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 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.
Unlike fractions with the same numerator.
38 and
35
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 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.
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.
43 ×
55 =
2015
and 25 ×
33 =
615
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.
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.
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?
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.
Cross multiplication method is limited to compare maximum two fractions only.
Cross multiplication method can be used to compare any two fractions whether they are unit fractions, like fractions or even unlike fractions.
Example 1: Compare unit fractions
15 and
17
Cross multiply both fractions
So, 1 × 7 = 7 and 1 × 5 = 5
So, 7 > 5
15 >
17
Example 2: Compare like fractions 165
and 115
Cross multiply both fractions
So, 16 × 5 = 80 and 11 × 5 = 55
So, 80 > 55
165 >
and 115
Example 3: Compare unlike fractions
43 and
25
Cross multiply both fractions
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.
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 and
75
∴ the ascending order can be written as
15 ,
35 ,
45 ,
65 and
75.
Descending order of fractions can be written as:
75 ,
65 ,
45 ,
35 and
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.
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.
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.
43 ×
2020 =
8060
24 ×
1515 =
3060
52 ×
3030 =
15060
75 ×
1212 =
8460
14 ×
1515 =
1560
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
