Ordering Fractions

Ordering fractions is arranging a group of fractions from least to greatest (ascending order) or from greatest to least (descending order).

Based on whether the fractions have the same denominator, different denominator, or different numerator and denominator, the steps for ordering vary.

With the Same Denominator

Let us order the fractions ${\dfrac{1}{9},\dfrac{7}{9},\dfrac{5}{9},\dfrac{4}{9},\dfrac{13}{9}}$ from greatest to least and least to greatest.

Identifying the Denominators

Here, the denominator is 9

Since the fractions have the same denominator, we can compare the numerators directly.

Comparing the Numerators

13 > 7 > 5 > 4 > 1

This gives the fractions from the highest to the lowest value. As a result, the least numerator gives the least fraction, and vice versa.

Ordering the Fractions

Ordering from greatest to least, we get

${\dfrac{13}{9} >\dfrac{7}{9} >\dfrac{5}{9} >\dfrac{4}{9} >\dfrac{1}{9}}$

Ordering from least to greatest, we get

${\dfrac{1}{9} <\dfrac{4}{9} <\dfrac{5}{9} <\dfrac{7}{9} <\dfrac{13}{9}}$

E.g.1.

With the Same Numerator

Now, let us order the following fractions having the same numerator ${\dfrac{7}{4},\dfrac{7}{2},\dfrac{7}{5},\dfrac{7}{3}}$

Identifying the Denominators

Here, the numerators of fractions are 7, which is the same.

Comparing the Denominators

5 > 4 > 3 > 2

This gives the fractions from the lowest to the highest value. As a result, the least denominator gives the highest fraction, and vice versa.

Ordering the Fractions

Ordering from greatest to least, we get

${\dfrac{7}{2} >\dfrac{7}{3} >\dfrac{7}{4} >\dfrac{7}{5}}$

Ordering from least to greatest, we get

${\dfrac{7}{5} <\dfrac{7}{4} <\dfrac{7}{3} >\dfrac{7}{2}}$

With Different Numerators and Denominators

To order fractions with different denominators and numerators, we need to make the denominators the same. That is achieved by converting them to equivalent fractions.

If we order the fractions ${\dfrac{2}{3},\dfrac{5}{9},\dfrac{1}{2},\dfrac{3}{4}}$

Identifying the Denominators

Here, the numerators and denominators are all different.

Converting them into Equivalent Fractions 

The denominators are 3, 9, 2, and 4. Their LCM is 36

Now, the fractions with a common denominator are

${\dfrac{2\times 12}{3\times 12}=\dfrac{24}{36}}$

${\dfrac{5\times 4}{9\times 4}=\dfrac{20}{36}}$

${\dfrac{1\times 18}{2\times 18}=\dfrac{18}{36}}$

${\dfrac{3\times 9}{4\times 9}=\dfrac{27}{36}}$

Comparing the Numerators 

27 > 24 > 20 > 18

Ordering the Fractions 

Now, writing the fractions in descending order,

${\dfrac{3}{4} >\dfrac{2}{3} >\dfrac{5}{9} >\dfrac{1}{2}}$

Also, in ascending order,

${\dfrac{1}{2} <\dfrac{5}{9} <\dfrac{2}{3} >\dfrac{3}{4}}$