Fractions: The word fraction is derived from ‘fractio’, a Latin word, which means ‘to break’. A fraction is a quantity used to express a part of the whole. When we divide a whole into parts, each part obtained is called a fraction. There are mainly two parts in a fraction, i.e. numerator and denominator.

There are different types of fractions: like, unlike, proper, improper, and mixed fractions, etc. We use the basic operations on fractions such as addition, subtraction, multiplication, and division in real life. In this article, we will learn everything about fractions.

Definition of Fractions

A fraction is a quantity used to express a part of the whole. When we divide a whole into parts, each part obtained is called a fraction.

Example:

When we cut an apple into two parts, then each part represents the fraction \(\left({\frac{1}{2}} \right)\) of whole.

There are mainly two parts in the fraction, and they are the numerator and the denominator. The topmost part of the fraction is known as the numerator, and the bottommost part of the fraction is called the denominator.

In the fraction: \(\frac{x}{y}\)
1. Numerator – The part which has taken \({\left( x \right) }\)
2. Denominator – Total parts of the whole \(\left( y \right)\)

Practice Exam Questions

Example:

\(\frac{5}{8}\) is the fraction, which is spelt as “five eights”
Here, \(5 – \)numerator and \(8 – \) denominator.

Types of Fractions

There are different types of fractions based on the relative values of the numerators and the denominators. Those are proper fractions, improper fractions, and mixed fractions.

Proper Fractions

A fraction is said to be a proper fraction when its value is less than one. In a proper fraction, the value of the numerator always has a lesser value as compared to the value of the denominator.

The condition of the proper fraction is \({\text{numerator}} < {\text{denominator}}.\)

Improper Fractions

A fraction is an improper fraction when its value is greater than or equal to one. In an improper fraction, the value of the numerator always has a larger value or equal as compared to the value of the denominator.

The condition of the improper fraction is the \({\text{numerator}} \geqslant {\text{denominator}}.\)

Mixed Fractions

A mixed fraction is a fraction combination of whole and part. The value of a mixed fraction is always greater than one. It has a natural number along with the fraction as shown below:

Unit Fractions

A fraction which has a numerator as one \(\left(1 \right)\) is called the unit fraction.

Like and Unlike Fractions

Fractions are said to be like fractions if they have the same denominator.

Example:

\(\frac{1}{4},\frac{2}{4},\frac{3}{4}\) all are having the same denominator \(4.\) So, they are known as like fractions.

Fractions are said to be unlike if they have different denominators.

Example:

\(\frac{1}{2},\frac{2}{3},\frac{3}{4}\) all are having different denominators. So, they are known as, unlike fractions.

Equivalent Fractions

Two or more fractions representing the equal part of a whole are called equivalent fractions. Multiplying or dividing the numerator and denominator with the same value does not alter the fraction.

The above figure gives examples of equivalent fractions. Here, all fractions show half of the shaded portion of the whole. So, they are known as equivalent fractions.

To find an equivalent fraction to a given fraction, we divide or multiply both the numerator and the denominator of the given fraction by the same non-zero number.

Simplest Form of Fractions

The simplest form of the fraction is the fraction that has no common factor between the numerator and denominator except one \(\left( 1 \right).\)
A fraction with H.C.F of numerator and denominator equal to one \(\left( 1 \right).\) said to be in the simplest form.
A fraction \(\frac{a}{b}\) is said to be in the simplest form if the H.C.F of \(a,b\) is one.

Example:

The simplest form of the fraction \(\frac{8}{{24}}\) is obtained by dividing both numerator and denominator by \(2\) consecutively as shown below:

Addition and Subtraction of Fractions

Addition and subtraction are two fundamental operations that are also applicable to fractions. Before we discuss the addition and subtraction of fractions, we must understand whether the fractions have the same denominators or different denominators because we follow different steps in each case.

Addition and Subtraction of Like Fractions

Like fractions have the same denominator. To add or subtract the like fractions, just add or subtract the values of the numerator, keeping the same denominator as it is.

Example: \(\frac{3}{7} + \frac{5}{7}\)

Here, given fractions have the same denominator \(7,\) so just add the numerators and keep the same denominator.
\(\frac{3}{7} + \frac{5}{7} = \frac{{3 + 5}}{7} = \frac{8}{7}\)

Addition and Subtraction of Unlike Fractions

Unlike fractions have different denominators. To add or subtract the unlike fractions, follow the steps mentioned below:

1. Obtain the L.C.M. of the denominators of the given fractions
2. Change the fractions with denominator as L.C.M.
3. Now, add or subtract the numerators.

