How to compare two fractions in numbers?

In order to determine which of the two fractions is larger or smaller, one has to compare them. Based on the numerator and the denominator and the kind of fractions given there are different methods and rules to compare fractions. They are:

1. Comparing Fractions with Same Denominators
2. Comparing Fractions With Unlike Denominators
3. Comparing Fractions using the Decimal Method
4. Comparing Fractions using Visualization
5. Comparing Fractions using the Cross Multiplication method

Comparing Fractions with Same Denominators

It is easy to find the greater or smaller fraction when the fractions have the same denominators. When comparing fractions, check whether the denominators are the same or not. If the denominators are equal, then the fraction with the bigger numerators is the bigger fraction. The fractions are equal if the numerators and denominators of both fractions are equal.

Example: Compare: 5/12 and 17/12.

Solution:

Step 1: First, observe the denominators of the given fractions, i.e., 5/12 and 17/12. Here, the denominators are the same for both fractions.

Step 2: Now, compare the numerators of the given fractions. We can observe that 17 > 5.

Step 3: We know that the fraction with the larger numerator is larger. Hence, 5/12 < 17/12.

Comparing Fractions With Unlike Denominators

To compare fractions with unlike denominators, we have to convert them to like denominators for which we have to find the Least Common Multiple (LCM) of the denominators. As the denominators are made equal, we can compare the fractions with ease.

Example: Compare: 1/4 and 2/3.

Solution: 

Step 1: First, observe the denominators of the given fractions, i.e., 1/4 and 2/3. Since the denominators are different make them equal by finding the LCM of 4 and 3. LCM(4,3) = 12.

Step 2: Now, let us convert the given fraction in such a way that they have the same denominators. So, multiply the first fraction with 3/3, i.e., 1/4 × 3/3 = 4/12. 

Step 3: Similarly, multiply the second fraction with 4/4, i.e., 2/3 × 4/4 = 8/12. Thus, the first fraction becomes 4/12 and the other becomes 8/12.

Step 4: Compare the obtained new fractions, i.e., 4/12 and 8/12. As the denominators are the same, we will compare the numerators. We can observe that 4 < 8.

Step 5: The fraction that has a large numerator is the larger fraction. So, 8/12 > 4/12. So, 1/4 > 2/3.

Note: Make a note that if the given fractions have the same numerators and different denominators, then we can compare them easily by looking at their denominators. The fraction that has a smaller denominator has a greater value, while the fraction that has a larger denominator has a smaller value. For example 6/2 > 6/5.

Comparing Fractions using the Decimal Method

In this method, one can compare fractions by finding the decimal values of the fractions and comparing them. For this, divide the numerator by the denominator, and thus the fraction is converted into a decimal. Finally, compare their decimal values. Let us understand this by going through an example.

Example: Compare 3/5 and 2/4. 

Solution:

 

Step 1: To write 3/5 and 2/4 in decimals, divide the numerator by the denominator. Divide 3 by 5, and 2 by 4.

Step 2: The obtained decimal values are 0.6 and 0.5.

Step 3: Finally, compare the decimal values. 0.6 > 0.5. The fraction that has a larger decimal value would be larger. Hence, 3/5 > 2/4.

Comparing Fractions using Visualization

Compared to any other method, comparing fractions using visualization is easier. Make two boxes such that the length and width of both are the same. The figure is given below shows models A and B, which represent two fractions. Then divide each model into equal parts equivalent to their respective denominators. Now, we can easily find that 2/6 < 2/4, as the 2/4 covers a larger shaded area compared to the 2/6. The smaller fraction occupies a lesser area of the same whole, while the larger fraction occupies a larger area of the same whole.

Comparing Fractions using the Cross Multiplication method

To compare fractions using cross multiplication, we have to multiply the numerator of one fraction with the other fraction’s denominator. Let us understand this by going through an example.

Example: Compare 3/8 and 4/5.

Observe the figure given below which explains the concept of cross multiplication better.

 

Step 1: Make a note that when we are performing cross multiplication to compare two fractions, we have to multiply the numerator of the first fraction with the second fraction’s denominator. We have to write the product on the side of the selected numerator. Here, the product is 3 × 5 = 15, which we write near the first fraction. 

Step 2: Similarly, when multiplying the second fraction’s numerator with the first fraction’s denominator, we have to write the product next to the second fraction. Here, the product is 4 × 8 = 32, which we will write near the second fraction.

Step 3: Now, compare both products, i.e., 15 and 32. Since 15 < 32, the respective fractions can be easily compared, i.e., 3/8 < 4/5. Hence, 3/8 < 4/5.

“Comparison of fractions” refers to the determination of the larger and the smaller fraction within a given set of fractions. While comparing fractions, a set of rules is followed to compare the numerator and denominator of a fraction, where the numerator is the number above the fractional bar and the denominator is the number below the fractional bar. We can determine the greater and smaller fractions by comparing any two fractions. Fractions can be compared even if they have different numerators and denominators. To understand the concept better, let’s go over the various ways to compare fractions.