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# Comparing and Ordering Fractions

In mathematics, a fraction is said be a number of the form $\frac{a}{b}$, where the number above the bar is called numerator and the number below the bar is known as denominator.

In early arithmetic, we often come across with the problems of comparing and ordering of fractions. We may require to compare two fractions, i.e. which one is bigger or smaller. While ordering of fractions includes finding which of the given fraction is greater or lesser than the others and place them in either ascending or descending order. To compare two fraction we use less than symbol (<) and greater than symbol (>). More than two fractions can also be written in ascending (smallest to biggest) or descending order (biggest to smallest).

Let us have a look at the following images :

First image shows fraction $\frac{3}{10}$, while second represents $\frac{4}{10}$. Here, since 3 is smaller than 4, hence $\frac{3}{10}$ is smaller than $\frac{4}{10}$.

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## Comparing Fractions

The concept of comparing two fractions is the process of determining their relative magnitude, i.e. finding which one is bigger or smaller. In other words, to find two fractions if they are equal to (=) each other, one fraction less than (<), or greater than (>) the other, is known as comparing.
Steps for Comparing Fractions

CASE I - Same Denominators
Step 1: If denominator of both fractions are same, then just consider the numerators of both fractions and compare.
Step 2: The fraction with bigger numerator will be bigger fraction. Similarly, the fraction having smaller numerator will be the smaller fraction.

CASE II - Different Denominators
Step 1: In case of both fractions have different denominators, we find least common denominator, i.e. least common multiple (LCM) of the denominators.
Step 2: Make equivalent fractions with the new denominator. This is done by multiplying numerator as well as denominator in each fraction by a number which makes denominator equal to the LCM.
Step 3: We have same denominators. Now, compare the numerators and decide smaller and bigger fraction.

### Solved Example

Question:  What is larger, $\frac{6}{3}$ or $\frac{7}{2}$
Solution:

Step 1: The least common multiple of 3 and 2 is 6.

Step 2: So, let us do multiplying to make each denominator equal to 6 :

$\frac{6\times2}{3\times2}$
= $\frac{12}{6}$

and

$\frac{7\times3}{2\times3}$ = $\frac{21}{6}$

Step 3: Compare the numerators

21 > 12

$\frac{6}{3}$ < $\frac{7}{2}$

So  $\frac{7}{2}$  is the larger fraction.

## Ordering Fractions

Ordering of the fractions involves arranging fractions in ascending or in descending order. In other words, when a set of fractions is arranged in either ascending or descending order, then this process is termed as ordering of fractions. This is almost similar process to the comparing, except for comparing is essentially done between two fractions.
Steps for Ordering Fractions:

Step 1: Find the LCM of all the denominators of given numbers.

Step 2:
Multiply numerator and denominator of each fraction by a number in order to make denominator equal to LCM.

Step 3: Now since all the denominators are same, we can compare numerators and write them in required order either ascending or descending.

### Solved Example

Question: Arrange fractions $\frac{3}{4}$, $\frac{5}{2}$ in ascending and in descending order .

Solution:

Step1: LCM of denominators (4, 2) = 4

Step 2: Multiplying to make each denominator equal to 4 :

$\frac{3\times1}{4\times1}$ = $\frac{3}{4}$

and

$\frac{5\times2}{2\times2}$ = $\frac{10}{4}$

Step 3:
Ordered the fractions:

In the ascending order

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

In the descending order

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

## Examples

Consider following solved examples for better understanding.

Example 1 : Determine which one is bigger?

(i) $\frac{3}{7}$ and $\frac{4}{7}$

(ii) $\frac{18}{35}$ and $\frac{23}{35}$

Solution : (i) $\frac{3}{7}$ and $\frac{4}{7}$

Denominators are same.

Since 3 < 4

Therefore, $\frac{3}{7}$ < $\frac{4}{7}$

Hence, $\frac{4}{7}$ is the bigger fraction.

(ii) $\frac{18}{35}$ and $\frac{23}{35}$

Here also, same denominators.

Since 18 < 23

So, $\frac{18}{35}$ < $\frac{23}{35}$

Therefore, fraction $\frac{23}{35}$ is bigger.
Example 2 : Determine which fraction is smaller between $\frac{5}{8}$ and $\frac{7}{6}$.

Solution : Different denominators

LCM of 8 and 6 = 24

Multiplying numerator and denominator of first fraction by 3, we get

$\frac{5 \times 3}{8 \times 3}$ = $\frac{15}{24}$

Multiplying numerator and denominator of second fraction by 4, we get

$\frac{7 \times 4}{6 \times 4}$ = $\frac{28}{24}$

Since, $\frac{15}{24}$ < $\frac{28}{24}$

Hence, $\frac{5}{8}$ < $\frac{7}{6}$
Example 3 : Arrange the following fractions in descending order.

$\frac{7}{8}$, $\frac{2}{3}$, $\frac{5}{6}$, $\frac{1}{2}$

Solution : LCM of 8, 3, 6, 2 = 24

Now

$\frac{7}{8}$ = $\frac{7 \times 3}{8 \times 3}$ = $\frac{21}{24}$

$\frac{2}{3}$ = $\frac{2 \times 8}{3 \times 8}$ = $\frac{16}{24}$

$\frac{5}{6}$ = $\frac{5 \times 4}{6 \times 4}$ = $\frac{20}{24}$

$\frac{1}{2}$ = $\frac{1 \times 12}{2 \times 12}$ = $\frac{12}{24}$

$\frac{21}{24}$ > $\frac{20}{24}$ > $\frac{16}{24}$ > $\frac{12}{24}$

or

$\frac{7}{8}$ > $\frac{5}{6}$ > $\frac{2}{3}$ > $\frac{1}{2}$
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