Example:\(\frac{2}{5} + \frac{1}{3}\)
L.C.M of\(\left({3,5} \right)\) is \(15.\)
\( \Rightarrow \frac{2}{5} \times \frac{3}{3} + \frac{1}{3} \times \frac{5}{5} = \frac{6}{{15}} + \frac{5}{{15}} = \frac{{6 + 5}}{{15}} = \frac{{11}}{{15}}\)

Multiplication of Fractions

To multiply given fractions, we need to multiply the numerator and denominators separately. The numerator of the first is multiplied with the numerator of the second, and so on. Similarly, the denominator of the first is multiplied by the denominator of the second, and so on. And finally, reduce the obtained fraction in the simplest form, if required. Let’s understand multiplication of fractions with an example.

Example:

The product of \(\frac{2}{3}\) and \(\frac{5}{7}\) is found by multiplying the numerators and denominators separately.
\( \Rightarrow \frac{2}{3} \times \frac{5}{7} = \frac{{10}}{{21}}\)

Division of Fractions

Dividing the fraction by another fraction means multiplying the first fraction with the reciprocal of the second fraction. Let’s learn the concept of division of fractions in detail.

For dividing a fraction by another fraction, follow the below steps:

1. Calculate the reciprocal of the second fraction. The reciprocal of the fraction is obtained by interchanging the denominator and numerator of the fraction.
2. Multiply the first fraction with the reciprocal of the second fraction. 
3. Find the simplest form of the fraction, if needed.

Example:

\(\frac{2}{7} \div \frac{3}{5}\)
Here, the reciprocal of the second fraction \(\left({\frac{3}{5}} \right)\) is obtained by interchanging denominator and numerator and is given by \(\left({\frac{5}{3.}} \right)\)
So, \(\frac{2}{7} \div \frac{3}{5}\) can be written as \(\frac{2}{7} \times \frac{5}{3} = \frac{{10}}{{21}}.\)

Solved Examples of Fractions

Q.1. Express the following mixed fraction to an improper fraction:
(a) \(2\frac{3}{4}\)
(b) \(7\frac{1}{9.}\)
Ans: Given mixed fractions are \(2\frac{3}{4}\) and \(7\frac{1}{9.}\)
To convert the mixed fractions to an improper fraction, multiply the denominator with the whole and add to the numerator keeping the same denominator.
Mixed fractions are converted to an improper fraction as follows:
\(\frac{{{\text{ }}\left({{\text{whole }} \times {\text{ denominator}}} \right) + {\text{ numerator }})}}{{{\text{ denominator }}}}\)
\( \Rightarrow \left( a \right)2\frac{3}{4} = \frac{{\left({2 \times 4} \right) + 3}}{4} = \frac{{8 + 3}}{4} = \frac{{11}}{4}\)
\( \Rightarrow \left( b \right)7\frac{1}{9} = \frac{{\left({7 \times 9} \right) + 1}}{9} = \frac{{63 + 1}}{9} = \frac{{64}}{9}\)

Q.2. To make a cake, \(1\frac{1}{2}\) cups of sugar are needed. How many cups of sugar is needed for making \(6\) cakes?
Ans: Given, to make a cake, \(1\frac{1}{2}\) cups of sugar is needed.
The number of cups of sugar needed to make \(6\) cakes is obtained by multiplying the number of cakes with the sugar needed for one cake.
\( = 1\frac{1}{2} \times 6\)
Convert the above-mixed fraction to an improper fraction by multiply the denominator with the whole and add to the numerator keeping the same denominator.
Mixed fractions are converted to an improper fraction as follows:
\(\frac{{{\text{ }}\left({{\text{whole }} \times {\text{ denominator}}} \right) + {\text{ numerator }})}}{{{\text{ denominator }}}}\)
\( = 1\frac{1}{2} = \frac{{\left( {1 \times 2} \right) + 1}}{2} = \frac{3}{2}\)
The amount of sugar required for making \(6\) cakes is given by
\( = \frac{3}{2} \times 6 = \frac{{18}}{2} = 9\)
Hence, to make the \(6\) cakes, we require \(9\) cups of sugar.

