Operation on Fractions: Definition, Classification, Examples

A fraction is a part of a whole quantity. When we divide a whole quantity into equal parts, each part is called a fraction. A fraction has two parts, the numerator and the denominator. A fraction is written as \(\frac{a}{b}, a\) and \(b\) are whole numbers, and \(b\) should not be zero. The different operations on fractions are addition, subtraction, multiplication, and division on fractions.

Operating fractions is different from operating whole numbers, natural numbers, and integers. Let’s look at the steps involved in performing the various mathematical operations on fractions. Continue reading to learn more!

Definition of Fractions

A fraction is a number that represents a part of the whole. The whole may be a single object or multiple objects. A fraction is written as \(\frac{x}{y}\) where \(x\) and \(y\) are whole numbers and \(y \neq 0\). Numbers such as \(\frac{1}{9}, \frac{2}{15}, \frac{14}{9}, 2 \frac{11}{7}\) are known as the fractions.

Classification of Fractions

The classification of fractions are as follows:

1. Proper Fractions and Improper Fractions

In a proper fraction, the numerator is always smaller than the denominator. For example, \(\frac{7}{10}, \frac{2}{4}, \frac{1}{6}, \frac{3}{20}, \frac{21}{25}\) are the proper fractions.

In an improper fraction, the numerator is always greater than or equal to the denominator. For example, \(\frac{13}{2}, \frac{17}{4}, \frac{6}{5}, \frac{15}{11}, \frac{10}{10}\) are the improper fractions.

2. Mixed Fractions

A mixed fraction is a fraction that is a combination of both whole and part fractions in the same fraction. A mixed fraction has a value that is always greater than one.

For example, \(1 \frac{1}{2}, 2 \frac{3}{4}, 5 \frac{5}{6}\) are the mixed fractions.

3. Like Fractions and Unlike Fractions

The group of two or more fractions with the same denominators or identical denominators are called like fractions. For examples, \(\frac{1}{6}, \frac{2}{6}, \frac{3}{6}, \frac{4}{6}, \frac{7}{6}\), etc., are all like fractions.

The group of two or more fractions that have different denominators are unlike fractions. For examples, \(\frac{1}{6}, \frac{2}{4}, \frac{2}{5}, \frac{4}{7}, \frac{5}{8}\), etc., are all unlike fractions.

Addition of Fractions

The addition is one of the fundamental operations that are also applicable to fractions. There are several methods of addition of fractions. Let us know about them.

Addition of Like Fractions

We know that the like fractions have identical denominators. To add the like fractions, add the values of the numerators, keeping the denominators the same.

Example: \(\frac{3}{5}+\frac{1}{5}\)

Here, given fractions have the same denominator, \(5\). We will keep the denominator the same for the result and add the numerator to find the numerator of the result.

Therefore, \(\frac{3}{5}+\frac{1}{5}=\frac{3+1}{5}=\frac{4}{5}\)

Addition of Unlike Fractions

We know that unlike fractions have different denominators. So, we need to convert the, unlike fractions into like fractions. It means the fractions must have identical denominators.

To add the unlike fractions, we need to follow some steps.

1. Find the LCM of the denominators of the given unlike fractions
2. Change the denominators into the obtained LCM. This process can change the numerators of the given, unlike fractions.
3. Now, add the numerators.

Example: \(\frac{4}{7}+\frac{2}{3}\)

LCM of \((7,3)\) is \(21\).
To make the denominator \(21\), we need to multiply the numerator and the denominator of \(\frac{4}{7}\) by \(3\). So, we get
\(\frac{4}{7} \times \frac{3}{3}=\frac{12}{21}\)
To make the denominator of \(\frac{2}{3}\) to \(21\), we need to multiply the numerator and the denominator by \(7\). So, we get
\(\frac{2}{3} \times \frac{7}{7}=\frac{14}{21}\)
Now, \(\frac{4}{7}, \frac{2}{3}\) are converted to like fractions that is \(\frac{12}{21}, \frac{14}{21}\) respectively.
Now, we can add them in a similar way to adding like fractions.
\(\frac{12}{21}+\frac{14}{21}=\frac{12+14}{21}=\frac{26}{21}\)

Subtraction of Fractions

We have discussed the addition of fractions. Similarly, we can subtract fractions. Let us see the subtraction of like and unlike fractions.

