Addition and Subtraction of Fractions: A fraction is a quantity that expresses a part of the whole. We use the addition and subtraction of fractions in our day-to-day life without even realising it. For example, dividing the amount of a bill among friends when we all visited a restaurant, dividing a pizza slice between our family members, scores of our exams are generally expressed in terms of fractions. 

The applications of fractions in everyday life are endless. They are of great importance and profoundly impact everything around us. In this article, we will provide detailed information on fractions and we will also learn how to add and subtract fractions. Scroll down to continue reading!

Definition of Fractions

If a certain quantity of rice is divided into four equal parts, each part so obtained is said to be one-fourth \(\left( {\frac{1}{4}} \right)\) of the whole quantity of the rice.

Similarly, if orange is divided into five equal parts, each part is one-fifth \(\left( {\frac{1}{5}} \right)\) of the whole orange. Now, if two parts of these five equal parts are eaten, three parts are left, and we say three-fifths \(\left( {\frac{3}{5}} \right)\) of the orange is left.

The numbers \(\left( {\frac{1}{4}} \right),\left( {\frac{1}{5}} \right),\left( {\frac{3}{5}} \right)\) discussed above, each represents a part of the whole quantity, which are called fractions. A fraction is a quantity that expresses a part of the whole.

Let us make the concept of fractions more clear:

Draw a circle with any suitable radius. Then, divide the circle into three equal parts (sectors).

Now, let’s have the look at the above-mentioned figure again:

If two parts of the three equal parts is shaded, we say two-thirds \(\left( {\frac{2}{3}} \right)\) of the circle is shaded and one-third \(\left( {\frac{1}{3}} \right)\) of the circle is not.

So, we can say that a fraction is a quantity that expresses a part of the whole. In a fraction \(\frac{a}{b},a\) is the numerator of the fraction and, \(b\) is the denominator of the fraction.

\({\rm{Fraction = }}\frac{{{\rm{ Numerator }}}}{{{\rm{ Denominator }}}}\)

Types of Fractions

Fraction can be classified into several types based on certain parameters. Below we have provided the types of fractions and their examples for your reference:

1. Proper Fraction: A fraction in which the numerator is less than the denominator is called a proper fraction. 
For example \(\frac{1}{3},\frac{7}{9},\frac{{13}}{{25}}\), etc., are proper fractions.

2. Improper Fraction: A fraction in which the numerator is greater than or equal to its denominator is called an improper fraction. 
For example \(\frac{8}{3},\frac{{17}}{9},\frac{{13}}{4}\) etc., are improper fractions.

3. Mixed Fraction: A mixture of a whole number and a proper fraction is called a mixed fraction or a mixed number. 
For example \(1\frac{1}{3},4\frac{7}{9},7\frac{{13}}{{25}}\), etc., are mixed fractions.

4. Unit Fraction: A proper fraction having \(1\) as a numerator and denominator as a positive integer is called a unit fraction. The unit function is nothing but the reciprocal of a positive integer.
For example \(\frac{1}{3},\frac{1}{9},\frac{1}{{25}}\) etc., are unit fractions.

5. Like and Unlike Fractions: Fractions with the same denominator are called like fractions. For example, \(\frac{1}{7},\frac{5}{7},\frac{6}{7}\) etc., are like fractions. 
Fractions with different denominators are called, unlike fractions. 
For example \(\frac{1}{7},\frac{5}{{71}},\frac{6}{{25}}\) etc., are unlike fractions.

6. Equivalent Fractions: If two or more fractions have the same value, they are called equivalent or equal fractions. 
For example \(\frac{1}{3},\frac{3}{9},\frac{6}{{18}}\)  and \(\frac{9}{{27}}\) are equivalent fractions as \(\frac{1}{3} = \frac{3}{9} = \frac{6}{{18}} = \frac{9}{{27}}\)

Reducing a Fraction to its Lowest Terms

A fraction is said to be in its lowest terms if its numerator and denominator have no common factor other than \(1\), or we can say that the numerator and denominator are co-prime.

To reduce a fraction into its lowest terms:

1. Find the HCF of its numerator and denominator.
2. Divide the numerator and denominator of the fraction by the HCF obtained in the previous step.

Let us understand this with the help of an example.

Consider the fraction \(\frac{8}{{10}}\). As the HCF of \(8\) and \(10\) is \(2\), divide both numerator and denominator by \(2\).

Thus, \(\frac{8}{{10}} = \frac{{8 \div 2}}{{10 \div 2}} = \frac{4}{5},\) which is the fraction in its lowest terms.

Addition and Subtraction of Fractions

Addition and subtraction are amongst two basic arithmetic operations that are also applicable to fractions. However, before adding or subtracting fractions, we must understand whether the fractions have the same or different denominators since we follow different steps in each case. Therefore, the association of these concepts while solving problems based on the addition and subtraction of fractions is an important criterion to obtain the desired answer.

Depending on the type of fractions, the following methods are available to add or subtract fractions.

1. Addition and Subtraction of Like Fractions

As we have already discussed the definition of like fractions, so it becomes easy to add and subtract like fractions, that is, the fractions with the same denominator. Just add or subtract the numerators and keep the same denominator.

