BMCC: Math Tutorials: Fractions:
Addition of Fractions
Quick links to topics on this page:
Adding Fractions with the Same Denominator
Adding Fractions with Different Denominators
Adding a Whole Number and a Fraction
Adding a Whole Number and a Mixed Number
Adding Two Mixed Numbers
Addition of fractions is fairly simple. You just need to remember that before two or more fractions can be added, they must all have the same denominator. If two fractions already have the same denominator, the sum of the two fractions is obtained by adding the two numerators, and placing that value over the common denominator in the original fractions. Shown in general terms,
For example:
That's all there is to it! Just remember that you should never add the denominators-- the denominator of the final sum should be the same as the denominator of the two fractions being added.
Also, after adding fractions, the resulting sum may be an improper fraction (the numerator may be equal to or larger than the denominator), which you may want to convert to a mixed number.
Before two fractions with different denominators can be added, each must be converted to an equivalent fraction so that they have a common denominator. The common denominator for the two original fractions will be the least common multiple of the two original denominators. You may want to review the sections on determining the least common multiple for two numbers, and finding equivalent fractions.
The process of adding two fractions with different denominators has three main steps:
Let's try an example. Add 
.
Step 1: We want to determine the least common multiple for the two denominators in these fractions, which are 3 and 4. You can verify for yourself that the least common multiple for the numbers 3 and 4 is 12 .
Step 2: Now we want to convert both fractions to their equivalent fractions with denominators of 12. You can verify for yourself that
Step 3: Add together the two fractions from step 2.
Note that 
is an improper fraction, which can be converted to the mixed number 
.
Another example: add 
.
Step 1: The denominators in the two fractions are 12 and 16, which have a least common multiple of 48.
Step 2: We want to convert the two fractions to their equivalent fractions with a denominator of 48. This gives
Step 3: Add the two fractions from step 2.
This is a proper fraction (value less than 1), and so does not need to be converted to a mixed number.
If a proper fraction (value less than 1; i.e., the numerator is smaller than the denominator) and a whole number are added, the sum is a mixed number which is a simple combination of the two numbers being added. For example,
To add an improper fraction to a whole number, the improper fraction should first be converted to a mixed number. Then you can add the whole number and the mixed number as described immediately below.
Adding a whole number and a mixed number results in a sum that is also a mixed number. To determine the whole number portion of the sum, you simply add the two whole numbers. The fractional portion of the sum will be exactly the same as the fractional portion of the original mixed number being added.
For instance, suppose we want to add 
. The whole number portion of the sum will be 5 + 2 , or 7 . The fractional portion of the sum will be 
. Thus,
Another example:
For a third example, let's show how to add a whole number and an improper fraction. Consider 
. As mentioned above, the first thing that must be done is to convert the improper fraction to a mixed number, like this:
Now we're back to adding a whole number and a mixed number like we've been doing in the previous two examples:
Adding two mixed numbers can be done in three main steps, but the process may also involve other procedures that we have covered in previous sections. The basic procedure for adding two mixed numbers is:
Easy example: Find the sum of 
.
Step 1: Add the fractional portions of the mixed numbers. The fractional portions are 
and 
, which when added result in a sum of 
.
Step 2: Add the whole number portions of the mixed numbers. The whole number portions are 2 and 4 , which have a sum of 6 .
Step 3: Combine the sums from the first two steps and, if necessary, reduce to simplest form. The sum from step 1 is and the sum from step 2 is 6 , which when combined yield a final sum of 
. Since is a proper fraction and is already in simplest form, no further simplification is necessary.
A more complicated example: Determine the sum of 
.
Step 1: Add the fractional portions of the two mixed numbers. The fractional portions are 
and 
. Since these have different denominators, they must both first be converted to equivalent fractions with a common denominator, and the common denominator should be the least common multiple of the two denominators. You can verify for yourself as an exercise that the least common multiple for the numbers 8 and 6 is 24. You can also verify for yourself that converting the two fractions to equivalent fractions with a denominator of 24 yields the following results:
and 
Now we just need to add the fractions together:
Note that 
is an improper fraction, but we'll take care of that in a bit.
Step 2: Add the whole number portions of the mixed numbers. The whole number portions are the numbers 1 and 2, which have a sum of 3 .
Step 3: Combine the sums from step 1 and step 2 into a mixed number, and simplify if necessary. Combining 3 and gives 
. Now we'll take care of the improper fraction. Our sum of can be thought of as 
. The improper fraction can be converted to the mixed number 
, and adding that to the whole number portion we already had gives the final sum:
The fraction 
is already in simplest form, and does not need further manipulation.
Next: Subtraction of Fractions
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