Addition of Fraction

Fractions are used to express a portion of a whole. A fraction is a number expressed with the form p/q, where q is not zero or one. A fraction is consist of two components, a numerator which is on the top, and the denominator which is on the bottom. Because a fraction has a numerator and a denominator that is separated by a bar, adding fractions is a little different than adding numbers. 

• The fractions are called like fractions when the denominators are the same. 

E.g. 2/5 and 3/5

• The fractions are called unlike fractions when the denominators are different. 

E.g. 4/7 and 5/ 11

Addition of fractions

Making the denominators equal is how fractions are added. Like fractions contain the same denominator, unlike fractions are converted to like fractions to simplify addition. In this section, we’ll understand more about adding fractions.

Addition of fractions with the same denominators

Writing the sum of the numerators over the same denominator is how you add fractions with the same denominators. Let’s use a circular model to add the fractions 1/5 and 3/5. In both fractions, the denominator is the same, these are called fractions.  

The addition of both fractions is shown in the diagram below. 1/5 denotes that one of the five portions is shaded. And 3/5 means that three out of every five pieces are shaded. So, if we want to determine how many portions are shaded in total in this model, we combine the two fractions (1/5 +3/5), which gives 4/5.

It’s easy to add fractions with the same denominators. Only the numerators of the given fractions should be added, and the common denominator should be kept. We retain the denominator at 5 and add the numerators in this case. 

2/4 + 1/4 = (2 +1)/4 = 3/4 is how we write it. As a result, the total is 3/4. Now, if we look at the graphic, we see that three out of the four parts are shaded, which may be expressed as 3/4 in fractional form.

Addition of fractions with different denominators

We have now learned how to add fractions with the same denominators. Let’s have a look at how to add fractions with different denominators now.  Fractions with different denominators are called, unlike fractions. 

The first step in converting such fractions to like fractions is to make the denominators common. This is accomplished by calculating the denominators’ least common multiple (LCM). Let’s look at the stages involved in adding the fractions 3/7 and 5/6.

• Step 1: Because the denominators of the provided fractions differ, we must calculate the LCM of 7 and 6 to equalize them. The LCM of 7 and 6 equals 42.

• Step 2: Multiply 3/7 by 6/6, (3/7) × (6/6) = 18/42, and 5/6 by 7/7, (5/6) × (7/7) = 35/42, to have like fractions of the same denominators.

• Step 3: Because the denominators are same, we can simply add the numerators and multiply by the common denominator. 18/42and 35/42 are two new fractions with the same denominators. As a result, 

18/42+ 35/42 equals (18 + 35)/42 = 53/42.

Addition of fraction and a whole number

Writing the provided fraction in its mixed form is an easy approach to add a whole integer and a correct fraction.  5 + 1/2 = 5½ = 11/2, 3 + 1/7 =3 1/7.

Let’s have a look at another way to add fractions to whole numbers.

Look at the example below: 3 + 1/2.

3 = 3/1 is the fractional representation of the whole number.

By matching the denominators, you could add them together unlike fractions. 

(3/1) + (1/2) = (3/1) × (2/2) + (1/2) = 6/2 + 1/2 = 7/2 = 3½.

Adding Fractions with Variables

We can apply the same technique to adding fractions with variables now that we’ve seen how to add fractions with like and unlike fractions. Consider the following example with the variable ‘y’: y/5 + 2y/5. Because they have the same denominator and y is common, these are like fractions. We may eliminate the common element and rewrite the equation as y/5 + 2y/5 = (1/5 + 2/5)y = 3y/5.

If we need to combine unlike fractions, such as y/2 + y/3, we use the LCM of the denominators to transform them into like terms. After that, we must remove the common variable and rebuild it as follows:

 LCM of (3 , 4) is 12

(y/3) = y/3 × (4/4) = 4y/12 

y/4 = (y/4 × (3/3) = 3y/12.

(4y/12) + (3y/12) = (4y + 3y)/12 = 7y/12 are two fractions having some common denominators. When we have various variables, we sometimes interpret them as unequal terms that can’t be reduced any further, such as x/5 + y/3.

Conclusion:

We have learned how to do the following during this lessons:

• Understand what a fraction is and the many types of fractions.

• Add fractions with the same denominator

• Add fractions with different denominators

• Add fractions with a whole number

• Add fractions with variables