### 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

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

### Example

Add: [latex]\Large\frac{1}{2}+\Large\frac{1}{3}[/latex]

Solution:

[latex]\Large\frac{1}{2}+\Large\frac{1}{3}[/latex] | |

Find the LCD of [latex]2[/latex], [latex]3[/latex]. | |

Change into equivalent fractions with the LCD [latex]6[/latex]. | [latex]\Large\frac{1\cdot\color{red}{3}}{2\cdot\color{red}{3}} +\Large\frac{1\cdot\color{red}{2}}{3\cdot\color{red}{2}}[/latex] |

Simplify the numerators and denominators. | [latex]\Large\frac{3}{6}+\Large\frac{2}{6}[/latex] |

Add. | [latex]\Large\frac{5}{6}[/latex] |

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

### Try It

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

### Try It

### Example

Add: [latex]\Large\frac{7}{12}+\Large\frac{5}{18}[/latex]

### 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.

### Think About It

Add [latex]\Large\frac{3}{4}+\Large\frac{1}{6}+\Large\frac{5}{8}[/latex]. 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: [latex]\Large\frac{7}{15}-\Large\frac{19}{24}[/latex]

### Try It

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

### Example

Add: [latex]-\Large\frac{11}{30}+\Large\frac{23}{42}[/latex]

### Try It

### Example

Subtract: [latex]\Large\frac{1}{2}-\left(-\Large\frac{1}{4}\right)[/latex]

## 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: [latex]\Large\frac{3}{5}+\Large\frac{x}{8}[/latex]

### Try It

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