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# Comparing Fractions

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 Sub Topics Fractions are the Numbers which are expressed in form of $\frac{p}{q}$, where ‘p’ and ‘q’ are the whole numbers and q ≠ 0. In this session we are going to learn about comparing Fractions. To compare fractions, we first see that all the fractions are greater than zero. Another thing to be remembered is that while comparing fraction, we first check that if the fraction is a proper fraction or improper fraction. If the fraction is improper it means that it is greater than one and if the fraction is proper fraction it is less than 1. So we conclude that a proper fraction is always smaller than the improper fraction. e.g, $\frac{3}{5}$ < $\frac{15}{7}$ Now if we come across the fractions, which are both proper or both improper fractions, then another thing which we can check is that are the denominators same or not. In case the denominators of the two fractions are same, we compare the numerators and conclude that the larger numerator represents the larger fraction and the smaller numerator represents the smaller fraction. e.g, $\frac{7}{8}$ < $\frac{12}{8}$ In case the denominators are different, but the numerators of the two fractions are same, we come to the conclusion that the fraction with smaller denominator represents the larger fraction and the fraction with larger denominator represents the smaller fraction. e.g, $\frac{3}{13}$ < $\frac{11}{6}$ Now we talk about the comparison of the two fractions, where neither of the situations satisfies. In such cases of comparison of the fractions, we simply cross multiply and the greater result represents the greater fraction. In this way the fractions are compared and arranged in ascending or descending order. e.g, $\frac{3}{13}$ ? $\frac{11}{6}$ $\frac{3*6}{13*6}$ ? $\frac{11*13}{6*13}$ $\frac{18}{78}$ ? $\frac{143}{78}$ $\frac{18}{78}$ < $\frac{143}{78}$

## Comparing Fractions with Different Denominators

Fractions are the Numbers which are expressed in form of $\frac{p}{q}$, where ‘p’ and ‘q’ are the whole Numbers and q ≠ 0. Fractions has two parts: numerator and denominator. In the above given equation 'p' is the numerator and 'q' is the denominator. Comparing fractions is to find the greatest and smallest among the given Set of fractions. Fractions with same denominators are easy to compare than fractions with different. So if the fractions with different denominators are to be compared then we must convert them to fractions with same denominators. In this session we are going to learn about fractions with different denominators.

For comparing fractions, first the denominators of the fractions should be made the same. Once the fractions are converted as fractions with same denominator, then its just enough to compare the numerator. If two fractions are compared then the smallest or greatest among them is found. If a set of fractions are given then the fractions are arranged in some order on a number line and compared. The comparison of fractions is done by using four comparison operators: less than(<), greater than(>), equal(=) and not equal(≠).

Now we see some of the steps for Comparing Fractions with different denominators.
1. First we take two fractions which have different denominators.
2. Then we multiply the numerator and denominator value of one fraction by the same number, so that both the fractions have same denominator values which will the LCM of the denominators.
3. After multiplication the same value in one fraction does not change the result because same numerator and denominator values give the result 1.
4. After that we easily compare the fractions.
Example:
Compare $\frac{5}{7}$ or $\frac{2}{5}$?
In the fractions $\frac{5}{7}$ and $\frac{2}{5}$, the denominators are the different.
Make the denominators same by taking LCM of them.
The LCM of 7 and 5 is 35.
$\frac{5}{7}$ ----> $\frac{5*5}{7*5}$ ----> $\frac{25}{35}$
$\frac{2}{5}$ ----> $\frac{2*7}{5*7}$ ----> $\frac{14}{35}$
Now, compare the numerators.
Among the numerator 25 and 14, 25 is the greatest.
So, $\frac{25}{35}$ is greater than $\frac{14}{35}$.
$\frac{25}{35}$ > $\frac{14}{35}$

## Comparing Unlike Fractions

Fractions are the Numbers which are expressed in form of $\frac{p}{q}$, where ‘p’ and ‘q’ are the whole Numbers and q ≠ 0. Fractions has two parts: numerator and denominator. In the above given equation 'p' is the numerator and 'q' is the denominator. Comparing Fractions is to find the greatest and smallest among the given Set of fractions. Like fractions are easy to compare than unlike fractions. So if the unlike fractions are to be compared then we must convert them to like fractions. In this session we are going to learn about unlike Fractions.