Q.3. An oil container contains \(7\frac{1}{2}\) litres of oil which are poured into \(2\frac{1}{2}\) litres bottles. How many bottles are needed to fill \(7\frac{1}{2}\) litres of oil?
Ans: Given, the total amount of oil in the container is \(7\frac{1}{2}\) litres.
The amount of oil each bottle holds is \(2\frac{1}{2}\) litres.
Let the total number of bottles required to be \(x.\)
According to the question, the total amount of oil in the container \( = \) Amount of oil in each bottle\( \times \)number of bottles.
\( \Rightarrow 7\frac{1}{2} = x \times 2\frac{1}{2}\)
\( \Rightarrow \frac{{15}}{2} = x \times \frac{5}{2}\)
\( \Rightarrow 15 = 5x\)
\( \Rightarrow x = \frac{{15}}{5} = 3\)
Hence, the total number of bottles required to fill \(7\frac{1}{2}\) litres of oil is \(3.\)

Q.4. Simplify: \(\frac{1}{5} \div \left({\frac{2}{7} \div \frac{3}{{14}}} \right).\)
Ans: Given \(\frac{1}{5} \div \left({\frac{2}{7} \div \frac{3}{{14}}} \right).\)
Dividing the fraction by another fraction means multiplying the first fraction with the reciprocal of the second fraction.
\( = \frac{1}{5} \div \left({\frac{2}{7} \times \frac{{14}}{3}} \right)\)
\( = \frac{1}{5} \div \frac{4}{3}\)
\( = \frac{1}{5} \times \frac{3}{4}\)
\( = \frac{3}{{20}}\)
Hence, the value of \(\frac{1}{5} \div \left({\frac{2}{7} \div \frac{3}{{14}}} \right)\) is \(\frac{3}{{20}}.\)

Q.5. Write the equivalent fraction to \(\frac{{36}}{{63}}\) with \(7\) as the denominator.
Ans: Given fraction is \(\frac{{36}}{{63}}\)
The equivalent fraction is obtained by dividing or multiplying the given fraction with a common factor.
Divide the numerator and denominator of the fraction by \(9.\)
\( = \frac{{\frac{{36}}{9}}}{{\frac{{63}}{9}}}\)
\( = \frac{4}{7}\)
The equivalent fraction of \(\frac{{36}}{{63}}\) is\( = \frac{4}{7.}\)

Summary

In this article, we have studied fractions and the definition of fractions, which tells the part of the whole area. We have also studied the different fractions, such as proper fractions, improper fractions and mixed fractions, and unit fractions.

This article also discussed equivalent fractions and the simplest form of fractions. In this article, we have discussed the basic operations of the fractions, such as addition and subtraction of fractions, multiplication of fractions and division of fractions using the solved examples.

FAQs on Fractions

Q.1. What are fractions? Explain with an example.
Ans: A fraction is a quantity that is used to express a part of the whole. When we divide a whole into parts, then each part is called a fraction.
Example: \(\frac{2}{3}\)

Q.2. What is numerator and denominator?
Ans: A fraction is a quantity that is used to express a part of the whole. The topmost part of the fraction is known as the numerator, and the bottommost part of the fraction is known as the denominator.
Example: \(\frac{a}{b}\)
1. Numerator – The part which has taken \(\left( a \right)\)
2. Denominator – Total parts of the whole \(\left( b \right)\)

Q.3. What are the types of fractions?
Ans: There are different types of fractions based on the value of the numerators and the denominators. They are:
1. Proper fractions
2. Improper fractions
3. Mixed fractions

Q.4. What are mixed fractions?
Ans: A mixed fraction is a fraction combination of whole and part. The value of a mixed fraction is always greater than one.
Example:\(2\frac{1}{3}\)

Q.5. What are equivalent fractions?
Ans: Two or more fractions representing the equal part of a whole are called equivalent fractions. Multiplying or dividing the numerator and denominator with the same value does not alter the fraction.
Example: \(\frac{2}{3} = \frac{4}{6} = \frac{8}{{12}} = \frac{{16}}{{24}}\)

So now we have reached the end of the article, and we hope the information provided to you on Fractions was helpful and answered all your queries. However, if you have further doubts, feel to use the comments section, and we will update you.