"Comparison of fractions" refers to the process of determining the larger or smaller fraction by comparing their numerators and denominators. This can be done for any fractions even if the fractions have different numerators and denominators.
Fractions
Before going through the concept of comparing two fractions, let us recall what a fraction is. A fraction is defined as the part of a whole that consists of two parts: the numerator and the denominator, where the numerator is the number above the fractional bar and the denominator is the number below the fractional bar.
Fraction RepresentationHow to Compare Two Fractions
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:
- Comparing Fractions with Same Denominators
- Comparing Fractions With Unlike Denominators
- Comparing Fractions using the Decimal Method
- Comparing Fractions using Visualization
- Comparing Fractions using the Cross Multiplication method
Learn all about Fractions : Fraction Tutorials
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: 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 converting the fractions into decimal 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:
Comparing fractions using Decimal methodStep 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.
To compare fractions visually we need to make two boxes of the same size are divided into parts based on the denominators of the fractions and shade the areas based on the numerators. Now, we can simply compare the shaded region to determine the larger or smaller fraction.
Comparing Fractions using VisualizationHere, we can easily see 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, while the larger fraction occupies a larger area of the same box.
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.
Steps to compare fractions Using Cross Multiplication method
Step 1: Multiply the numerator of the first fraction with the second fraction's denominator. And write the product on the side of the selected numerator.
Step 2: Similarly, multiply the second fraction's numerator with the first fraction's denominator, and write the product on the side of the selected numerator.
Step 3: Now, compare both products and the fraction on the side with greater product will be bigger than the other fraction.
Example: Compare 3/8 and 4/5.
Observe the figure given below which explains the concept of cross multiplication better.
Comparing fractions using cross multiplication methodStep 1: Multiply the numerator of the first fraction with the second fraction's denominator. Here, the product is 3 × 5 = 15, which we write near the first fraction.
Step 2: Multiply the second fraction's numerator with the first fraction's denominator. 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.
Solved Examples on Comparison of Fractions
Example 1: Which of the following fractions is larger: 6/11 or 8/15?
Solution:
Given fractions: 6/11 and 8/15
The denominators of the given fractions are different. So, find out the LCM of the denominators, i.e., LCM(11, 15) = 165.
Now, multiply 6/11 with 15/15 and 8/15 with 11/11.
6/11 × 15/15 = 90/165
8/15 × 11/11 = 88/165
Compare the numerators now, as the denominators are the same.
So, 90 > 88, i.e., 90/165 > 88/165.
Hence, 6/11 > 8/15, i.e., 6/11 is the larger fraction.
Example 2: Which of the given fractions is smaller: 13/85 or 21/85?
Solution:
Given fractions: 13/85 and 21/85.
The denominators of the given fractions are the same. So, compare the numerators of the given fractions.
13 < 21.
So, 13/85 < 21/85.
Hence, 13/85 is the smaller fraction.
Example 3: Compare the fractions 4/25 and 33/100.
Solution:
To compare the given fractions, find their decimal values. So, divide 4 by 25 and 33 by 100.
4/25 = 0.16
33/100 = 0.33
From the decimal values, we can conclude that 0.33 > 0.16. So, 33/100 is greater than 4/25.
Therefore, 33/100 is greater than 4/25.
Example 4: Mrunal Murthi was asked to prove that the given fractions are equal: 30/90 and 25/75. Can you prove the given statement using the LCM method?
Solution:
Given fractions: 30/90 and 25/75.
The denominators of the given fractions are different. So, find out the LCM of the denominators, i.e., LCM(90, 75) = 450.
Now, multiply 30/90 with 5/5 and 25/75 with 6/6.
30/90 × 5/5 = 150/450
25/75 × 6/6 = 150/450
Compare the numerators now, as the denominators are the same.
So, 150 = 150, i.e., 150/450 = 150/450.
Thus, 30/90 = 25/75, i.e., both the given fractions are equal.
Hence, proved.
Example 5: Which of the following fractions is larger: 27/41 or 27/67?
Solution:
Given fractions: 27/41 and 27/67.
Here, the numerators of both fractions are the same but the denominators are different.
We know that the fraction that has a smaller denominator has a greater value, while the fraction that has a larger denominator has a smaller value.
Here, 41 < 67.
So, 27/41 > 27/67
Therefore, 27/41 is the larger fraction.
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