Comparing fractions – Definition, Methods, and examples

Understanding fractions is far more important than being adept at calculations. You must know what you are calculating and how you can bring those arithmetic calculations in a Mathematics notebook to the real world. The concept of comparing fractions is one such strategy that allows you to imply fractional calculations to your everyday life. Read on to discover how to compare fractions with like or unlike denominators and numerators.

Definition 

Comparison of fractions is to consider fractions to see how similar or how different they are. We can compare two or more fractions. 

However, we must understand what a fraction represents. A fraction has two numbers. The upper number is the numerator, and the bottom number is the denominator. Thus, a fraction represents a part of the whole. So, when we divide a whole into one or more parts, every part is a fraction of the whole. 

The denominator denotes the total number of equal parts in a collection or whole. At the same time, the numerator tells us about the number of equal parts taken.

For example, we have a circle. We can write it as 1. When we divide a circle into two parts and consider one part, we can represent this as 1/2. On dividing the circle into three equal parts and considering two parts, we write 2/3. But when we consider only a part of this circle, we will write it as 1/3.

How do you compare fractions?

There are several methods to compare fractions and determine the larger and the smaller fraction from the given two or more fractions. We can compare fractions using various graphical methods or models. These methods help us visualize larger fractions. 

Certain rules will help you in comparing and ordering fractions. You may also come across conditions where both the numerators and the denominators are equal. Such fractions are equal. The following conditions are when either the numerator or the denominator or both are different.

Comparing Fractions with Same Denominators

It is very easy to compare fractions with the same denominators. We can easily determine the greater or the smaller fraction. So, once we check that the denominators are the same, we have to check out the numerators to draw comparisons between the given fractions. With the same denominators, the fraction with a greater numerator is greater than the other fraction.

For example, we will compare 6/15 and 16/15

Step 1: We will observe the denominators of the fractions. Here we have two fractions with the same denominator: 6/15 and 16/15. 

Step 2: Next, we will compare the numerators of the given fractions. In the example, we can note that 16 > 6.

Step 3: So, the fraction that has the larger numerator is larger than the other fraction. Thus, 6/15 < 16/15.

Comparing Fractions with Unlike Denominators

If you know how to compare fractions with like denominators, you just have an additional step for comparing fractions with unlike denominators. You will have to convert the given fractions with unlike denominators to like denominators. You can do this by finding the Least Common Multiple (LCM) of the unlike denominators. Once you have fractions with the same denominators, you can compare them easily by checking out the numerators. 

For example, we will compare 1/4 and 3/5.

Step 1: We will check out the denominators of the given fractions. Here we have unlike denominators in the given fractions: 1/4 and 3/5. So, we will find the LCM of 4 and 5. The LCM of 4 and 5 is 20.

Step 2: Next, we will convert the given fractions in such a way that we have like denominators. We will multiply the first fraction by 5/5. So, 1/4 × 5/5 = 5/20.

Step 3: Then, we will multiply the second fraction by 4/4. So, 3/5 × 4/4 = 12/20.

Step 4: Now, the two fractions have the same denominators, so we can compare the numerators. Thus, between 5/20 and 12/20, we can see that 12> 5.

Step 5: As 12/20 > 5/20. Therefore, 3/5 > ¼. 

Comparing Fraction with the Same Numerators

We can also compare fractions with the same numerators and different denominators. We have to consider the denominators to conclude which fraction is greater.  

When given two fractions with the same numerator, the fraction with a smaller denominator is larger in value than the fraction with a larger denominator.  

For example, we have 2/5 and 2/7.

Step 1: Since the numerators are the same, we will observe the denominators.

Step 2: We can see that 7 > 5. So, the fraction 2/5 is greater than the fraction 2/7 as the fraction with a larger denominator is smaller when the numerators are the same.

Decimal Method of Comparing Fractions

The decimal method of comparing fractions is helpful when we have fractions that differ both in terms of numerator and denominator. So, in this method, we convert the fraction into decimal form and compare the decimal values. To convert fractions into decimals, we will divide the numerator by the denominator. 

For example, we will compare 6/8 and 3/9.

Step 1: First, we will convert the fraction into a decimal form. So, 6/8 in decimals will be 0.75, and 3/9 will be 0.33.

Step 2: Now, we can easily compare the decimal values. We can see that 0.75 > 0.33.

