectively.
So, we can say 16 is greater than 11.
Therefore, fraction 165 with numerator 16 will also be greater than the fraction 115 with numerator 11.
165 is greater than 115.


Example 2: Compare fractions 114, 214, 94.
114, 214 and 94 fractions have the same denominators as 4.
So, only numerators will be compared as these are like fractions.
The numerators of 114, 214 and 94 fractions are 11, 21 and 9.
So, 21 is the largest number followed by 11 and then 9.
It can be written as 9 < 11 < 21 .
So, the fractions can be written as 94 < 114 < 214 .

Comparing unlike fractions

Unlike fractions have different denominators and their numerators may or may not be the same.

Example

Unlike fractions with the same numerator.
38 and 35

Example

Unlike fractions with different numerators.
43 and 25

While making comparisons of unlike fractions, the denominator is compared only among the fractions.
Again, the denominators are compared the same as the normal numbers are compared. So, greater the denominator, smaller will be the fraction. Smaller the denominator, greater the fraction will be.
There are two methods that can be used to compare unlike fractions. The both methods differ depending upon whether the numerator is the same or different.

Let’s see how these two types of unlike fractions are compared.

1. Unlike fractions with same numerator

To compare unlike fractions with same numerator, denominator of each fraction is compared.
The fraction with smaller denominator is greater than the other fraction which has greater denominator.
In other way around, the fraction with greater denominator is smaller than the other fraction which has smaller denominator

Example

Example 1: Compare fractions 35 and 38
35 and 38 are unlike fractions with the same numerator.
So, only denominators 5 and 8 will be compared.
We can say, 8 > 5
Therefore, 38<35
Or 38 is less than 35
Or 35 is greater than 38


Example 2: Compare fractions 95 and 96
The numerator in both fractions is the same, so compare denominators 5 and 6 only.
We can say, 6 > 5
Therefore, 96<95
Or, 95>96

2. Unlike fractions with different numerator

There are two ways to compare unlike fractions with different numerator. Let’s look at both of them with an example.

First way to compare fractions

In unlike fractions with different denominators, first unlike fractions are converted into like fractions using LCM.
Take the LCM of denominators of all fractions.
Then multiply the numerator and denominator of every fraction with such a number that makes the denominator equal to the number obtained in LCM.

Example

Compare fractions 43 and 25.
To compare unlike fractions with different denominators 3 and 5, first take LCM of 3 and 5.
LCM of 3 and 5 = 15
Now multiply 43 by 5 and 25 by 3 to make their denominators equal to LCM 15.
4 3 × 5 5 = 20 15
and 2 5 × 3 3 = 6 15
Hence, in both fractions, now the denominators are equal to LCM 15.
As both fractions have the same denominators which is 15, now they can be compared using numerator only.
So, 20 > 6
2015>615
i.e.43> 25

Second way to compare fractions

Comparing unit fractions

Unit fractions are the fractions with value of numerator as one and denominator can be any positive integer.

Example

15, 17

In other words, unit fractions are unlike fractions where the numerator is always the same i.e. one and the denominator is different.
Therefore, unit fractions can be compared with one of the above method used in comparing unlike fractions with the same numerator.
As we have already seen above, this method uses only denominators to compare the fractions.
A fraction with greater denominator is smaller in unit fraction and a fraction with smaller denominator will be greater in unit fraction.

Example

Compare fractions 15, 17.
Fractions 15 and 17 are unit fractions with denominators 5 and 7 respectively.
Unit fractions are compared by finding the greatest denominator, so denominator 7 is greater than 5.
15>17

What is cross multiplication method to compare fractions?

In cross multiplication method to compare unlike fractions with different numerators, fractions are cross multiplied. Then the two values obtained after the multiplication are compared to check which fraction is greater or smaller.

Note

Cross multiplication method is limited to compare maximum two fractions only.

Note

Cross multiplication method can be used to compare any two fractions whether they are unit fractions, like fractions or even unlike fractions.

