Adding Fractions

How to Add Fractions

Fractions are added by first making them equal and then adding the numerators. Each fraction is thus first expanded, so that all the fractions to be added have the same denominator. The numerators of the fractions with the same denominator are then added, while the common denominator remains unchanged.

In the following, we will proceed step by step and show by means of examples the addition of fractions with the same denominator, then the addition of fractions with different denominators, and finally the addition of mixed fractions.

How to Add Fractions With the Same Denominator

If the fractions to be added already have the same denominator, i.e., they all have the same denominator, then only the numerators of the fractions to be added must be added. The common denominator remains unchanged. In this way, you finally get the sum of the fractions.

In this example, both fractions have the same denominator, i.e., both have the same number below the fraction line: Both fractions here represent a certain number of quarters. Therefore, they have the same denominator. To add the two fractions, only the two numbers, i.e., the two numerators above the fraction bar, have to be added.

How to Add Fractions With Different Denominators

Fractions are unequally named if the numbers below the fraction bar, i.e., the denominators of the fractions to be added, are different. For the addition of fractions, unequal-named fractions must first be made equal-named, just as with the subtraction of fractions. Once they have the same name, i.e., the same denominator, only the numerators above the fraction must be added, and the common denominator must remain.

The two fractions to be added here initially have different denominators 3 and 4. To add them, they must first be made equal. To do this, both fractions must be transformed so that they have the same denominator, i.e., a common denominator. Transforming means that the fractions are transformed in such a way that their value does not change. There are several ways of transforming fractions, which are described on the introductory page on the topic Fractions.

Make fractions have the same denominator

Expand

Expanding a fraction is a transformation in which the value of the fraction, i.e., the fraction number, is not changed. This is because the represented fraction is only divided into smaller sections, i.e., the fraction or the division is refined.

Making the same denominator using the example

Now the two fractions with the same denominator can be added, as shown in the example below:

The left fraction is expanded with the denominator 4 of the right fraction. The numerator and denominator of the left-hand fraction are therefore multiplied by 4.

The right fraction is expanded with the denominator 3 of the left fraction. The numerator and denominator of the right-hand fraction are therefore multiplied by 3.

Now the two fractions with the same name can be added, as in the example:

Note

The homonymous making described here is based on expanding the two fractions so that the two different denominators are finally multiplied by each other. However, this often leads to the fact that the values of the expanded fractions can become very large, which makes the subsequent calculations more time-consuming. Therefore, to make them equal, the smallest common denominator (main denominator) of the fractions should be determined. The main denominator is the lowest common multiple (LCM) of the denominators, which is frequently less than the multiplication of the two denominators. You can read more about the lowest common denominator under Fractions.

How to Add Mixed Fractions

Mixed fractions are composed of an integer and an ordinary fraction. They are also called mixed numbers. To add mixed fractions, the whole number is first converted into the corresponding fraction so that the two fractions can then be added together. For this purpose, as with every addition of fractions, they must be made equal if necessary in order to finally add the numerators, with the denominator remaining the same.

The integer part of the two mixed fractions, i.e., the two in each case, was here converted into 6 thirds each and added to the associated fraction. The mixed fractions were thus converted into improper fractions. Fractions are called improper fractions if the numerator is greater than the denominator.

Conversion of mixed fractions into improper fractions

Conversion using the example

The two mixed fractions from the above example are thus converted into improper fractions as follows.

The mixed number on the left is converted as follows: The integer 2 is multiplied by the denominator 3 and added to the previous numerator 1.

The mixed number on the right is converted as follows: The integer 2 is multiplied by the denominator 3 and added to the previous numerator 2.

Addition of the two fractions

Since the two transformed fractions already have the same denominator, they can now be added.

Conversion of improper fraction to mixed fraction

Finally, the improper fraction of the result is calculated back into a mixed fraction by checking how many times the denominator fits into the numerator. This is then the whole number of the mixed fraction. The remainder is noted as a fraction with the existing denominator.

Video on Adding Fractions

Last but not least, here is a video on the subject of ‘adding fractions’ by Math Antics. After an introduction, the video first explains how to add fractions with the same denominator. Later, Math Antics explains how to add fractions with different denominators.