Step 3: Thus, we can write the fraction forms of the decimal numbers. The fraction that has a larger decimal value is larger than the other. Therefore, 6/8 > 3/9.

Comparing Fractions by Cross Multiplication

Cross multiplication is one of the most commonly used methods for comparing fractions. In this approach, we multiply the numerator of a fraction with the denominator of the other fraction. The following example will help you understand how we can compare two fractions using the cross multiplication method.

For example, we will compare 1/4 and 3/5. 

Step 1: In cross multiplication, we will cross multiply the given fractions and compare them. So, we have to multiply the first fraction’s numerator with the second fraction’s denominator. We will write the product obtained next to the first fraction’s numerator. Here, 1 × 5 = 5. So, we will write 5 next to the first fraction.

Step 2: Similarly, we will multiply the second fraction’s numerator with the first fraction’s denominator. We will write the product obtained near the numerator of the second fraction. So, 3 × 4 = 12. 

Step 3: Now, we can easily compare products 12 and 5. Since 5 < 12, we can compare their respective fractions. Here, 1/4 < 3/5. 

Real-Life Examples

Every day we come across a comparison of fractions in our day-to-day life. Here are some common examples of fractions that we compare in real-life.

• Equal slices of pizza
• Pieces of chocolate
• Division of fruits into equal parts
• When we divide cakes into pieces

Examples of Comparison of Fractions

The following examples will help you understand how to compare fractions.

Example 1: Which one of the following fractions is the largest?

1. 20/36
2. 3/5
3. 8/9
4. 5/6

Solution: Since we can see that the above-mentioned fractions have different numerators and denominators, we will use the decimal method to compare fractions.

So, we will convert the fractions into decimal forms. 

20/36 will be 0.55

3/5 will be 0.6

8/9 will be 0.88

5/6 will be 0.83

Thus, we can see that 0.88 is the largest. So, the fraction 8/9 is the largest. 

Example 2: Which one of the following fractions is the smallest?

1. 2/4
2. 6/18
3. 4/7
4. 8/15

Solution: Since we can see that the fractions mentioned above have different numerators and denominators, we will use the decimal method to compare fractions.

So, we will convert the fractions into decimal forms. 

2/4 will be 0.50

6/18 will be 0.33

4/7 will be 0.57

8/15 will be 0.53

Thus, we can see that 0.33 is the smallest. So, the fraction 6/18 is the smallest.

Example 3: Emma, Lily, and Bella are eating a cake. They ate the following amount of cake: 

1. Emma ate 2/5 
2. Lily ate 2/ 8
3. Bella ate 1/4

Who ate the maximum amount of cake among the three girls? Which of the three girls ate the same amount of cake?

​Solution: Since, here again, we have different numerators and denominators, we will convert the fractions into decimals to compare them.

Emma ate 2/5 = 0.4

Lily had 2/8 = 0.25

Bella has 1/4 = 0.25

Thus, we can say that Emma ate the maximum amount of cake among the three while Lily and Bella had the same amount of cake.

Frequently Asked Questions

1. What does it mean to compare fractions?

Ans. When you’re comparing fractions, you’re looking at how two or more numbers relate to each other.

An example of comparing fractions would be:

6/8 = .75

This means that 6/8 is greater than .75

2. What are some tricks to compare fractions?

Ans. If the numerators of two fractions are the same, you can subtract them and get an easier fraction to work with.

If the denominators of two fractions are the same, you can multiply them together and get a more accessible fraction to work with.

If both numerators and denominators are the same, you can add or subtract them together and get a more accessible fraction to work with!

3. How do you compare fractions in math?

Ans. Fractions can be compared by adding or subtracting the numerators and denominators.
For example, if you wanted to compare 1/3 with 2/5, you would add their numerators (1 + 2) and their denominators (3 + 5), which would give you 6/8.
This means that 1/3 > 2/5 since it’s larger than 6/8

4. What are the rules for comparing fractions?

Ans. There are two rules for comparing fractions:

1) The fraction with the larger numerator is greater.

2) The fraction with the larger denominator is greater.

5. What are the steps to comparing fractions?

Ans. The steps for comparing fractions are:

1. Subtract the numerators from the denominators.
2. If the result is positive, then the two fractions are equal.
3. If the result is a negative value, then the two fractions are not equal.