Let us understand the addition and subtraction of like fractions with the help of the examples given below.

(i). Add \(\frac{5}{7}\) and \(\frac{3}{7}\)

a. Add only the numerators,

b. Keep the denominator as it is.

\(\frac{5}{7} + \frac{3}{7} = \frac{{5 + 3}}{7} = \frac{8}{7}\)

(ii). Subtract \(\frac{9}{{17}}\) and \(\frac{3}{{17}}\)

a. Subtract only the numerator of the subtrahend,

b. Keep the denominator as such.

\(\frac{9}{{17}} – \frac{3}{{17}} = \frac{{9 – 3}}{{17}} = \frac{6}{{17}}\)

2. Addition and Subtraction of Unlike Fractions

For adding and subtracting unlike fractions, follow these steps.

a. Find the LCM of the denominators.

b. Convert the fractions into equivalent fractions with this LCM as the common denominator.

c. Now, add or subtract the equivalent fractions.

Let us understand the addition and subtraction of unlike fractions with the help of the examples given below.

1. Add \(\frac{5}{4}\) and \(\frac{1}{6}\)

 Find the LCM of the denominators \(4\) and \(6\).

LCM of \(4\) and \(6 = 12\)

Now, \(\frac{5}{4} = \frac{{5 \times 3}}{{4 \times 3}} = \frac{{15}}{{12}}\) and \(\frac{1}{6} = \frac{{1 \times 2}}{{6 \times 2}} = \frac{2}{{12}}\)

So, \(\frac{{15}}{{12}} + \frac{2}{{12}} = \frac{{17}}{{12}}\)

2. Subtract \(\frac{2}{3}\) from \(\frac{4}{5}\)

Find the LCM of the denominators \(3\) and \(5\)

Since, both \(3\) and \(5\) are prime numbers; their LCM will be:

LCM \( = 3 \times 5 = 15\)

Now, \(\frac{2}{3} = \frac{{2 \times 5}}{{3 \times 5}} = \frac{{10}}{{15}}\) and \(\frac{4}{5} = \frac{{4 \times 3}}{{5 \times 3}} = \frac{{12}}{{15}}\)

So, \(\frac{{12}}{{15}} – \frac{{10}}{{15}} = \frac{2}{{15}}\)

3. Addition and Subtraction of Mixed Fractions

We have provided the steps one needs to follow in order to add and subtract the mixed fractions below:

1. Convert the mixed fraction into an improper fraction.
2. Convert the improper fraction obtained into like fractions.
3. Keep the denominator the same, add or subtract the numerators of the equivalent fractions and obtain a single fraction.
4. Reduce the fraction into its lowest terms, if required, and then convert it again into a mixed fraction.

Solve \(1\frac{3}{7} – \frac{1}{6}\)

\( = \frac{{10}}{7} – \frac{1}{6}\)

\( = \frac{{10 \times 6}}{{7 \times 6}} – \frac{{1 \times 7}}{{6 \times 7}} = \frac{{60}}{{42}} – \frac{7}{{42}}\)

\( = \frac{{60 – 7}}{{42}} = \frac{{53}}{{42}} = 1\frac{{11}}{{42}}\)

Properties of Addition and Subtraction of Fractions

Some of the properties of addition and subtraction of fractions are as follows:

1. The sum and difference of \(0\) and a fraction is the fraction itself.
2. The change in the order of addends does not affect the sum.
3. The change in the order of minuend and subtrahend affects the difference.
4. Even after changing the grouping of addends, the sum remains the same.
5. When the grouping of numbers is changed while subtracting, the difference will change.

Applications of Fractions in Real Life

Fractions surround our everyday activities. So let us look at some of the examples of fractions in real life:

1. Think about a time you go to a restaurant with colleagues and the waiter brings a single bill. To divide the total amongst the colleagues, you use fractions.
2. Fractions are used in calculating probability. They are used to attribute a numerical value to the likelihood of the occurrence of an event.
3. Fractions are frequently used to analyse the performance of a particular player and team.
4. We use fractions to understand our body mass index to determine whether we are in a healthy range of body mass or not.
5. Different fractions of liquids are mixed in the right amounts to get the best outcome to make drinks like mocktails.
6. The shutter speed of a camera is calculated using fractions.
7. Scores of tests and exams are generally expressed as fractions.

