Adding and Subtracting Fractions

What about adding and subtracting fractions that do not have the same denominator? First, rewrite the fractions so they do have the same denominators, and then add them like we did above!

Here are some examples.

Consider the sum $\frac{1}{3} + \frac{1}{4}$. Since the denominators are different, we cannot (yet) combine the numerators. Follow these steps!

• First Step: Find a Common Denominator
  • Note that you can use any common denominator. This may be quicker, but you will probably have to reduce your answer at the end.
  • Using the least common denominator will keep the numbers somewhat small.
• Second Step: Add or Subtract the Common Denominator Fractions
• Possible Third Step: Reduce the final result in lowest terms

For our example, we can use the least common denominator, namely LCM(3,4) = 12.

• First Step. $\frac{1}{3} = \frac{4}{12}$ and $\frac{1}{4} = \frac{3}{12}$
• Second Step. $\frac{4}{12} + \frac{3}{12} = \frac{4+3}{12} = \frac{7}{12}$

Here is another example. Let's try to subtract $3/8$ from $5/6$. Again, we see two different denominators. So our first step is to find a common one. Let's use the "easy" common denominator $8\times 6 = 48$. We will find $3/8 = 18/48$ and $5/6=40/48$. Our next step is to subtract these expressions:

$\frac{40}{48} - \frac{18}{48} = \frac{40-18}{48} = \frac{22}{48}$

If we want to express this answer in lowest terms, we can write:

$\frac{22}{48} = \frac{11}{24}$