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

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.

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

Example

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

Show Solution

Solution:

	[latex]\Large\frac{7}{12}+\Large\frac{5}{18}[/latex]
Find the LCD of [latex]12[/latex] and [latex]18[/latex].
Rewrite as equivalent fractions with the LCD.	[latex]\Large\frac{7\cdot\color{red}{3}}{12\cdot\color{red}{3}} +\Large\frac{5\cdot\color{red}{2}}{18\cdot\color{red}{2}}[/latex]
Simplify the numerators and denominators.	[latex]\Large\frac{21}{36}+\Large\frac{10}{36}[/latex]
Add.	[latex]\Large\frac{31}{36}[/latex]

Because [latex]31[/latex] is a prime number, it has no factors in common with [latex]36[/latex]. The answer is simplified.

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.

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]

Show Solution

Solution:

	[latex]\Large\frac{7}{15}-\Large\frac{19}{24}[/latex]
Find the LCD.	[latex]15[/latex] is ‘missing’ three factors of [latex]2[/latex] [latex]24[/latex] is ‘missing’ a factor of [latex]5[/latex]
Rewrite as equivalent fractions with the LCD.	[latex]\Large\frac{7\cdot\color{red}{8}}{15\cdot\color{red}{8}} -\Large\frac{19\cdot\color{red}{5}}{24\cdot\color{red}{5}}[/latex]
Simplify each numerator and denominator.	[latex]\Large\frac{56}{120}-\Large\frac{95}{120}[/latex]
Subtract.	[latex]-\Large\frac{39}{120}[/latex]
Rewrite showing the common factor of [latex]3[/latex].	[latex]-\Large\frac{13\cdot 3}{40\cdot 3}[/latex]
Remove the common factor to simplify.	[latex]-\Large\frac{13}{40}[/latex]

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

Example

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

Show Solution

Solution:

	[latex]\Large\frac{1}{2}-\left(-\Large\frac{1}{4}\right)[/latex]
Find the LCD of [latex]2[/latex] and [latex]4[/latex].
Rewrite as equivalent fractions using the LCD [latex]4[/latex].	[latex]\Large\frac{1\cdot\color{red}{2}}{2\cdot\color{red}{2}} - (--\Large\frac{1}{4})[/latex]
Simplify the first fraction.	[latex]\Large\frac{2}{4}-\left(-\Large\frac{1}{4}\right)[/latex]
Subtract.	[latex]\Large\frac{2-\left(-1\right)}{4}[/latex]
Simplify.	[latex]\Large\frac{3}{4}[/latex]

One of the fractions already had the least common denominator, so we only had to convert the other fraction.

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]

Show Solution

Solution:
The fractions have different denominators.

	[latex]\Large\frac{3}{5}+\Large\frac{x}{8}[/latex]
Find the LCD.
Rewrite as equivalent fractions with the LCD.	[latex]\Large\frac{3\cdot\color{red}{8}}{5\cdot\color{red}{8}} +\Large\frac{x\cdot\color{red}{5}}{8\cdot\color{red}{5}}[/latex]
Simplify the numerators and denominators.	[latex]\Large\frac{24}{40}+\Large\frac{5x}{8}[/latex]
Add.	[latex]\Large\frac{24+5x}{40}[/latex]

We cannot add [latex]24[/latex] and [latex]5x[/latex] since they are not like terms, so we cannot simplify the expression any further.

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

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