Solved Examples on Addition and Subtraction of Fractions

Q.1. Add \(\frac{{20}}{{30}} + \frac{9}{{30}} \cdot \)
Ans: Given \(\frac{{20}}{{30}} + \frac{9}{{30}}\)
Since both are like fractions,
\(\frac{{20}}{{30}} + \frac{9}{{30}} = \frac{{20 + 9}}{{30}}\)
\( = \frac{{29}}{{30}}.\)

Q.2. Simplify \(5 – \left( {\frac{8}{{11}} – \frac{6}{{11}}} \right) \cdot \)
Ans: Given \(5 – \left( {\frac{8}{{11}} – \frac{6}{{11}}} \right) \cdot \)
As \(\frac{8}{{11}}\& \frac{6}{{11}}\) are like fractions, we can write
\(5 – \left( {\frac{8}{{11}} – \frac{6}{{11}}} \right) = 5 – \frac{{8 – 6}}{{11}}\)
\( = 5 – \frac{2}{{11}}\)
\( = \frac{{5 \times 11}}{{1 \times 11}} – \frac{2}{{11}}\)
\( = \frac{{55}}{{11}} – \frac{2}{{11}}\)
\( = \frac{{55 – 2}}{{11}} = \frac{{53}}{{11}}\)

Q.3. From a rope \({\rm{10}}\frac{{\rm{1}}}{{\rm{2}}}{\rm{m}}\) long \({\rm{5}}\frac{{\rm{5}}}{{\rm{8}}}{\rm{m}}\) is cut off. Find the length of the remaining rope.
Ans. Total length of the rope \({\rm{10}}\frac{{\rm{1}}}{{\rm{2}}}{\rm{m}}\) From the total length \({\rm{5}}\frac{{\rm{5}}}{{\rm{8}}}{\rm{m}}\) cut off.
Length of the remaining rope \( = 10\frac{1}{2} – 5\frac{5}{8} = \frac{{21}}{2} – \frac{{45}}{8}\)
\( = \frac{{21 \times 4}}{{2 \times 4}} – \frac{{45 \times 1}}{{8 \times 1}} = \frac{{84}}{8} – \frac{{45}}{8}\)
\( = \frac{{84 – 45}}{8} = \frac{{39}}{8}\)
\({\rm{ = 4}}\frac{{\rm{7}}}{{\rm{8}}}{\rm{m}}\)

Q.4. Solve \(3\frac{8}{9} + 8\frac{3}{4}.\)
Ans: \(3\frac{8}{9} + 8\frac{3}{4} = \frac{{35}}{9} + \frac{{35}}{4}\)
LCM of \(4\) and \(9 = 36\)
\(\frac{{35 \times 4}}{{9 \times 4}} + \frac{{35 \times 9}}{{4 \times 9}} = \frac{{140}}{{36}} + \frac{{315}}{{36}}\)
\( = \frac{{140 + 315}}{{36}} = \frac{{455}}{{36}} = 12\frac{{23}}{{36}}.\)

Q.5. Manu spends \(\frac{2}{5}\) of his money and is left with \(₹30\). How much money did he initially have?
Ans: Manu spends \(\frac{2}{5}\) of his money.
Therefore, money left with him \( = \left( {1 – \frac{2}{5}} \right)\) of his money \(= \frac{3}{5}\) of his money.
Given, \(= \frac{3}{5}\) of his money \(=₹30\)
Initially, he had \(₹30 \times \frac{5}{3} = ₹50\)

Summary

A fraction is a quantity that expresses a part of the whole. Fractions can be divided into 4 types namely, proper fraction, improper fraction, mixed fraction, and unit fraction. Fractions are often utilised to analyse the performance of a particular player and team. While adding or subtracting fractions one must keep in mind if the fractions have the same or different denominators.

We also learned that for fractions with the same denominator, we subtract the numerators and for fractions with different denominators we take the LCM of the denominators. We then convert the given fractions to the same denominators and subtract them as like fractions.

FAQs on Addition and Subtraction of Fractions

Q.1. What are the steps of adding and subtracting fractions?
Ans: The first thing is to check and see if the fractions have the same denominator. If they don’t have the same denominator, then convert them to equivalent fractions with the same denominator with the help of LCM. Once they have the same denominator, add or subtract the numerators and finally write the answer with the new numerator over the denominator.

Q.2. How do you solve fraction addition problems?
Ans: Firstly, make sure the denominators are the same. Then, add the numerators and put the answer over the denominator. Reduce the fraction into its lowest terms, if required.

Q.3. How do you add and subtract fractions with different denominators?
Ans: To add or subtract fractions with different denominators, follow these steps.
1. Find the LCM of the denominators.
2. Convert the fractions into equivalent fractions with LCM as the common denominator.
3. Now, add or subtract the equivalent fractions.

Q.4. Why do you need a common denominator to add or subtract fractions?
Ans: If the denominators are the same, then it says that the whole portion is the same for both the fractions, and only the parts of the whole are differing. When the whole portion is the same, it is easy to compare, add or subtract the parts. That is why we need a common denominator.

Q.5. How do you add and subtract fractions examples?
Ans: Add \(\frac{{51}}{{73}}\) and \(\frac{{31}}{{73}}\)
1. Add only the numerators
2. Keep the denominator as such
\(\frac{{51}}{{73}} + \frac{{31}}{7} = \frac{{51 + 31}}{{73}} = \frac{{82}}{{73}}\)
Subtract \(\frac{5}{{27}}\) and \(\frac{3}{{27}}\)
1. Subtract only the numerator of the subtrahend,
2. Keep the denominator as such.
\(\frac{5}{{27}} – \frac{3}{{27}} = \frac{{5 – 3}}{{27}} = \frac{2}{{27}}\)

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