Methods to Compare Order Like, Unlike & Unit Fractions

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 $\frac{9}{4}<\frac{11}{4}<\frac{21}{4}$.

$\frac{3}{8}$ and $\frac{3}{5}$

$\frac{4}{3}$ and $\frac{2}{5}$

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, $\frac{9}{5}>\frac{9}{6}$

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. $\frac{4}{3}>\frac{2}{5}$

$\frac{1}{5}$, $\frac{1}{7}$

Compare fractions $\frac{1}{5}$ and $\frac{1}{7}$.
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.
∴ $\frac{1}{5}>\frac{1}{7}$

Example 1: Compare unit fractions $\frac{1}{5}$ and $\frac{1}{7}$
Cross multiply both fractions

So, 1 × 7 = 7 and 1 × 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 × 5 = 80 and 11 × 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 × 5 = 20 and 3 × 2 = 6
So, 20 > 6
$\frac{4}{3}>\frac{2}{5}$

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

Arrange unlike fractions $\frac{4}{5},\frac{4}{3},\frac{4}{7},\frac{4}{6},\frac{4}{9}$ 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.
The denominators in ascending order 3 < 5 < 6 < 7 < 9
3 is the smallest and 9 is the largest denominators.
Therefore, the fraction with denominator 3 will be the greatest fraction.
$\frac{4}{3}>\frac{4}{5}>\frac{4}{6}>\frac{4}{7}>\frac{4}{9}$
∴ descending order can be written as
$\frac{4}{3},\frac{4}{5},\frac{4}{6},\frac{4}{7},\frac{4}{9}$
Ascending order will be the reverse
$\frac{4}{9},\frac{4}{7},\frac{4}{6},\frac{4}{5},\frac{4}{3}$

Arrange $\frac{4}{3},\frac{2}{4},\frac{5}{2},\frac{7}{5},\frac{1}{4}$ fractions into ascending order.
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.
$\frac{4}{3}\times \frac{20}{20}=\frac{80}{60}$
$\frac{2}{4}\times \frac{15}{15}=\frac{30}{60}$
$\frac{5}{2}\times \frac{30}{30}=\frac{150}{60}$
$\frac{7}{5}\times \frac{12}{12}=\frac{84}{60}$
$\frac{1}{5}\times \frac{15}{15}=\frac{15}{60}$
These fractions become like fractions with the same denominator of 60 and now compare their numerator.
15 < 30 < 80 < 84 < 150
Or $\frac{15}{60}<\frac{30}{60}<\frac{80}{60}<\frac{84}{60}<\frac{150}{60}$
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:
$\frac{5}{2}>\frac{7}{5}>\frac{4}{3}>\frac{2}{4}>\frac{1}{4}$