Comparing Fractions

We compare fractions to find which of the given fractions is greater, smaller, or equivalent.  

With Same Denominators

When fractions have the same denominator, comparing them is easy. We just look at the numerator. The fraction with the larger numerator is the larger fraction. 

Let us compare the fractions ${\dfrac{5}{7}}$ and ${\dfrac{2}{7}}$

Identifying the Denominators

Here, both denominators are 7

Comparing the Numerators

5 > 2

Thus, ${\dfrac{5}{7}}$ > ${\dfrac{2}{7}}$

With Different Denominators

When fractions have different denominators, we need to find a common denominator before comparing them. A common denominator is a shared multiple of the denominators of the given fractions.

Common Denominator Method

To compare the fractions with different denominators, we first convert them to like fractions by finding their LCM (Least Common Multiple).

Finding the LCD involves identifying the smallest number that both denominators share as a multiple. Next, we convert each fraction to an equivalent fraction with the LCD as the new denominator. 

Let us compare the fractions ${\dfrac{1}{6}}$ and ${\dfrac{5}{9}}$

Finding the LCD

Here, the denominators are 6 and 9

The LCM of 6 and 9 is 18

Converting each Fraction to an Equivalent Fraction with LCD as the Denominator

${\dfrac{1}{6}}$ = ${\dfrac{1\times 3}{6\times 3}}$ = ${\dfrac{3}{18}}$

${\dfrac{5}{9}}$ = ${\dfrac{5\times 2}{9\times 2}}$ = ${\dfrac{10}{18}}$

Comparing the Numerators

10 > 3

Thus, ${\dfrac{10}{18}}$ > ${\dfrac{3}{18}}$

⇒ ${\dfrac{5}{9}}$ > ${\dfrac{1}{6}}$

Using the Decimal Values

Here, we convert each fraction to its equivalent decimal values and then compare the decimal values. 

Let us compare between ${\dfrac{2}{5}}$ and ${\dfrac{3}{8}}$

Dividing the Numerators by the Denominators

${\dfrac{2}{5}}$ = 0.4

${\dfrac{3}{8}}$ = 0.375

Comparing the Decimal Values

0.4 > 0.375

Thus, ${\dfrac{2}{5}}$ > ${\dfrac{3}{8}}$

Using Cross Multiplication

Another method to compare fractions with different denominators is cross-multiplication. This method eliminates the need to find a common denominator.

Here, we multiply the numerator of one fraction with the denominator of the other and vice versa. The fraction with the larger product is the larger fraction. It is also known as the butterfly method.

If we compare ${\dfrac{4}{9}}$ and ${\dfrac{2}{7}}$

Cross-Multiplying and Comparing the Product

Thus, ${\dfrac{4}{9}}$ > ${\dfrac{2}{7}}$

Visually

We use various graphical models to visualize fractions, which helps us to compare them.

Using Bar-Model

Comparing both fractions ${\dfrac{5}{8}}$ and ${\dfrac{7}{12}}$ by their shaded region gives us the greater or smaller fraction. 

${\dfrac{5}{8}}$ > ${\dfrac{7}{12}}$ as ${\dfrac{5}{8}}$ covers a larger shaded area of the same whole than ${\dfrac{7}{12}}$. 

Using Number Line

We can also compare fractions on a number line. Here, we first convert them into like fractions and then plot them on the number line. 

Let us compare between ${\dfrac{5}{8}}$ and ${\dfrac{7}{12}}$

With Same Numerators

To compare fractions with the same numerators, we compare them by their denominators. The fraction with the greatest denominator is the smallest, and vice versa.

Let us compare ${\dfrac{3}{8}}$ and ${\dfrac{3}{14}}$

Identifying the Numerators

Here, both the numerators are 3

Comparing the Denominators

8 < 14

Thus, ${\dfrac{3}{14}}$ < ${\dfrac{3}{8}}$