To add or subtract fractions with different denominators, we first must write them as equivalent fractions having the same denominator. We’ll use the techniques from the previous section to find the LCM of the denominators of the fractions. Recall that we call this the LCD (the least common denominator). We only use the denominators of the fractions, not the numerators, when finding the LCD.

Then we can use the Equivalent Fractions Property to algebraically change a fraction to an equivalent one. Remember, two fractions are equivalent if they have the same value. The steps for finding the LCD and the Equivalent Fractions Property are repeated below for reference.

Once we have converted two fractions to equivalent forms with common denominators, we can add or subtract them by adding or subtracting the numerators. Try the examples and practice problems below to refresh these skills.

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

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.

The following video provides two more examples of how to subtract 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.

When we use the Equivalent Fractions Property, there is a quick way to find the number you need to multiply by to get the LCD. Write the factors of the denominators and the LCD just as you did to find the LCD. The “missing” factors of each denominator are the numbers you need.

The LCD, [latex]36[/latex], has [latex]2[/latex] factors of [latex]2[/latex] and [latex]2[/latex] factors of [latex]3[/latex].

Twelve has two factors of [latex]2[/latex], but only one of [latex]3[/latex] —so it is ‘missing‘ one [latex]3[/latex]. We multiplied the numerator and denominator of [latex]\Large\frac{7}{12}[/latex] by [latex]3[/latex] to get an equivalent fraction with denominator [latex]36[/latex].

Eighteen is missing one factor of [latex]2[/latex] —so you multiply the numerator and denominator [latex]\Large\frac{5}{18}[/latex] by [latex]2[/latex] to get an equivalent fraction with denominator [latex]36[/latex]. We will apply this method as we subtract the fractions in the next example.

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]