Adding Fractions

The addition of fractions is a little different from the normal addition of numbers since a fraction has a numerator and a denominator which is separated by a bar. The addition of fractions can be easily done if the denominators are equal. While like fractions have common denominators, unlike fractions are converted to like fractions to make addition easier. Let us explore more about adding fractions, and how to add two fractions in this article.

Fractions are part of a whole. Before moving to the addition of fractions, let us quickly revise what are fractions. Fractions are made up of two parts, the numerator and the denominator. A general representation of a fraction is a/b, where 'a' is the numerator, 'b' is the denominator, and 'b' cannot be zero. For example, 2/3, 14/5, 6/7, 28/9, and 21/43. Just like other numbers, we can perform the arithmetic operations of addition, subtraction, multiplication, and division on fractions. The addition of fractions means finding the sum of two or more fractions. Now, let us learn the basic steps of the addition of fractions with the help of the following example.

Example: Add 1/4 + 2/4

Solution: Let us add these fractions using the following steps.

• Step 1: Check if the denominators are the same. (Here, the denominators are the same, so we move to the next step)
• Step 2: Add the numerators and place the sum over the common denominator. This means (1 + 2)/4 = 3/4
• Step 3: Simplify the fraction to its lowest form, if needed. Here, it is not needed. So, the sum of the given fractions is, 1/4 + 2/4 = 3/4

There are different types of fractions in Mathematics. While adding fractions we need to check whether they are like fractions or unlike fractions. Like fractions are a group of fractions with a common denominator, while unlike fractions are a group of fractions having different denominators. While learning about the addition of fractions, we might come across the following scenarios.

• Addition of fractions with same denominators: 3/4 + 1/4
• Addition of fractions with different denominators: 3/5 + 1/2
• Addition of fractions with whole numbers: 1/2 + 2
• Adding fractions with variables: 3/5y + 1/4y

Now, let us learn more about the above cases in detail.

Adding fractions with the same denominators is done by writing the sum of the numerators over the common denominator. Let us understand how to add fractions with the same denominator with the help of an example.

Example: Add the fractions 2/4 + 1/4

Solution: We can see that the denominators of the given fractions are the same. These fractions are called like fractions.

The addition of like fractions can be done by adding the numerators of the given fractions and retaining the common denominator. In this case, we keep the denominator as 4, and we add the numerators. This can be expressed as 2/4 + 1/4 = (2 +1)/4 = 3/4. This gives the sum as 3/4.

We just learned how to add fractions with like denominators. Now let us understand how to do the addition of fractions with different or unlike denominators. When the denominators are different, the fractions are called unlike fractions. In such fractions, the first step is to convert them to like fractions so that the denominators become common. This is done by finding the Least Common Multiple (LCM) of the denominators. Let us understand this with the help of the following example.

Example: Add the fractions 1/3 and 3/5.

Solution: We will use the following steps to add these fractions.

• Step 1: Since the denominators in the given fractions are different, we find the LCM of 3 and 5 to make them the same. LCM of 3 and 5 = 15.
• Step 2: Now, multiply 1/3 with 5/5, (1/3) × (5/5) = 5/15, and 3/5 with 3/3, (3/5) × (3/3) = 9/15, which will convert them to like fractions with the same denominators.
• Step 3: Now, the denominators are the same, so we simply add the numerators and write the sum over the common denominator. The new fractions with common denominators are 5/15 and 9/15. So, 5/15 + 9/15 = (5 + 9)/15 = 14/15.

An easy way to add a whole number and a proper fraction is to combine and express them as a mixed fraction. For example, 5 + 1/2 can be combined and expressed as 5½ = 11/2. Similarly, 3 + 1/7 = \(3\frac{1}{7} \) = 22/7. However, there is another method for adding fractions with whole numbers. Let us understand that with the help of the following example.

