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Adding and Subtracting Fractions

It is easiest to work with proper or improper fractions when adding or subtracting fractions. If the fraction is mixed, simply convert it to an improper fraction before you start. There are three simple steps to adding or subtracting fractions.

  • Step 1: Make sure the denominators are the same (lowest common denominator).
  • Step 2: Add or subtract the numerators and put the answers over the same deonominator
  • Step 3: Simplify the fraction (if needed).

 Example: \(\frac{3}{4}-\frac{1}{4}\)


Since both deonominators are the same, go to step 2: 


Now, simplify the fraction:  


As stated in step 1, before you can add or subtract fractions, the fractions need to have a common denominator. If the denominators are not the same, you can either use the least common denominator method to make them the same, or you can multiply both parts of each fraction by the denominator of the other.
This latter method always works, but you will often need to simplify the fraction afterwards.
Now, take a look at the fractions \(\frac{1}{3} \text{ and } \frac{1}{6}\). If we wanted to add or subtract these two fractions, we would need a common denominator. As stated above, we could multiply both parts of each fraction by the denominator of the other:
\(\frac{1}{3}\times\frac{6}{6} = \frac{6}{18}\) and \(\frac{1}{6}\times\frac{3}{3}=\frac{3}{18}\)
Now we can add or subract these two fraction. However, this is not the lowest common denominator. To find the lowest common denominator for the two fractions \(\frac{1}{3}\) and \(\frac{1}{6}\), we would have to list the mulitples of 3 and the multiples of 6:
\(\frac{1}{3}\)  has multiples of 3: 3, 6, 9, 12, 15, 18, 21,...
 has multiples of 6: 6, 12, 18, 24, 30, 36,...
Notice that the smallest number that is the same is the numer 6, so the lowest common denominator for \(\frac{1}{3}\) and \(\frac{1}{6}\) is 6. Now, the fraction \(\frac{1}{3}\) can be written as \(\frac{2}{6}\) by using equivalent fractions, and we can now add/subtract the two fractions \(\frac{2}{6}\) and \(\frac{1}{6}\). 
Example: Find the lowest common denominator and equivalent fractions for \(\frac{3}{4}\) and \(\frac{5}{12}\).  
\(\frac{3}{4}\)  has multiples of 4: 4,8,12,16,20,24,28,....
\(\frac{5}{12}\) has multiples of 12: 12, 24, 36, 48, 60, 72,....
The lowest common denominator for \(\frac{3}{4}\) and \(\frac{5}{12}\) is 12. The equivalent fractions would be: 
\(\frac{3}{4}\times\frac{3}{3}=\frac{9}{12}\) and \(\frac{5}{12}\times \frac{1}{1}=\frac{5}{12} \)
Example:  \(\frac{1}{5}-4\frac{5}{7}\)
First convert the mixed fraction into an improper fraction: \(\frac{4 \times 7+5}{7}=\frac{33}{7} \). 
Now, what we really want is \(\frac{1}{5}-\frac{33}{7}\).   
Notice that the the denominators are different. The lowest common denominator would be 35. The equivalent fractions are:  
\(\frac{1}{7} \times \frac{7}{7}=\frac{7}{35}\) and   \(\frac{33}{5}\times\frac{5}{5}=\frac{165}{35}\)    
Therefore, \(\frac{7}{35}-\frac{165}{35}=\frac{158}{35}\)