Comparing Fractions

Comparing Fractions: When it comes to fractions, we usually compare two or more fractions. In fact, we experience fractions in our day to day life. A simple example, if you cut an apple in two parts, it is also a fraction. Basically, comparing two fraction means to determine the large and the smaller fraction among them. Let us you more about fractions and their comparison. In this article you get details on fractions definition, how to compare fractions, fraction comparison rules, the decimal method of comparing fractions, and solved examples for quicker and better understanding.

We, at Embibe, strongly recommend that you learn the basic concept behind fractions, because it is an important aspect when it comes to algebra. it will help you get a better understanding of algebra if you thoroughly read the basics of fractions and how to compare it and what are unlike and like fractions. Read full article to get complete knowledge.

Define Fractions

A fraction is said to be a number that represents a part of the whole. That whole can be a single object or a group of objects. A fraction is written as \(\frac{p}{q}\), where \(p\) and \(q\) are whole numbers and \(q \ne 0\). Numbers such as \(\frac{1}{2},\frac{2}{3},\frac{4}{5},\frac{{11}}{7}\) are known as the fractions. 

The number below the division line is called the denominator. It describes us how many equal parts a whole is divided into. The number above the line is called the numerator. It tells us how many equal parts are taken.

Examples: \(\frac{3}{7},\frac{5}{{10}} = \frac{{11}}{4}\)etc. are some of the examples of the fractions.

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How to Compare Fractions?

Now that you have got the understanding of what is a fraction? Let us tell you how to compare fractions. As you already know, a fraction is nothing but a part of a whole object. So, if you break the glass dish into several pieces, can you still say that each element represents the fraction? 

Yes, you can. Reasonably, each element can still be called a fraction of the glass dish, but a fraction comes with a rule in mathematics. The rule is “each of the parts has to be equal.” Thus, a fraction has two parts; they are called the numerator and the denominator. 

Now, let’s discuss what comparing fractions mean when the two bits are compared to find out which is greater or smaller. The real-time examples of comparing fractions include various activities like checking discounted prices while shopping, reaching sales of a particular product, medical prescriptions by the doctor, scores of tests and exams, etc. So let us go through the different methods of comparing fractions with the help of examples to understand the concept better. Again, comparing fractions or fractions is what you experience or deal with in your daily life. If you focus enough, you can easily get a practical understanding of the same everyday while doing normal house chores and mathematical calculations.

Comparing Fractions Rules

There are a few rules that we have to follow when comparing the fractions:

1. When the denominators of the fraction are the same, the fraction with the smaller numerator is the smaller fraction, and the fraction with the greater numerator is considered a greater fraction. 
2. When the numerators are equal, the fractions are considered equivalent.
When the fractions have the same numerator, the smaller denominator is considered, the more significant fraction.

Comparing Fractions with the Same Denominators

Definition of like Fractions :Two or more fractions having the same dominator are known as ‘like fractions’.
Example: \(\frac{3}{{11}},\frac{{15}}{{11}} = \frac{{ – 9}}{{11}}\) are ‘like fractions’.

Comparing Like Fractions

In this method, you have to check whether the denominators are the same or not. If the denominators are the same, then the fraction with the bigger numerator is the more considerable fraction. The fraction with the smaller numerator is the smaller fraction. If both the numerators and the denominators are equal, then the fractions are also identical. For example: Let us compare \(\frac{6}{{17}}\) and \(\frac{{16}}{{17}}\).

1. Look for the denominators of the given fractions: \(\frac{6}{{17}}\) and \(\frac{{16}}{{17}}\). Here the denominators are the same.
2. Next, compare the numerators \(16 > 6\).
3. Now, the fraction with the larger numerator would be larger. 
4. Hence, \(\frac{6}{{17}} < \frac{{16}}{{17}}\).

Comparing Fractions with Unlike Denominators

Definition of an Unlike Fraction: When two or more fractions have a different denominator, it is known as ‘unlike fractions’.

Example: \(\frac{3}{5},\frac{7}{{11}}, – \frac{2}{4},\frac{{39}}{{14}}\) are unlike fractions.

Comparing Unlike Fractions

To compare the fractions with unlike denominators, we advise you to start by finding the least common denominator (LCM) to make denominators the same. When the denominators are converted into the same denominators, then the fraction with the larger numerator is the more significant fraction—for example, \(\frac{1}{2}\) and \(\frac{2}{5}\).