Subtraction of Like Fractions

Subtraction of like fractions is similar to the addition of the like fractions. To subtract the like fractions, subtract the values of the numerators, keeping the denominators the same.

Example: \(\frac{3}{4}-\frac{2}{4}\)

Here, given fractions have the same denominator \(4\). We will keep the denominator the same for the result and subtract the numerators to find the numerator of the result.

Therefore, \(\frac{3}{4}-\frac{2}{4}=\frac{3-2}{4}=\frac{1}{4}\)

Subtraction of Unlike fractions

Here, we will follow the same steps which we followed in the addition of unlike fractions. But instead of adding numerators, we will subtract them.

Let us learn the steps:

1. Find the LCM of the denominators of the given unlike fractions
2. Change the denominators into the obtained LCM. This process can change the numerators of the given, unlike fractions.
3. Now, subtract the numerators.

Example: \(\frac{5}{8}-\frac{3}{64}\)

LCM of \((8,64)\) is \(64\).
To make the denominator \(64\), we need to multiply the numerator and the denominator of \(\frac{5}{8}\) by \(8\). So, we get \(\frac{5}{8} \times \frac{8}{8}=\frac{40}{64}\)

Now, \(\frac{40}{64}\) and \(\frac{3}{64}\) are like fractions as they have the same denominators.

Therefore, \(\frac{40}{64}-\frac{3}{64}=\frac{37}{64}\)

Multiplication of Fractions

The numerators and denominators of two fractions are multiplied separately when they are multiplied. The first fraction’s numerator will be multiplied by the second’s numerator, and the first fraction’s denominator will be multiplied by the second’s denominator. In the end, we will reduce the fraction to its lowest form if it is required.

Example:

The product of \(\frac{2}{3}\) and \(\frac{1}{2}\) is determined by multiplying the numerators and denominators separately.
\(\Rightarrow \frac{2}{3} \times \frac{1}{2}=\frac{2}{6}\)
Here, \(\frac{2}{6}\) is not in its simplest form as the HCF of the numerator and the denominator is not \(1\)
HCF \((2, 6)=2\)
We can reduce the fraction to its lowest form by dividing the numerator and the denominator by \(2\). Thus, \(\frac{2 \div 2}{6 \div 2}=\frac{1}{3}\)

Division of Fractions

Dividing the fraction by another fraction is multiplying the first fraction by the reciprocal of the second fraction. Let us learn the following steps of the division of fractions.

1. We will keep the first fraction the same, and we need to determine the reciprocal of the second fraction. 
2. Change the division sign by multiplication sign and multiply the first fraction with the reciprocal of the second fraction. 
3. Find the simplest form of the fraction, if needed.

Example: \(\frac{4}{15} \div \frac{2}{3}\)

Here, the reciprocal of the second fraction \(\left(\frac{2}{3}\right)\) is obtained by interchanging denominator and numerator and we get, \(\frac{3}{2}\).
So, \(\frac{4}{15} \div \frac{2}{3}\) can be written as \(\frac{4}{15} \times \frac{3}{2}=\frac{12}{30}\).

Here, \(\frac{12}{30}\) is not in its simplest form as the HCF of the numerator and the denominator is not \(1\)
HCF \((12,30)=6\)
We can reduce the fraction to its lowest form by dividing the numerator and the denominator by \(6\). Thus, \(\frac{12 \div 6}{30 \div 6}=\frac{2}{5}\).

Solved Examples on Operation on Fractions

Q.1. A stick of length \(5 \mathrm{~m}\) is cut into the small sticks of length \(\frac{1}{5} \mathrm{~m}\) each. How
many small sticks can be cut?
Ans: The number of small rods \(=5 \div \frac{1}{5}\)
\(=5 \times \frac{5}{1} \) (Reciprocal of \(\frac{1}{5}\) is \(5\))
\(=25\) small sticks

Q.2. Solve \(\frac{9}{12}-\frac{7}{12}\)
Ans: We have \(\frac{9}{12}-\frac{7}{12}\)
Here, given fractions have the same denominator \(12\). We will keep the denominator the same for the result and subtract the numerators to find the numerator of the result.
\(\frac{9}{12}-\frac{7}{12}=\frac{9-7}{12}\)
\(=\frac{2}{12}\)
\( = \frac{{2 \div 2}}{{12 \div 2}}\) (Dividing the numerator and denominator by their HCF \(2\))
\(=\frac{1}{6}\)
Hence, the obtained fraction is \(\frac{1}{6}\).