For Comparing Fractions, first the unlike fractions has to be converted to like fractions. Once the fractions are converted as fractions with same denominator, then its just enough to compare the numerator. If two fractions are compared then the smallest or greatest among them is found. If a set of fractions are given then the fractions are arranged in some order on a number line and compared. The comparison of fractions is done by using four comparison operators: less than(<), greater than(>), equal(=) and not equal(≠).

Some examples of comparing unlike fractions are given below:

Example 1:
Which is the greatest $\frac{5}{7}$ or $\frac{2}{5}$?
In the fractions $\frac{5}{7}$ and $\frac{2}{5}$, the denominators are the different.
Make the denominators same by taking LCM of them.
The LCM of 7 and 5 is 35.
$\frac{5}{7}$ ----> $\frac{5*5}{7*5}$ ----> $\frac{25}{35}$
$\frac{2}{5}$ ----> $\frac{2*7}{5*7}$ ----> $\frac{14}{35}$
Now, compare the numerators.
Among the numerator 25 and 14, 25 is the greatest.
So, $\frac{25}{35}$ is greater than $\frac{14}{35}$.
$\frac{25}{35}$ > $\frac{14}{35}$

Example 2:
Arrange the fractions in the order of lowest to greatest. Also find the greatest and the smallest among them. $\frac{2}{3}$, $\frac{9}{2}$, $\frac{3}{2}$, $\frac{8}{3}$?
In all the fractions the denominators are different.
Make the denominators same by taking LCM of them.
The LCM of 2 and 3 is 6.
$\frac{2}{3}$ ----> $\frac{2*2}{3*2}$ ----> $\frac{4}{6}$
$\frac{9}{2}$ ----> $\frac{9*3}{2*3}$ ----> $\frac{27}{6}$
$\frac{3}{2}$ ----> $\frac{3*3}{2*3}$ ----> $\frac{9}{6}$
$\frac{8}{3}$ ----> $\frac{8*2}{3*2}$ ----> $\frac{16}{6}$
Compare the numerators and arrange them in ascending order.
$\frac{4}{6}$, $\frac{9}{6}$, $\frac{16}{6}$, $\frac{27}{6}$
The smallest among the fraction is $\frac{4}{6}$.
The greatest among the fraction is $\frac{27}{6}$.

## Comparing Fractions with Same Denominator

Fractions are the Numbers which are expressed in form of $\frac{p}{q}$, where ‘p’ and ‘q’ are the whole Numbers and q ≠ 0. Fractions has two parts: numerator and denominator. In the above given equation 'p' is the numerator and 'q' is the denominator. Comparing Fractions is to find the greatest and smallest among the given Set of fractions. While Comparing Fractions we have to see if the denominators of the fractions are same i.e, like fractions. Like fractions are easy to compare than unlike fractions. So if the fractions are unlike then they are converted to like fractions. In this session we are going to learn about comparing Fractions with same denominator.

For comparing fractions with same denominator its just enough to compare the numerator. If two fractions are compared then the smallest or greatest among them is found. If a set of fractions are given then the fractions are arranged in some order on a number line and compared. The comparison of fractions is done by using four comparison operators: less than(<), greater than(>), equal(=) and not equal(≠).

Some examples of comparing fractions with same denominator are given below:

Example 1:
Which is the greatest $\frac{3}{7}$ or $\frac{8}{7}$?
In the fractions $\frac{3}{7}$ and $\frac{8}{7}$, the denominators are the same.
Compare the numerators.
Among the numerator 3 and 8, 8 is the greatest.
So, $\frac{8}{7}$ is greater than $\frac{8}{7}$.
$\frac{8}{7}$ > $\frac{8}{7}$

Example 2:
Arrange the fractions in the order of lowest to greatest. Also find the greatest and the smallest among them. $\frac{2}{5}$, $\frac{9}{5}$, $\frac{11}{5}$, $\frac{3}{5}$, $\frac{8}{5}$?
In all the fractions the denominators are the same.
Compare the numerators and arrange them in ascending order.
$\frac{2}{5}$, $\frac{3}{5}$, $\frac{8}{5}$, $\frac{9}{5}$, $\frac{11}{5}$
The smallest among the fraction is $\frac{2}{5}$.
The greatest among the fraction is $\frac{2}{5}$.