Session 3: Comparing and Ordering Fractions – Methods, Interactives and Examples

In our everyday life, we often come across situations where we have to compare two or more fractions. Comparing two fractions means determining the larger and the smaller fraction among them. Let us go through the different methods of comparing fractions.

Comparing Fractions with Like Denominators

1) Compare \(\frac{1}{7}\) and \(\frac{3}{7}\).

The diagram shows \(\frac{1}{7}\) and \(\frac{3}{7}\). In the first circle, one part out of seven parts is shaded. In the second circle, three parts out of seven parts are shaded. Clearly, the shaded part in the second circle is more than the first circle. Thus, the circle that has a more shaded part (the fraction with a larger numerator) will always be greater.

Here, \(\frac{1}{7}\)<\(\frac{3}{7}\).

Thus, if two fractions have the same denominator, then to compare two like fractions, it is just enough to compare their numerators. The fraction with the larger numerator is the larger fraction, and the fraction with the smaller numerator is the smaller fraction.

Steps to Compare Fractions with Like Denominators

To compare fractions with like denominators, we may use the following steps:

Step 1: Compare numerators.

Step 2: The fraction with a larger numerator is larger.

Comparing Fractions with Like Numerators

2) Compare \(\frac{3}{5}\) and \(\frac{3}{10}\).

The diagram shows \(\frac{3}{5}\) and \(\frac{3}{10}\). In the first circle, three parts out of five parts are shaded. In the second circle, three parts out of ten parts are shaded. Clearly, the shaded part in the first circle is more than the second circle. Thus, the one that has more shaded part (the fraction with a smaller denominator) will be greater.

Here, \(\frac{3}{5}\)>\(\frac{3}{10}\).

Thus, if two fractions have the same numerator but different denominators, the fraction with the larger denominator is the smaller fraction, and the fraction with the smaller denominator is the larger fraction.

Steps to Compare Fractions with Like Numerators

To compare fractions with like numerators, we may use the following steps:

Step 1: Compare denominators.

Step 2: The fraction with a larger denominator is smaller.

Comparing Fractions with Unlike Numerators and Denominators

When we compare things, be it numbers, objects or people, we must remember that they should be of the same quality, value or strength.

3) Compare \(\frac{1}{4}\) and \(\frac{1}{5}\).

The first set of diagrams shows \(\frac{1}{4}\) and \(\frac{1}{5}\). Here it is impossible to compare and say which is bigger though \(\frac{1}{5}\) seems bigger because the two bars are not the same size. 

In the second diagram, both the bars are of the same size; we can see that \(\frac{1}{4}\) is greater than \(\frac{1}{5}\). So we must remember that if we compare parts of two things that are not the same in size, we may end up giving the wrong answer. 

From this, we can also understand that while comparing one part of two diagrams that have been divided into a different number of parts, like \(\frac{1}{4}\) and \(\frac{1}{5}\), the one with less number of parts (the fraction with a smaller denominator will always be greater). Here, \(\frac{1}{4}\) is greater than \(\frac{1}{5}\).

Note: While comparing unlike fractions, make sure the wholes are the same size, and only the parts are divided differently.

Steps to Compare Fractions with Unlike Numerators and Denominators

To compare fractions with unlike numerators and denominators, we may use the following steps:

Step 1: Find the LCM of the denominators of the given fractions.

Step 2: Convert each fraction to its equivalent fraction with the denominator equal to LCM obtained in the above step. 

Step 3: Compare the numerators of the equivalent fractions.

Step 4: The fraction with a larger numerator is larger. 

4) Which is larger \(\frac{6}{7}\) or \(\frac{8}{9}\)?

Let us first find the LCM of 7 and 9. We have, 

∴ LCM of 7 and 9 =3×3×7=63.

Now, we convert the given fraction to equivalent fractions with denominator 63. We have, 

and

We know that 56>54. 

Comparing Fractions Using Cross Multiplication

In this method, we cross multiply the numerator of one fraction with the other fraction’s denominator.

5) Compare \(\frac{2}{3}\) and \(\frac{5}{8}\).

By cross-multiplication, we have 

Because 16 is greater than 15, we know that \(\frac{2}{3}\) is greater than \(\frac{5}{8}\).   

Remember that to find out which of two fractions is larger, cross-multiply and place the two products, in order, under the two fractions. The larger product is always under the larger fraction.

Note: When cross-multiplying to compare fractions and find out which is greater, make sure to start with the numerator of the first fraction.

Ordering Fractions

Sometimes, we need to write fractions in order from least to greatest or from greatest to least. This becomes very simple if the fractions have the same denominator.

6) Write in order from least to greatest. 

Since all of these fractions have a common denominator, use the numerators and arrange them from the smallest numerator to the largest numerator.

The answer is .

To order fractions that do not have a common denominator, rewrite all fractions using the lowest common multiple (LCM).

7) Order the given fractions from greatest to least (descending order).

Let us first find the LCMs of the denominators: 

We have, 

∴LCM(3,4,9,12)=2×2×3×3=36

Now, we convert each fraction to its equivalent fraction with denominator 36.  

We know that 

Comparing and Ordering Fractions – Examples

Example 1

Of the trees in the park, \(\frac{3}{8}\) were pines, and \(\frac{3}{5}\) were spruce. Were there more pines or more spruce trees in the park?

We know that the fraction with the smaller denominator is larger out of two fractions with the same numerator. 

Hence, there are more spruce trees in the park.    

Example 2

Karen ate \(\frac{1}{7}\) parts of the cake at the party while Ria ate \(\frac{2}{11}\) of it. Who ate less?

To know who ate less, we will compare \(\frac{1}{7}\) and \(\frac{2}{11}\).

We have, 

LCM(7,11)=7×11=77

Converting each fraction into an equivalent fraction 77 as its denominator, we have 

Hence, Karen ate lesser cake than Ria. 

Example 3

Compare \(1\frac{3}{4}), \(1\frac{5}{6}) and \(\frac{1}{3}).

First, we convert the given fractions into like fractions, i.e. fractions having the common denominator. For this, we first find the LCM of the denominators of the given fractions. 

LCM(3,4,6)=12

Now we convert each fraction into equivalent fractions with 12 as the denominator. 

The equivalent fraction of \(1\frac{3}{4})=\(\frac{7}{4}), with denominator .

The equivalent fraction of \(1\frac{5}{6})=\(\frac{6}{11}), with denominator .

The equivalent fraction of \(\frac{1}{3}), with denominator .

We know, 

Example 4

Priya lives \(\frac{3}{4}\) km from the school. Divya lives \(\frac{2}{5}\) km from school, and Shaheen lives \(\frac{1}{2}\) km from the school. Who lives closest to school?

We observe that the given fractions neither have a common numerator nor a common denominator. So, first, we convert them into like fractions. For this, we find the LCM of the denominators. 

LCM(4,5,2)=20.

Now, we convert the given fractions into equivalent fractions with the denominator as 20. 

The equivalent fraction of \(\frac{3}{4}\) with denominator .    

The equivalent fraction of \(\frac{2}{5}\) with denominator .

The equivalent fraction of \(\frac{1}{2}\) with denominator .

We know, 

Hence, Divya lives closest to school.  

Remember this!

1. If two fractions have the same denominator, the fraction with the larger numerator is larger.  2. If two fractions have the same numerator, the fraction with the larger denominator is smaller.