Addition of Fractions

What are Fractions?

One day Alaina gave a chocolate bar to Cristopher. John saw this and started shouting “give me one, give me one.” Alaina did not have another one. So she asked them to share the chocolate bar equally. John ran to the kitchen and brought a knife and cut the bar into two equal pieces. He took one and gave the other to his brother Cristopher. As both the pieces were of equal size, they did not quarrel over the size of the piece they received.


The next day, John was given some sums of division by his maths teacher in the school. He solved all of them except the following: 

i) 1 ÷ 2            ii) 1 ÷ 3          iii) 2 ÷ 5


He was confused and did not know how to divide 1 by 2, 1 by 3 or 2 by 5. Can we divide 1 by 2? Yes, we can! I divided by 2 is \[\frac{1}{2}\]. It means that one part out of the two equal parts. In the same way, 1 divided by 3 will be written as \[\frac{1}{3}\] and it means that one part out of the three equal parts. 2 divided by 5 means two parts out of the five equal parts and can be written as \[\frac{2}{5}\].


The numbers such as \[\frac{1}{2}, \frac{1}{3}, \frac{2}{5}\] are called fractions. In a faction \[\frac{5}{12}\], 12 is called the denominator and it means the total number of equal into which the whole number has been divided. 5 is called a numerator and it is a number of equal parts which have been taken.


Therefore, 5/12 represents five-twelfth of the whole. Hence, we can define function as:


A Fraction is a number that represents a part of the whole.


Types of Fractions

Types of Fraction 

Definition

Example

Proper Fraction

A fraction whose numerator is greater than zero but less than its denominator is called a proper fraction. 

\[\frac{2}{3}, \frac{3}{7}, \frac{13}{128}\]

Improper Fraction

A fraction whose numerator is equal to or greater than its denominator is called an improper fraction. 

\[\frac{13}{5}, \frac{9}{7}, \frac{215}{89}\]

Mixed Fraction 

A number which consists of two parts: a natural number and a proper number is called a mixed fraction. 

\[5\frac{3}{7}, 7\frac{13}{128}\]

Like Fraction

Two or more fractions having the same denominator are called like factions.

\[\frac{3}{7}, \frac{6}{7}, \frac{11}{7}, \frac{25}{7}\]

Unlike Fraction

Two or more fractions having different denominators are called unlike fractions.

\[\frac{1}{2}, \frac{2}{5}, \frac{4}{7}, \frac{8}{9}\]


Addition of Fractions 

Now you must be wondering how to solve fractions addition? By the end of this section, we would know how to add fractions with unlike denominators and addition of like fractions. First we will start with addition of like fractions. Addition of fractions examples are also given to properly guide you.


Addition of Like Fractions

For the addition of like fractions we just need to add the numerators of the fractions as the denominations of all the fractions would be the same. Here is an example of addition of like fractions.


Adding fractions examples:


Example 1) Add \[\frac{2}{18}\] and \[\frac{12}{18}\]


Solution 1) The addition of like fractions is done in the following ways:


\[\frac{2}{18}\] + \[\frac{12}{18}\]


= \[\frac{2 + 12}{18}\]


= \[\frac{14}{18}\]


Example 2) Add \[\frac{3}{21}\] and \[\frac{7}{21}\]


Solution 2) The addition of like fractions is done in the following ways:


\[\frac{3}{21}\] and \[\frac{7}{21}\]


= \[\frac{3 + 7}{21}\]


= \[\frac{10}{21}\]


Now, I’m sure that you know how to solve fractions addition or rather I should say the addition of like fractions. Now that we know how the addition of like fractions is done, we can move onto and learn how to add fractions with unlike denominators.


How to Add Fractions With Unlike Denominators?

Adding fractions with unlike denominators examples are a little different from the addition of like fractions. Let us see how?


Adding fractions examples:


Example 1) Add \[\frac{11}{4}, \frac{35}{6}, \frac{3}{8}\]


Solution 1) Since the denominator of the fractions are different from each other, we first have to make them equal and we can do that by finding the L.C.M.  of 4, 6 and 8. The L.C.M. would be 24. Therefore,


\[\frac{11 + 35 + 3}{24} = \frac{49}{24}\]


Example 2) Add \[\frac{5}{8}\] and \[\frac{9}{12}\]


Solution 2) Again, first we have to find out the L.C.M. of 8 and 12 which will be 24. Therefore, 


\[\frac{5 + 9}{24} = \frac{14}{24}\]


Addition of Mixed Fractions

Addition of mixed fractions includes a bit more of process than addition of like fractions or addition of fractions with unlike denominator. 


Adding fractions examples:


Example 1) Add \[5\frac{4}{2}\] and \[2\frac{3}{3}\]


Solution 1) For the addition of mixed fraction we need to follow few steps, these are:


Step 1: Multiply the denominator with the whole number then add the result with the numerator.

2 x 5 = 10 + 4 = 14                                      3 x 2 = 6 + 3 = 9


Step 2; The final outcome will become the numerator and the denominator will remain the same.

\[\frac{14}{2}\]                                              \[\frac{9}{3}\]

                      

Step 3: Find the L.C.M. if the denominators are different. So the L.C.M. of 2 and 3 will be 6, therefore, 


\[\frac{14 + 9}{6} = \frac{23}{6}\]

FAQ (Frequently Asked Questions)

Question 1) What is the Use of Fraction?

Answer 1) The word ‘Fraction’ is derived from a Latin term ‘Fractus’ which means to break. It is from here that words like ‘fracture’, ‘fragment’, ‘fractal’, ‘diffract’ etc are generated. Division can be represented in two ways that is fractions or decimals. We use fraction to represent one or more equal parts of something. Now, for example if there are 5 pastries and you have eaten 3 of them. Then we use fraction to represent the total number of pastries and the number of pastries you have eaten. Therefore, ⅗ is how we will represent it. We can use fractions in our day to day life for various uses.

Question 2) What are the Basic Properties of Fractions?

Answer 2) Few basic properties of fractions are:

  1. The value of a fraction does not change if the numerator and the denominator of a fraction are multiplied by the same (non-zero) number.

  2. The value of a fraction does not change if the numerator and the denominator of a fraction are divided by the same (non-zero) number.

  3. Dividing the numerator and denominator of a fraction by the same number is usually called cancelling.