Adding Fractions with Different Denominators
The first step to adding fractions with unlike denominators is to transform them so they have equal denominators. Once this first step is done, then it will be easy to add the fractions.
To achieve this, find the least common denominator (LCD). For instance, the least common denominator of
^{2}/
_{5} and
^{1}/
_{10} is 10, since 10 is the lowest number that can be divided by both 5 and 10 that results to a whole number.
Below are other examples:
The LCD of
^{4}/
_{5} and
^{1}/
_{3} is 15.
The LCD of
^{3}/
_{4},
^{5}/
_{16} and
^{11}/
_{20} is 80.
The LCD of
^{2}/
_{9},
^{1}/
_{8} and
^{5}/
_{12} is 72.
After finding the LCD, the next thing to do is write down the equivalent fractions using the common denominator. You get the numerator of the equivalent fraction by dividing the common denominator by the fraction's original denominator and multiplying it by the original numerator. Then, you just replace the original denominator with the common denominator. You do this for every fraction you are adding.
Below are the conversions using the earlier examples:
^{4}/_{5} and ^{1}/_{3} become ^{12}/_{15} and ^{5}/_{15}
In this example the common denominator is 15, which is the product of the denominators. The LCD can sometimes be found by multiplying the two denominators. So, ^{4}/_{5} is equal to ^{12}/_{15} (15 divided by 5 equals 3, and 3 multiplied by 4 equals 12), and ^{1}/_{3} is equal to ^{5}/_{15} (15 divided by 3 equals 5, and 5 x 1 equals 5).
Then, we add the equivalent fractions: ^{12}/_{15} + ^{5}/_{15} = ^{17}/_{15} or 1 ^{2}/_{15}
^{3}/_{4}, ^{5}/_{16} and ^{11}/_{20} become ^{60}/_{80}, ^{25}/_{80} and ^{44}/_{80}
In this example, 80 is the lowest number that can be divided by each denominator that results in a whole number.
So, the sum of these 3 fractions is: ^{60}/_{80} + ^{25}/_{80} + ^{44}/_{80} = ^{129}/_{80} or 1 ^{49}/_{80}
^{2}/_{9}, ^{1}/_{8} and ^{5}/_{12} become ^{16}/_{72}, ^{9}/_{72} and ^{30}/_{72}
So, ^{16}/_{72} + ^{9}/_{72} + ^{30}/_{72} = ^{55}/_{72}.
To recap, in order to add fractions with the same denominators you have to follow 3 simple steps:
- Find the Least Common Denominator (LCD)
- Get the equivalent fractions
- Add the fractions that now have the same denominators
Adding Mixed Fractions with the Different Denominators
When adding mixed fractions you need to convert the mixed fraction into an improper fraction. The equivalent improper fraction will have the same denominator. To get the numerator, multiply the denominator by the whole number and add the original numerator to it.
Here is an example of a mixed fraction converted into its equivalent improper fraction:
2 ^{1}/_{4} = ^{(4 x 2) +1}/_{4}
^{(8)+1}/_{4} = ^{9}/_{4}
If the denominator of the resulting improper fraction is NOT equal to the denominators of the other fractions being added, then you need to find the LCD of the fractions before you can proceed to adding them.
Here's an example of adding mixed fractions with different denominators:
1 ^{1}/_{4} + 2 ^{3}/_{8} = ^{(4 x 1)+1}/_{4} + ^{(8 x 2)+3}/_{8}
^{(4)+1}/_{4} + ^{(16)+3}/_{8} = ^{5}/_{4} + ^{19}/_{8}
First, we converted the mixed fractions into improper fractions.
^{10}/_{8} + ^{19}/_{8} = ^{29}/_{8} or 3 ^{5}/_{8}
Then, we found the LCD (in this example it is 8) of the fractions and proceeded with adding them.
When adding a mixed fraction and a fraction with different denominators, the first step is to convert the mixed fraction into an improper fraction, then proceed with the steps in adding fractions with different denominators. Simplify the sum when needed.
^{2}/_{3} + 2 ^{2}/_{5} = ^{2}/_{3} + ^{(5 x 2)+2}/_{5}
^{2}/_{3} + ^{(10)+2}/_{5} = ^{2}/_{3} + ^{12}/_{5}
^{10}/_{15} + ^{36}/_{15} = ^{46}/_{15} or 3 ^{1}/_{15}