Adding Fractions

Fractions are used to represent a part of a whole. 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 is done by making the denominators 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 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. Some of the examples of fractions are 2/3, 14/5, 6/7, 28/9, 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 which are given below with the help of an example of adding 1/4 + 2/4.

• Step 1: Make the denominators the same. (Here, the denominators are already the same, so we move to the next step)
• Step 2: Add the numerators and place the sum over the common denominator. This implies (1 + 2)/4 = 3/4.
• Step 3: Simplify the fraction to its lowest form, if needed.

So, the sum of the given fractions is 3/4.

You must have studied about types of fractions. There are two types of fractions - like fractions and unlike fractions. Like fractions are a group of fractions with the common denominator, while unlike fractions are the group of fractions having different denominators. To learn about the addition of fractions, there are four cases that might come up to you. Those are explained below:

• 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 add the fractions 2/4 and 1/4 using a rectangular model. The denominator in the given fractions is the same. These fractions are called like fractions. The following figure represents the addition of both the fractions. Here 2/4 indicates that 2 out of 4 parts are shaded blue. And 1/4 indicates that 1 out of 4 parts is shaded. So, if we want to know the total number of parts that are shaded in this model, we add the two fractions (2/4 +1/4), which is equal to 3/4.

The addition of fractions with the same denominators is simple. We only need to add the numerators of the given fractions and retain the common denominator. In this case, we keep the denominator as 4, and we add the numerators. We write it as 2/4 + 1/4 = (2 +1)/4 = 3/4. This gives the sum as 3/4. Now, if we observe the figure, we can see that out of the 4 parts, 3 parts are shaded and in the fractional form, this can be represented 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 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 see the steps to be followed if we want to add the fractions 1/3 and 3/5.

• 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 write the given fraction in its mixed form. 5 + 1/2 = 5½ = 11/2, 3 + 1/7 = \(3\frac{1}{7} \) = 22/7. Let us look at another method for adding fractions with whole numbers.

• Consider the following example: 3 + 1/2.
• Write the whole number in fractional form, i.e. 3 = 3/1.
• Add them like unlike fractions by making the denominators the same. This implies, (3/1) + (1/2) = (3/1) × (2/2) + (1/2) = 6/2 + 1/2 = 7/2 = 3½.

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. Consider this example with 'y' as the variable, y/5 + 2y/5. These are like fractions since they have the same denominator and y is common. We can take the common factor out and rewrite it as: y/5 + 2y/5 = (1/5 + 2/5)y = 3y/5.

Similarly, if we have to add unlike fractions like y/2 + y/3, we take the LCM of the denominators and convert them into like terms. 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

y/3 = (y/3 × (2/2) = 2y/6.

We got two fractions with common denominators, (3y/6) + (2y/6) = (3y + 2y)/6 = 5y/6. In some cases, when we have different variables, 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

Check these interesting articles related to the concept of adding fractions in math.

• Subtraction of Fractions
• Multiplying Fractions
• Division of Fractions

How to do Addition of Fractions?

The process of addition of fractions is a little different from normal addition. The first step in adding fractions is to see that the denominators of both the fractions should be the same. This is made possible by multiplying the numerator and denominator of each of the fractions with a suitable constant. Finally, after the denominators are equal for both the fractions, the numerators are added, and the sum of the numerator with the common denominator is the result of the addition of fractions. Let us understand this with a simple example: 1/2 + 2/3 = (1/2 × 3/3) + (2/3 × 2/2) = 3/6 + 4/6 = (3 + 4)/6 = 7/6.

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. Only then we can add fractions.

How to Perform Addition of Fractions With Whole Numbers?

To add a fraction with a whole number, 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.

How to do Addition of Fractions With Different Denominators?

The fractions with different denominators can be added by making the denominators common. This is possible by multiplying the numerator and denominator of each of the fractions with a suitable constant. To add the fractions 3/5 + 4/3, we need to multiply both the fractions with a number which 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.

How Do you Add Three 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. 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.

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 the Commutative Property of Addition of Fractions?

The commutative property of the addition of fractions is the same as the commutative property of the addition of whole numbers or natural numbers. With fractions, we have, a/b + c/d = c/d + a/b. Further, the next step for the addition of these fractions is the same as the normal addition of fractions. This commutative property of addition applies for normal fractions, mixed fractions, and also for the addition of fractions with a whole number.

What is Subtraction and Addition of Fractions?

In the subtraction and addition of fractions, first, the denominators of the fractions should be made equal. The process starts with the LCM 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.