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Introduction

What a fraction is and what the types of fractions are, have been explained in detail in the chapter Fractions and its types.

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 \(\frac{16}{5}\) and \(\frac{11}{5}\).
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 \(\frac{16}{5}\) and \(\frac{11}{5}\) respectively.
So, we can say 16 is greater than 11.
Therefore, fraction \(\frac{16}{5}\) with numerator 16 will also be greater than the fraction \(\frac{11}{5}\) with numerator 11.
∴ \(\frac{16}{5}\) is greater than \(\frac{11}{5}\).


Example 2: Compare fractions \(\frac{11}{4}, \frac{21}{4}, \frac{9}{4}\).
\(\frac{11}{4}\), \(\frac{21}{4}\) and \(\frac{9}{4}\) fractions have the same denominators as 4.
So, only numerators will be compared as these are like fractions.
The numerators of \(\frac{11}{4}\), \(\frac{21}{4}\) and \(\frac{9}{4}\) 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 \(\frac{9}{4}\) < \(\frac{11}{4}\) < \(\frac{21}{4}\).

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.
\(\frac{3}{5}\) and \(\frac{3}{8}\)

Example

Unlike fractions with different numerators.
\(\frac{4}{3}\) and \(\frac{2}{5}\)

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 \(\frac{3}{5}\) and \(\frac{3}{8}\)
\(\frac{3}{5}\) and \(\frac{3}{8}\) are unlike fractions with the same numerator.
So, only denominators 5 and 8 will be compared.
We can say, 8 > 5
Therefore, \(\frac{3}{8}\) < \(\frac{3}{5}\)
Or \(\frac{3}{8}\) is less than \(\frac{3}{5}\)
Or \(\frac{3}{5}\) is greater than \(\frac{3}{8}\)


Example 2: Compare fractions \(\frac{9}{5}\) and \(\frac{9}{6}\)
The numerator in both fractions is the same, so compare denominators 5 and 6 only.
We can say, 6 > 5
Therefore, \(\frac{9}{6}\) < \(\frac{9}{5}\)
Or, \(\frac{9}{5}\) > \(\frac{9}{8}\)

2. Unlike fractions with different numerator

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 \(\frac{4}{3}\) and \(\frac{2}{5}\).
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 \(\frac{4}{3}\) by 5 and \(\frac{2}{5}\) by 3 to make their denominators equal to LCM 15.
\(\frac{4}{3} \times \frac{5}{5} = \frac{20}{15}\)
and \(\frac{2}{5} \times \frac{3}{3} = \frac{6}{15}\)
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
∴ \(\frac{20}{15}\) > \(\frac{6}{15}\)
i.e.\(\frac{4}{3}\) > \(\frac{2}{5}\)

Comparing unit fractions

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

Example

\(\frac{1}{5}, \frac{1}{7}\)

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 \(\frac{1}{5}, \frac{1}{7}\).
Fractions \(\frac{1}{5}\) and \(\frac{1}{7}\) 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.
∴ \(\frac{1}{5}\) > \(\frac{1}{7}\)

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 \(\frac{1}{5}\) and \(\frac{1}{7}\)
Cross multiply both fractions
So, 1 x 7 = 7 and 1 x 5 = 5
So, 7 > 5
\(\frac{1}{5}\) > \(\frac{1}{7}\)


Example 2: Compare like fractions \(\frac{16}{5}\) and \(\frac{11}{5}\)
Cross multiply both fractions
So, 16 x 5 = 80 and 11 x 5 = 55
So, 80 > 55
\(\frac{16}{5}\) > \(\frac{11}{5}\)


Example 3: Compare unlike fractions \(\frac{4}{3}\) and \(\frac{2}{5}\)
Cross multiply both fractions
So, 4 x 5 = 20 and 3 x 2 = 6
So, 20 > 6
\(\frac{4}{3}\) > \(\frac{2}{5}\)

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 \(\frac{4}{5}, \frac{6}{5}, \frac{1}{5}, \frac{3}{5}\) and \(\frac{7}{5}\) in ascending order.
The fractions \(\frac{4}{5}, \frac{6}{5}, \frac{1}{5}, \frac{3}{5}\) and \(\frac{7}{5}\) 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:
\(\frac{1}{5}\) < \(\frac{3}{5}\) < \(\frac{4}{5}\) < \(\frac{6}{5}\) < \(\frac{7}{5}\)
∴ the ascending order can be written as \(\frac{1}{5}, \frac{3}{5}, \frac{4}{5}, \frac{6}{5}, \frac{7}{5}\).
Descending order of fractions can be written as:
\(\frac{7}{5}, \frac{6}{5}, \frac{4}{5}, \frac{3}{5}, \frac{1}{5}\).

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 \(\frac{4}{5}, \frac{4}{3}, \frac{4}{7}, \frac{4}{6}\) and \(\frac{4}{9}\) 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:
\(\frac{4}{9}\) > \(\frac{4}{7}\) > \(\frac{4}{6}\) > \(\frac{4}{5}\) > \(\frac{4}{3}\)
∴ descending order can be written as
\(\frac{4}{3}, \frac{4}{5}, \frac{4}{6}, \frac{4}{7}, \frac{4}{9}\)
Ascending order can be written as
\(\frac{4}{9}, \frac{4}{7}, \frac{4}{6}, \frac{4}{5}, \frac{4}{3}\)

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
\(\frac{4}{3}, \frac{2}{4}, \frac{5}{2}, \frac{7}{5}\) and \(\frac{1}{4}\).
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
Next, multiple each fraction with the LCM 60 to make the denominators same for all fractions.
\(\frac{4}{3}\) X \(\frac{20}{20}\) = \(\frac{80}{60}\)
\(\frac{2}{4}\) X \(\frac{15}{15}\) = \(\frac{30}{60}\)
\(\frac{5}{2}\) X \(\frac{30}{30}\) = \(\frac{150}{60}\)
\(\frac{7}{5}\) X \(\frac{12}{12}\) = \(\frac{84}{60}\)
\(\frac{1}{4}\) X \(\frac{15}{15}\) = \(\frac{15}{60}\)
So, the new fractions can be written as:
\(\frac{80}{60}, \frac{30}{60}, \frac{150}{60}, \frac{84}{60}, \frac{15}{60}\)
These fractions become like fractions with the same denominator of 60 and now compare their numerator.
15 < 30 < 80 < 84 < 150
\(\frac{1}{4}\) < \(\frac{2}{4}\) < \(\frac{4}{3}\) < \(\frac{7}{5}\) < \(\frac{5}{2}\)
Therefore, these fractions can be arranged in ascending order as:
\(\frac{1}{4}\) < \(\frac{2}{4}\) < \(\frac{4}{3}\) < \(\frac{7}{5}\) < \(\frac{5}{2}\)
Also, these fractions can be arranged in descending order as: \(\frac{5}{2}\) > \(\frac{7}{5}\) > \(\frac{4}{3}\) > \(\frac{2}{4}\) > \(\frac{1}{4}\)