Example

Example 1: Compare unit fractions 15 and 17
Cross multiply both fractions
Cross multiply fractions 1/5 and 1/7
So, 1 × 7 = 7 and 1 × 5 = 5
So, 7 > 5
15>17


Example 2: Compare like fractions 165and 115
Cross multiply both fractions
Cross multiply fractions 16/5 and 11/5
So, 16 × 5 = 80 and 11 × 5 = 55
So, 80 > 55
165>115


Example 3: Compare unlike fractions 43 and 25
Cross multiply both fractions
Cross multiply fractions 4/3 and 2/5
So, 4 × 5 = 20 and 3 × 2 = 6
So, 20 > 6
43>25

Ordering of fractions

Ordering of fractions are meant by arranging the fractions in ascending order or descending order.
Fractions can be written in an order after the comparison of fractions are completed.
Let’s see how fractions can be arranged for like fractions, unlike fractions with same numerator and unlike fractions with different numerators with the following examples.

1. Ordering of like fractions

Ordering of like fractions can be done after the fractions have been compared using the method described in comparing like fractions, which compares the numerator only.

Example

Write 45, 65, 15, 35 and 75 in ascending order.
The fractions 45, 65, 15, 35 and 75 are like fractions as each fraction has the same denominator.
So, compare their numerator.
1 < 3 < 4 < 6 < 7
Or we can write fractions as:
15 , 35 , 45 , 65 , 75
∴ the ascending order can be written as 15 , 35 , 45 , 65 , 75 .
Descending order of fractions can be written as:
75 , 65 , 45 , 35 , 15 .

2. Ordering of unlike fractions with same numerator

Ordering of unlike fractions with same numerator is done after comparing of unlike fractions with same numerator.

Example

Arrange unlike fractions 45 , 43 , 47 , 46 , 49 in ascending order.
Compare the denominator as all fractions have the same numerator i.e. 4.
The fraction with smaller denominator is greater for unlike fractions.
9 > 7 > 6 > 5 > 3
Or we can write fractions as:
49 > 47 > 46 > 45 > 43
∴ descending order can be written as
43 , 45 , 46 , 47 , 49
Ascending order can be written as
49 , 47 , 46 , 45 , 43

3. Ordering of unlike fractions with different numerator

Ordering of unlike fractions with different numerator is done after comparing of unlike fractions with different numerator method.

To arrange the unlike fractions with different numerators in ascending order or descending order, first change unlike fractions into like fractions.
Take the LCM of all denominators, then multiple each fraction with LCM number to make the denominators same for all fractions.

Example

Arrange these fractions into ascending order
43 , 24 , 52 , 75 , 14
These fractions are unlike fractions with different numerators.
Take the LCM of denominators of all fractions i.e. 3, 4, 2, 5 and 4
LCM of 3, 4, 2, 5 and 4 = 60
Multiply each fraction with the number to get the required LCM 60.
4 3 × 20 20 = 80 60
2 4 × 15 15 = 30 60
5 2 × 30 30 = 150 60
7 5 × 12 12 = 84 60
1 4 × 15 15 = 15 60
So, the new fractions can be written as:
8060 , 3060 , 15060 , 8460 , 1560
These fractions become like fractions with the same denominator of 60 and now compare their numerator.
15 < 30 < 80 < 84 < 150
Or 1560 < 3060 < 8060 < 8460 < 15060
Therefore, these fractions can be arranged in ascending order as:
14 < 24 < 43 < 75 < 52
Also, these fractions can be arranged in descending order as:
52 > 75 > 43 > 24 > 14

Frequently Asked Questions

1) What are like fractions in maths?

Like fractions are those fractions which have the same denominators, for example:
4 7 , 10 7 and 22 7 are like fractions.

2) What are unlike fractions in maths?

Unlike fractions are those fractions which have different denominators, for example:
7 4 , 1 2 and 4 5 are unlike fractions.

Solved Examples

1) Compare fractions 1 9 and 1 10 .

1 9 and 1 10 are like fractions because they have same numerators.
So, compare their denominators
9 < 10
1 9 > 1 10

2) Which is larger 7 8 or 1 3 .

First find the LCM of 8 and 3
LCM of 8 and 3 is 24.
Convert 7 8 and 1 3 to equivalent fractions with denominator 24.
7 8 × 3 3 = 21 24
1 3 × 8 8 = 8 24
21 24 > 8 24
7 8 > 1 3

3) Compare the following fractions using symbols > or < or = 3 6 and 4 8

Reduce the fractions into its the lowest terms.
3 ÷ 3 6 ÷ 3 = 1 2
4 ÷ 4 8 ÷ 4 = 1 2
3 6 = 4 8

4) Compare 4 3 and 7 4

Cross multiply both fractions
So, 4 × 4 = 16
3 × 7 = 21
21 > 16
7 4 > 4 3

5) Arrange the following fractions into ascending order 1 2 , 7 3 , 2 5 , 3 4 and 1 5 .