Example: Add 3 + 4/5

Solution: Let us add these numbers using the following steps:

• Step 1: In this method, we change the whole number to its fraction form by writing 1 as its denominator. Here, 3 is the whole number and this can be written as 3/1
• Step 2: Now, 3/1 can be added to 4/5, that is, 3/1 + 4/5. We will add these by making the denominators the same because they are unlike fractions. This implies, (3/1) + (4/5) = (3/1) × (5/5) + (4/5) × (1/1) = 15/5 + 4/5 = 19/5 = \(3\frac{4}{5} \)

Now that we have seen the addition of fractions with like and unlike fractions, we can extend the same concept for adding fractions with variables. Let us understand this with the help of the following example.

Example: Add y/5 + 2y/5 where 'y' is the variable.

Solution: Let us add these fractions using the following steps:

• Step 1: The given fractions, y/5 + 2y/5 are like fractions since they have the same denominator and we can see that 'y' is common.
• Step 2: We can take the common factor out and rewrite it as: y/5 + 2y/5 = (1/5 + 2/5)y = 3y/5
• Step 3:Therefore, the sum of y/5 + 2y/5 = 3y/5

Now, let us learn how to add unlike fractions using the following example.

Example: Add y/2 + y/3

Solution: Let us add the fractions using the following steps.

• Step 1: Since the given fractions, y/2 + y/3 are unlike fractions, we will take the LCM of the denominators and convert them into like fractions.
• Step 4: Next, we need to take the common variable out and rewrite it as follows: LCM (2, 3) = 6; y/2 = (y/2) × (3/3) = 3y/6 and y/3 = (y/3 × (2/2) = 2y/6
• Step 5: We got two fractions with common denominators, (3y/6) + (2y/6) = (3y + 2y)/6 = 5y/6. Therefore, the sum of y/2 + y/3 = 5y/6

It should be noted that in some cases, when we have different variables, like 'x' and 'y', they are treated as unlike terms and cannot be simplified further, for example, x/2 + y/3

Tips and Tricks on Addition of Fractions

The following points are helpful and should be remembered while working with the addition of fractions:

• For unlike fractions, we do not add the numerators and denominators directly. 1/5 + 2/3 ≠ 3/8
• To add unlike fractions, first, convert the given fractions to like fractions by taking the LCM of the denominators.
• Add the numerators and retain the common denominator to get the sum of the fractions.

☛ Related Topics

• Subtraction of Fractions
• Multiplying Fractions
• Division of Fractions
• Like Fractions and Unlike Fractions
• Adding Fractions Calculator

How to Add Fractions?

The process of addition of fractions is a little different from normal addition of whole numbers. The first step in adding fractions is to check if the denominators of the given fractions are the same. Then we use the following procedure to add them.

• If the fractions have common denominators then we can easily add the numerators and keep the same denominator to get the sum. For example, 2/4 + 1/4 = (2 + 1)/4 = 3/4
• If the denominators are different, we make the denominators equal by converting them to equivalent fractions by finding the LCM of the denominators. Then the addition can be done. For example, 1/2 + 2/3 = (1/2 × 3/3) + (2/3 × 2/2) = 3/6 + 4/6 = (3 + 4)/6 = 7/6 = \(1 \dfrac{1}{6}\)

What is the Rule for Adding Fractions?

The basic rule for the addition of fractions is to make the denominators of the fractions the same. If the fractions have the same denominator we can simply add the numerators keeping the same denominator. However, if the denominators are different, we need to convert them to like fractions with the same denominators. This is done by writing their equivalent fractions by taking the LCM of the denominators. Once they are converted to like fractions, the fractions can be added easily because we just need to work with the numerators while we keep the same denominator.

How to Add Fractions with Whole Numbers?

To add a fraction with a whole number, we first convert the whole number into a fraction. For example, if we need to add 3 and 1/2, the whole number 3 can be easily converted into a fraction like 3/1 and added to the other fraction. Let us see how this works. (3/1) + (1/2) = (3/1) × (2/2) + (1/2) = 6/2 + 1/2 = 7/2 = 3½. Another way to add fractions and whole numbers is to simply combine and express them as mixed fractions. For example, 6 + 1/2 can be combined and written as \(6 \dfrac{1}{2}\)

How to Add Fractions with Different Denominators?