Q.3. Harsha was given \(\frac{3}{5}\) of a basket of apples. What portion of apples has been left in the basket?
Ans: Harsha was given \(\frac{3}{5}\) of a basket of apples.
We need to find the fraction of apples was left in the basket.
So, \(1-\frac{3}{5}=\frac{1}{1}-\frac{3}{5}\)
\(=\frac{1 \times 5}{1 \times 5}-\frac{3}{5}=\frac{5}{5}-\frac{3}{5}\)
\(=\frac{5-3}{5}=\frac{2}{5}\)
Hence, the fraction of apples left in the basket was \(\frac{2}{5}\).

Q.4. Solve \(\frac{5}{6} \times \frac{9}{10}\).
Ans: Given \(\frac{5}{6} \times \frac{9}{10}\)
The product of \(\frac{5}{6}\) and \(\frac{9}{10}\) is determined by multiplying the numerators and denominators separately.
\(\frac{5}{6} \times \frac{9}{10}=\frac{5 \times 9}{6 \times 10}=\frac{45}{60}\)
HCF \((45,60)=15\)
Therefore, \(\frac{45 \div 15}{60 \div 15}=\frac{3}{4}\).

Q.5. Solve \(2 \frac{3}{5}+\frac{4}{5}+1 \frac{2}{5}\)
Ans: Given, \(2 \frac{3}{5}+\frac{4}{5}+1 \frac{2}{5}\)
\(=\frac{10+3}{5}+\frac{4}{5}+\frac{5+2}{5}\)
Here, given fractions have the same denominator, \(5\). We will keep the denominator the same for the result and add the numerator to find the numerator of the result.
\(=\frac{13}{5}+\frac{4}{5}+\frac{7}{5}\)
\(=\frac{13+4+7}{5}=\frac{24}{5}\)

Summary

In this article, we studied fractions, the different types of fractions, and the fundamental operations of fractions like addition, subtraction of like and unlike fractions. One can perform basic arithmetic operations like addition, subtraction, multiplication, and division on fractions. A fraction can be defined as a number that represents a part of the whole. A fraction is represented as \(\frac{x}{y}\) where \(x\) and \(y\) are whole numbers. Some of the examples of fractions are \(\frac{2}{3}, \frac{3}{16}, etc.

FAQs on Operations on Fractions

Q.1. What are the four operations of fractions?
Ans: The four operations of fractions are:
1. Addition
2. Subtraction
3. Multiplication
4. Division

Q.2. How will you add like fractions?
Ans: To add two or more like fractions, we may follow the following steps:
Step 1: Get the fractions.
Step 2: Add the numerators of all fractions.
Step 3: Maintain the common denominator of all fractions.
Step 4: Write down the fraction as \(\frac{\text { Result in Step 2 }}{\text { Result in Step 3. }}\).

Q.3. How to add or subtract, unlike fractions?
Ans: 1. Find the LCM of the denominators of the given unlike fractions
2. Change the denominators into the obtained LCM. This process can change the numerators of the given, unlike fractions.
3. Now, add or subtract the numerators.

Q.4. How to multiply two fractions?
Ans: The numerators and denominators of two fractions are multiplied separately when they are multiplied. The first fraction’s numerator will be multiplied by the second’s numerator, and the first fraction’s denominator will be multiplied by the second’s denominator. In the end, we will reduce the fraction to its lowest form if it is required. For example, \(\frac{2}{3} \times \frac{4}{7}=\frac{2 \times 4}{3 \times 7}=\frac{8}{21}\)

Q.5. How to divide a fraction by a fraction?
Ans: 1. We will keep the first fraction the same, and we need to determine the reciprocal of the second fraction.
2. Change the division sign by multiplication sign and multiply the first fraction with the reciprocal of the second fraction.
3. Find the simplest form of the fraction, if needed.

We hope this detailed article on operations on fractions helped you in your studies. If you have any doubts, queries or suggestions regarding this article, feel to ask us in the comment section and we will be more than happy to assist you. Happy learning!