Addition of Fractions
Fractions are used to represent a part of a whole and they show a way to split a number into equal parts. 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 main aim is to make the denominators equal, and then the addition can be carried out easily. While like fractions have common denominators, unlike fractions are converted to like fractions to make addition easier. Let us explore more about the addition of fractions in this lesson.
Introduction to 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 numbers, the numerator and the denominator. A general representation of a fraction is a/b, where a is the numerator and 'b' is the denominator. Some of the examples of fractions are 2/3, 4/5, 6/7, 8/9, 21/43.
Just like other numbers, we can perform the arithmetic operations of addition, subtraction, multiplication, and division on fractions. Let us see the various steps involved.
How to Add Fractions?
The basic steps that are followed to add fractions are given below. For example, to add 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 on the common denominator. (1 + 2)/4 = 3/4.
- Simplify the fraction to its lowest terms, if needed.
- So, the sum of the given fractions is 3/4.
Addition of Fractions
We have learnt the addition of whole numbers (For example: 2 + 4 = 6). Similarly, we can add fractions. However, in fractions the denominator of the fractions is not added. We need to take the LCM of the denominator, before performing the addition process. The addition of fractions involves different formats like:
- 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
- Addition of fractions with variables: 3/5y + 1/4y
Now, let us learn the different formats of the addition of fractions.
Addition of Fractions with Same Denominators
Let us add the fractions 1/5 and 2/5 using a rectangular model. The denominator in the given fractions is the same. These fractions are called like fractions. The following figure represents both the fractions in the same model. Here 1/5 indicates that 1 out of 5 parts is shaded yellow. And 2/5 indicates that 2 out of 5 parts are shaded green. So, if we want to know the total number of parts that are shaded in this model, we add the two fractions (1/5 +2/5).
Adding 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 5, and we add the numerators. We write it as 1/5 + 2/5 = (2 +1)/5 = 3/5. This gives the sum as 3/5. Now, if we observe the figure, we can see that out of the 5 parts, 3 parts are shaded and in the fractional form, this can be represented as 3/5.
Addition of Fractions with Different Denominators
We just learned how to add fractions with like denominators. Now let us understand how to add 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 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.
Addition of 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 = \(3\frac{1}{7} \) = 22/7. Let us look at the following step for another example for addition of fractions.
- Consider the following example: 3 + 1/2
- Convert the whole number to a 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
Addition of Fractions with Variables
Now that we have seen the addition of fractions with like and unlike fractions, we can extend the same concept to add fractions with variables. Consider this example with 'y' as the variable, 1y/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 1y/2 + 1y/3, we take the LCM of the denominators and convert them to like terms. Next, we need to take the common variable out and rewrite it as follows: LCM 2, 3 = 6, y/2 = (1y/2) × (3/3) = 3y/6, (1y/3 × (2/2) = 2y/6, This brings to us, 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, (1x/2) + (1y/3).
Important Notes
The following points are helpful and should be remembered while working with 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.
FAQs on Addition of Fractions
How to Add Fractions?
The process of addition of fractions is a little different from normal addition. The first step in the addition of fractions is to see that the denominators of both the fractions are 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.
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 Do you Add 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, in this case is 15. The numerator and denominator of the first fraction 3/5 is multiplied by 3, and the numerator and denominator of the second fraction 4/3 is multiplied by 5. Hence, we have (3/5 × 3/3) + (4/3 × 5/5) = (9/15) + (20/15) = (9 + 20)/15 = 29/15= \(1\frac{14}{15}\).
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 the 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, 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 = \(1\frac{19}{30}\).
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 the Commutative Property of Addition of Fractions?
The commutative property of addition of fractions is the same as the commutative property of addition of whole numbers or natural numbers. 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 Addition and Subtraction of Fractions?
In the addition and subtraction 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.
How To Solve Mixed Fractions?
Mixed fractions can be solved by converting them into improper fractions. This can be done by multiplying the denominator with the whole number and then adding the numerator to it. Let us convert a mixed fraction 3½ into an improper fraction. 3½ = {(2 × 3) + 1} /2= (6 + 1)/2 = 7/2.