## Adding and Subtracting Fractions With Different Denominators

### Learning Outcomes

• Add and subtract fractions with different denominators
• Add and subtract fractions with different denominators that contain variables
• Identify and use fraction operations

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators.

### Add or subtract fractions with different denominators

1. Find the LCD.
2. Convert each fraction to an equivalent form with the LCD as the denominator.
3. Add or subtract the fractions.
4. Write the result in simplified form.

### Example

Add: $\Large\frac{1}{2}+\Large\frac{1}{3}$

Solution:

 $\Large\frac{1}{2}+\Large\frac{1}{3}$ Find the LCD of $2$, $3$. Change into equivalent fractions with the LCD $6$. $\Large\frac{1\cdot\color{red}{3}}{2\cdot\color{red}{3}} +\Large\frac{1\cdot\color{red}{2}}{3\cdot\color{red}{2}}$ Simplify the numerators and denominators. $\Large\frac{3}{6}+\Large\frac{2}{6}$ Add. $\Large\frac{5}{6}$

Remember, always check to see if the answer can be simplified. Since $5$ and $6$ have no common factors, the fraction $\Large\frac{5}{6}$ cannot be reduced.

### Try It

Watch the following video to see more examples and explanation about how to add two fractions with unlike denominators.

### Example

Add: $\Large\frac{7}{12}+\Large\frac{5}{18}$

### Try It

You can also add more than two fractions as long as you first find a common denominator for all of them. An example of a sum of three fractions is shown below. In this example, you will use the prime factorization method to find the LCM.

Add $\Large\frac{3}{4}+\Large\frac{1}{6}+\Large\frac{5}{8}$.  Simplify the answer and write as a mixed number.

What makes this example different than the previous ones? Use the box below to write down a few thoughts about how you would add three fractions with different denominators together.

## Subtracting Fractions

When you subtract fractions, you must think about whether they have a common denominator, just like with adding fractions. Below are some examples of subtracting fractions whose denominators are not alike.

### Example

Subtract: $\Large\frac{7}{15}-\Large\frac{19}{24}$

### Try It

The following video provides two more examples of how to subtract two fractions with unlike denominators.

### Example

Add: $-\Large\frac{11}{30}+\Large\frac{23}{42}$

### Example

Subtract: $\Large\frac{1}{2}-\left(-\Large\frac{1}{4}\right)$

## Adding and Subtracting Fractions that Contain Variables

In the next example, one of the fractions has a variable in its numerator. We follow the same steps as when both numerators are numbers.

### Example

Add: $\Large\frac{3}{5}+\Large\frac{x}{8}$

### Try It

Watch the following video to see more examples of how to add and subtract fractions with unlike denominators that contain variables.