The fractions with different denominators can be added by making the denominators common. This is done by multiplying the numerator and denominator of each of the fractions with a suitable number such that all the fractions become like fractions. To add the fractions 3/5 + 4/3, we need to multiply both the fractions with a number that makes the denominators equal. For this, we need the LCM of the denominators, which is 15 in this case. The numerator and denominator of the first fraction 3/5 have to be multiplied by 3, and the numerator and denominator of the second fraction 4/3 have to be multiplied by 5. Hence, we have (3/5 × 3/3) + (4/3 × 5/5) = (9/15) + (20/15) = (9 + 20)/15 = 29/15 = \(1 \dfrac{14}{15}\)

How to Add 3 Fractions with Different Denominators?

The addition of three fractions is the same as the addition of two fractions with different denominators. First of all, we need the LCM of all three denominators. Accordingly, the denominators of all the three fractions are made common by multiplying the numerator and denominator of each of the fractions with a suitable number so that they are converted to like fractions. Now, once the denominators are common, the numerators are added to get the sum of the fraction. Let us understand this with the help of this addition problem: 2/3 + 4/5 + 1/6. The LCM of 3, 5, and 6 is 30. Now, we will multiply each fraction with the suitable number to make their denominators common: (2/3 × 10/10) + (4/5 × 6/6) + (1/6 × 5/5) = (20/30) + (24/30) + (5/30) = (20 + 24 + 5)/30 = 49/30 = \(1 \dfrac{19}{30}\)

What is the Identity Element For the Addition of Fractions?

The identity element for addition is 0, which means, for any real number 'a', a + 0 = a. Similarly, for the addition of fractions, the identity element is 0. For a fraction of the form a/b, we have a/b + 0 = 0 + a/b = a/b. The use of the identity element for addition does not change the value of the fraction.

What is Subtraction and Addition of Fractions?

In the subtraction and addition of fractions, first, the denominators of the fractions should be made equal. If the denominators are the same, we can simply add or subtract the fractions easily. However, if the fractions have different denominators, then the process starts with the LCM (Least Common Multiple) of the denominators. Then, the fractions are multiplied with a suitable number which makes all the denominators equal. Finally, the numerators are added or subtracted as per the question and the new denominator remains the same.

How to Add Fractions with the Same Denominators?

In order to add fractions with the same denominators, we can simply add the numerators and keep the denominator the same. For example, let us add 3/7 + 2/7. Since the fractions have the same denominators, we just need to add the numerators. So, this will be 3/7 + 2/7 = (3 + 2)/7 = 5/7

How to Add Improper Fractions?

In order to add improper fractions, we use the same rules of adding fractions. For example, let us add 8/3 + 7/3. Since the fractions have the same denominators, we just need to add the numerators. So, this will be 8/3 + 7/3 = (8 + 7)/3 = 15/3 = 5

How to Add Mixed Fractions Step by Step?

Adding mixed numbers follows the same rules of adding fractions. The only extra step is to convert the mixed fractions to improper fractions. Let us understand this with an example. Let us add \(6 \dfrac{1}{2}\) + \(3 \dfrac{3}{4}\) using the following steps.

• Step 1: To add \(6 \dfrac{1}{2}\) + \(3 \dfrac{3}{4}\), let us convert these mixed fractions to improper fractions. This will be 13/2 + 15/4
• Step 2: Now, we will use the basic rules of addition. Here, the denominators are different, so we will convert these to their equivalent fractions so that their denominators become the same.
• Step 3: The LCM of 2 and 4 is 4. Now, we will multiply each fraction with the suitable number to make their denominators common: (13/2 × 2/2) + (15/4 × 1/1) = (26/4) + (15/4) = (26 + 15)/4 = 41/4 = \(1 0\dfrac{1}{4}\)