# Adding and Subtracting Fractions with Unlike (Different) Denominators

Adding and subtracting two fractions with different denominators takes some extra steps compared to adding and subtracting two fractions with the same denominator. In this lesson, we will try to explain how to add and subtract two fractions with different denominators by following three basic steps, and you can then skip them and complete the procedure directly.

Adding and subtracting two fractions with different denominators requires prior knowledge of the least common multiple, so if you are not familiar with this algebraic concept, I suggest you visit this lesson first.

Now we will learn how to add fractions with different denominators. When adding fractions, the denominators of those fractions must be the same. But they are not always the same.

For example, the two fractions and can be added because they have the same denominators.

But the fractions and cannot be added at once, because these fractions have different denominators. In such cases, rewrite each fraction to its equivalent fraction with a denominator which is equal to the Least Common Multiple of the two deniminators. As a result of these actions, fractions that had different denominators turn into fractions that have the same denominators. And we already know how to add such fractions.

By the same way, we subtract two fractions.

### Note:

Let and represent two fractions and, , represent whole numbers.

And let .

Convert the two fractions to its equivalent fractions:

Subtract the two fraction by add the two equivalent fractions.

To make it easier to add fractions with different denominators, you can use the following step-by-step instructions:

1. Find the LCM of the denominators of fractions;

2. Divide the LCM by the denominator of each fraction and get an additional multiplier for each fraction;

3. Multiply the numerators and denominators of fractions by their additional factors;

4. Add fractions that have the same denominators;

5. Always make sure that the resulting fraction is reduced to the lowest term. That is, the gcf of the numerator and denominator is 1.

## Examples of Adding and Subtracting Fractions with Different Denominators

#### Example1:

Add and subtract the fractions and

#### Solution:

The two fractions and with unlike denominators. Therefore, we must convert each fraction to its equivalent fraction. But, first we must find the least common multiple of 6 and 4. that is, finding the least number that divides 6 and 4 together

Thus, and

Once their denominators are equal, add (subtract) the fractions by adding (subtracting) their numerators and then keeping the common denominator.

The fraction is in the lowest term because

The fraction is in the lowest term because

#### Example2:

Add and subtract the two fractions and

#### Solution:

The two fractions and are unlike denominators. Therefore, we must convert each fraction to its equivalent fraction. But, first we must find the least common multiple of 16 and 4. that is, finding the least number that divides 16 and 4 together

In this case, we notice that one of the denominators is a multiple of the other, here we only convert the fraction with the smallest denominator to a fraction whose denominator is the denominator of the fraction with the largest denominator. Which, we convert only the fraction to its equivalent fraction.

Once their denominators are equal, add (subtract) the fractions by adding (subtracting) their numerators and then keeping the common denominator.

The fraction is in the lowest term because

The fraction is in the lowest term because

#### Example3:

Find the value of the expression:

#### Solution:

To add three fractions with different denominators we apply the same rule to add two fractions with different denominators, we first find the LCM of 3, 4 and 5.

Now we can convert each fraction to its equivalent fraction with denominator is 60.

Thus, and and

Once their denominators are equal, add the fractions by adding their numerators and then keeping the common denominator.

The fraction is in the lowest term because

#### Example4:

Add and subtract the two fractions and

#### Solution:

The denominators are coprime, meaning that the lcm is the product of the two denominators. This means that:

Once their denominators are equal, add (subtract) the fractions by adding (subtracting) their numerators and then keeping the common denominator.

The fraction is in the lowest term because

The fraction is in the lowest term because