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.

What is the Addition of Fractions?

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.

How to Add Fractions?

Now, let us learn the basic steps of addition of fractions which are given below with the help of an example of adding 1/4 + 2/4.

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

Addition of Fractions with Same Denominators

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 are 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).

The addition of fractions with the same denominators is simple. We only need to add the numerators of the given fractions and retain the same 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.

Adding Fractions with Unlike Denominators

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 then copy 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.

Adding Fractions with Whole Numbers

An easy way to add a whole number and fraction is to write the given fraction in its mixed form. 5 + 1/2 = 5½ = 11/2, 3 + 1/7 = 317317 = 22/7. Let us look at another method for adding fractions with whole numbers.

Consider the following example: 3 + 1/2.
Convert the whole number to the fractional form. 3 = 3/1.
Add them like unlike fractions by making the denominators the same. (3/1) + (1/2) = (3/1) × (2/2) + (1/2) = 6/2 + 1/2 = 7/2 = 3½.

Adding Fractions with Variables

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.

Adding Fractions Tips and Tricks:

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 same denominator to get the sum of the fractions.

Formula For Adding Fractions :

Before learning the formula for adding fractions, let us recall what are fractions. In mathematics, fractions can be defined as the parts of a whole. A fraction can be a portion or section of any quantity out of a whole, where, the whole can be any number, a specific value, or a thing. Examples are 1/2, 3/7,4/5, 6/6, etc. There are different types of fractions depending upon their form. These are

Proper Fraction
Improper Fraction
Unit Fraction
Mixed Fraction
Equivalent Fractions
Like Fractions, and
Unlike Fractions.

What Is Formula for Adding Fractions?

The formulas for adding fractions vary in each of the following situations: Adding fractions with whole numbers, adding fractions with like denominators, adding fractions with different denominators, and adding fractions with variables. These formulas are:

Adding fractions with whole numbers
(a+bc)(a+bc) = (a×c)+bc(a×c)+bc
Adding fractions with like denominators
(ab+db)(ab+db)=(a+d)b(a+d)b
Adding fractions with different denominators
(ab+cd)(ab+cd)=(a×d+b×cb×d)(a×d+b×cb×d)
Adding fractions with variables
abx+cbx=(ab+cb)x=(a+cb)xabx+cbx=(ab+cb)x=(a+cb)x

where

a, b, c, and, d are constants
x is variable.

Let us see the applications of formula for adding fractions in the solved examples below.

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Examples Using Formula for Adding Fractions

Example 1: Using the formula for adding fractions, add 3/2 and 5.

Solution:

To find: addition of 3/2 and 5.

We have

(a+bc)(a+bc) = (a×c)+bc(a×c)+bc

Adding Fractions = (2×5)+32(2×5)+32

= (10+3)/2

= 13/2

Answer: The addition of 3/2 and 5 is 13/2.

Example 2: If the difference between two fractions is 2/3 and one fraction is 6/5, find the original fraction.

Solution:

To find: The original fraction.

Using formula for adding fractions,

Adding Fractions = a×d+b×cb×da×d+b×cb×d

Adding Fractions = 2×5+6×33×52×5+6×33×5

= (10+18)/15

= 28/15

Answer: The original fraction is 28/15.