All fractions have different denominator.
So, take LCM to make equivalent fractions.
Take LCM of their denominators 2, 3, 5, 4 and 5.
LCM = 60
1 2 × 30 30 = 30 60
7 3 × 20 20 = 140 60
2 5 × 12 12 = 24 60
3 4 × 15 15 = 45 60
1 5 × 12 12 = 12 60
Arrange them in ascending order
12 60 , 24 60 , 30 60 , 45 60 , 140 60
1 5 , 2 5 , 1 2 , 3 4 , 7 3

6) Arrange the following fractions into descending order 3 8 , 2 5 , 7 4 , 4 5 and 1 2 .

All fractions have different denominator.
So, take LCM to make like fractions.
Take LCM of their denominators 8, 5, 4, 5 and 2.
LCM = 40
3 8 × 5 5 = 15 40
2 5 × 8 8 = 16 40
7 4 × 10 10 = 70 40
4 5 × 8 8 = 32 40
1 2 × 20 20 = 20 40
Arrange them in descending order
70 40 , 32 40 , 20 40 , 16 40 , 15 40
7 4 , 4 5 , 1 2 , 2 5 , 3 8

Worksheet 1 on Comparing of Fractions

Put an appropriate sign >, < or = in the following boxes.

1) 8 3 4 5
2) 9 7 9 7
3) 11 12 13 11
4) 0 7 4 7
5) 4 6 6 4
6) 7 5 0 4
7) 2 3 11 12
8) 14 15 1 2
9) 1 6 6 12
10) 3 20 5 20
Worksheet 2 on Ordering of Fractions

A) Arrange the following fractions in ascending order.

1) 5 9 , 3 9 , 4 9 , 7 9 , 0 9 .

2) 9 4 , 3 4 , 7 4 , 10 4 , 6 4 .

3) 2 5 , 3 4 , 4 5 , 2 3 , 6 5 .

4) 7 15 , 4 10 , 7 8 , 5 3 , 2 4 .

B) Arrange the following fractions in descending order.

1) 2 8 , 6 4 , 3 2 , 1 5 , 4 3 .

2) 7 4 , 1 2 , 4 3 , 3 2 , 1 4 .

3) 1 8 , 4 8 , 0 8 , 9 8 , 7 8 .

4) 6 11 , 17 11 , 10 11 , 18 11 , 5 11 .

Worksheet 3 on Comparing and Ordering of Fractions

Multiple choice questions

1) Which is the greatest fraction?

a) 1 2

b) 1 3

c) 1 4

d) 1 5

2) Which is the smallest fraction?

a) 3 4

b) 4 3

c) 1 2

d) 3 2

3) In comparing like fractions, greater the numerator, ___________ will be the fraction.

a) equivalent

b) equal

c) smaller

d) greater

4) 4 3 = x 12

a) 14

b) 12

c) 16

d) 1

5) In comparing two unlike fractions with the same numerator, the fraction with greater denominator is ___________ than the other fraction.

a) greater

b) equal

c) equivalent

d) smaller

6) The correct ascending order of fractions 1 4 , 1 2 , 1 3 , 1 5 is

a) 1 5 , 1 4 , 1 3 , 1 2

b) 1 5 , 1 2 , 1 4 , 1 3

c) 1 5 , 1 3 , 1 4 , 1 2

d) 1 5 , 1 4 , 1 2 , 1 3

7) 5 9 is ___________ 40 72

a) equal to

b) greater than

c) smaller than

d) equivalent to

8) The correct descending order of fractions 3 5 , 3 8 , 3 7 , 3 11 is

a) 3 5 , 3 7 , 3 8 , 3 11

b) 3 11 , 3 8 , 3 7 , 3 5

c) 3 11 , 3 7 , 3 5 , 3 8

d) 3 11 , 3 8 , 3 5 , 3 7

9) Choose the correct symbol for x 1 7 ___________ 1 8

a) <

b) >

c) ≥

d) =

10) The lowest term of fraction 14 25

a) 7 5

b) 14 25

c) 5 7

d) 1

MCQ Answer Key Hide Show
1. a
2. c
3. d
4. c
5. d
6. a
7. d
8. b
9. b
10. b
Last updated on: 27-07-2024