1. Look for the denominators of the given fractions, \(\frac{1}{2}\) and \(\frac{2}{5}\) here, the denominators are not the same.
2. Next, take out the LCM of \(2 & 5\), so \({\mathop{\rm LCM}\nolimits} (2,5) = 10\).
3. Here, \(\frac{1}{2} = \frac{1}{2} \times \frac{5}{5}\) and \(\frac{2}{5} = \frac{2}{5} \times \frac{2}{2}\).
4. Now, compare the fractions, \(\frac{5}{{10}}\) and \(\frac{4}{{10}}\) so, the denominators are the same. 
5. We will compare the numerators, \({\rm{5 > 4}}{\rm{.}}\).
6. Comparing the fractions, \(\frac{5}{{10}} > \frac{4}{{10}}\) . The fraction with a larger numerator is the larger fraction. Thus, \(\frac{5}{{10}} > \frac{4}{{10}}\). Therefore, \(\frac{1}{2} > \frac{2}{5}\)

If the denominators are different and the numerators are the same, you can easily compare fractions by looking at their denominators. The fraction with a smaller denominator has a greater value. The fraction with a larger denominator has a smaller value. 

For example, \(\frac{2}{3} > \frac{2}{6}\)

Decimal Method of Comparing Fractions

In this method, you have to compare the decimal values of the fractions. First, the numerator is divided by the denominator, and then the fraction is converted into a decimal. Then, the decimal values are compared.

Example: \(\frac{4}{5}\) and \(\frac{6}{8}\)

1. First, write the given fractions \(\frac{4}{5}\) and \(\frac{6}{8}\) in decimal form. \(\frac{4}{5} = 0.8\) and \(\frac{6}{8} = 0.75\).
2. Now, compare the decimal values, \(0.8 > 0.75\).
3. Here, the fraction with the larger decimal value is the larger fraction. 
4. Therefore, \(\frac{4}{5} > \frac{6}{8}\).

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Comparing Fractions Using Cross Multiplication

In this method, the numerator of one fraction is cross multiplied by the other fraction’s denominator. For example, the arrows have indicated the same in the given diagram below.

Example: When we cross multiply, we get \(4\) and \(6\).

1. Now, the numbers \(4\) and \(6\) are the numerators we get if you have expressed \(\frac{1}{2}\) and \(\frac{3}{4}\) with the common denominator \(8\). 

2. Next, the new fractions with the same denominators will be \(\frac{4}{8}\) and \(\frac{6}{8}\). 

3. So, the number 6 is the greater numerator, \(\frac{4}{8} < \frac{6}{8}\). 

4. Therefore, \(\frac{1}{2} < \frac{3}{4}\).

Solved Examples – Comparing Fractions

Q.1. Compare the two fractions \(\frac{4}{7}\) and \(\frac{2}{7}\).
Ans: Given, \(\frac{4}{7}\) and \(\frac{2}{7}\)
We can see that the denominators in the given fractions are the same.
Here, we will follow the rule of when the denominators of the fraction are the same, the fraction with the smaller numerator is the smaller fraction, and the fraction with the greater numerator is considered a greater fraction. 
So, compare the numerators \(4 > 2\)
Hence, \(\frac{4}{7} > \frac{2}{7}\)

Q.2. Compare the two given fractions: \(\frac{6}{{13}}\) and \(\frac{6}{{20}}\)
Ans: Given \(\frac{6}{{13}}\) and \(\frac{6}{{20}}\)
We can see that the numerators in the given fractions are the same.
Here, we will follow the rule that when the fractions have the same numerator, the smaller denominator is considered, the greater fraction.
So, compare the denominators \(13 > 20\)
Hence, \(\frac{6}{{13}} > \frac{6}{{20}}\)

Q.3. Compare the given fractions using the cross multiplication method: \(\frac{3}{8}\) and \(\frac{5}{{10}}\)
Ans: Given \(\frac{3}{8}\) and \(\frac{5}{{10}}\)
We will use the cross multiplication method,
So, \(\frac{3}{8} \to \frac{5}{{10}}\) which means multiply \(3 \times 10 = 30\)
Now, \(\frac{3}{8} \leftarrow \frac{5}{{10}}\) which means multiply \(5 \times 8 = 40.\)
Here, \(30 < 40\)
Hence, \(\frac{3}{8} < \frac{5}{{10}}\)

Q.4. Arrange the fractions \(\frac{5}{6},\frac{{11}}{{16}},\frac{{13}}{{18}}\) in ascending order
Ans: We will take out the LCM of the denominators first,
\(6,16,18 = 2 \times 3 \times 8 \times 3 = 144\)
Now, write the fractions as equivalent like fractions.
\(\frac{5}{6} = \frac{{5 \times 24}}{{6 \times 24}} = \frac{{120}}{{144}},\frac{{11}}{{16}} = \frac{{11 \times 9}}{{16 \times 9}} = \frac{{99}}{{144}},\frac{{13}}{{18}} = \frac{{13 \times 8}}{{18 \times 8}} = \frac{{104}}{{144}}.\)
So, \(99 < 104 < 120 \Rightarrow \frac{{99}}{{144}} < \frac{{104}}{{144}} < \frac{{120}}{{144}} \Rightarrow \frac{{11}}{{16}} < \frac{{13}}{{18}} < \frac{5}{6}\)
Hence, the given fraction in ascending order are \(\frac{{11}}{{16}},\frac{{13}}{{18}},\frac{5}{6}\)

Q.5. Which is bigger: \(\frac{4}{8}\) or \(\frac{6}{{12}}\) Compare using the decimal method.
Ans: We can use a calculator \(4 \div 8\) and \(6 \div 12\)
Now, we get
\(\frac{4}{8} = 0.5\) and \(\frac{6}{{12}} = 0.5\)
So, both the fractions are equal \(\frac{4}{8} = \frac{6}{{12}}\)
Hence, \(\frac{4}{8} = \frac{6}{{12}}\)

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Summary

In this article, you have learned how to compare the fractions, a recap on what fractions are, and discussed the rules of the fractions. Later we learnt to compare the like fractions and the unlike fractions. 

Also, we had a glance at comparing fractions using cross multiplications and decimal method along with the solved examples and some of the FAQ. Readers can compare fractions tricks to solve the questions quickly.

Frequently Asked Questions

Q.1. Which fractions are bigger?
Ans: To compare fractions with unlike denominators, convert them to equivalent fractions with the same denominator. If denominators are the same, you can compare the numerators. The fraction with the bigger numerator is the larger fraction.

Q.2.What function has comparing fractions?
Ans: Comparing fractions is easier if the denominators of the given fractions are the same. 
Hence, if there is a group of like fractions, they can be compared easily. For example, when you compare two fractions \(\frac{{21}}{{50}}\) and \(\frac{{37}}{{50}}\), you have to compare the numerators only.

Q.3. What are the tricks of comparing fractions?
Ans: Given two fractions, you can use a little trick to understand which fraction is larger.
Example: Compare \(\frac{3}{8}\) and \(\frac{4}{9}\)
Multiply the numerator of the first fraction (the top number in the fraction) by the denominator (the bottom number in the fraction) of the second fraction. 
Then compare the two answers.

Q.4. What is the meaning of comparing fractions?
Ans: Comparing fractions means if you want to identify that one fraction is less than, greater than or equal to another. So we use symbols as we use with whole numbers <, > or = 

Q.5. Give some compare fractions examples.
Ans: An examples of comparing fractions is listed below:
Example1: \(\frac{4}{5}\) and \(\frac{6}{8}\)
1. First, write the given fractions \(\frac{4}{5}\) and \(\frac{6}{8}\) in decimal form. \(\frac{4}{5} = 0.8\) and \(\frac{6}{8} = 0.75\).
2. Now, compare the decimal values, \(0.8 > 0.75\).
3. Here, the fraction with the larger decimal value is the larger fraction.
4. Therefore \(\frac{4}{5} > \frac{6}{8}\)
Example2: When we cross multiply, we get \(4\) and \(6\). 
1. Now, the numbers \(4\) and \(6\) are the numerators we get if you have expressed \(\frac{1}{2}\) and \(\frac{3}{4}\) with the common denominator \(8\). 
2. Next, the new fractions with the same denominators will be \(\frac{4}{8}\) and \(\frac{6}{8}\). 
3. So, the number \(6\) is the greater numerator, \(\frac{4}{8} < \frac{6}{8}\) 
4. Therefore, \(\frac{1}{2} < \frac{3}{4}\)

Q.6. What are the two easy methods used for comparing the fractions?
Ans: The two methods which are used to compare the fractions easily are
1. Decimal method and 
2. Cross Multiplication method

Q.7. What are the uses of Compare Fractions?
Ans: Comparing fractions helps us to understand whether the given fractions are less than or greater than, or equal to each other. Like we compare the whole numbers using the symbols of \(<, >\), or \(=\) , We use the same signs to comparing the fractions.

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