Addition of Fractions with Same Denominators

Adding two fractions that have the same denominator is straightforward. We just add the numerators of each fraction together and keep the same denominator.

Example 

Add \frac{2}{3} to \frac{5}{3}

\frac{2}{3}+\frac{5}{3}=\frac{2+5}{3}=\frac{7}{3}

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Addition of Fractions with Unlike Denominators

Say we want to add \frac{1}{4} to \frac{1}{2}

\frac{1}{4}+\frac{1}{2}

We can’t just add the numerators and denominators to find the answer, because the denominators of both fractions are not equal.

The trick is to find a common (same) denominator for both fractions that will allow us to properly add them.

What number can we choose that both 2 and 4 will divide into evenly?

Our best choice would be 4 because we always want to​ choose the smallest such number.

We could choose any number we want that is divisible by both 2 and 4, such as 8, 12, 16, and it will work. It just saves us time if we choose the smallest one, as we will see shortly.

\frac{1}{4} already has a 4 in the denominator, so our job is to make \frac{1}{2} into an equivalent fraction that has a 4 in the denominator.

The equivalent fraction to \frac{1}{2} with a 4 in the denominator is \frac{2}{4}.

Now, we are in a position to add them together

\frac{1}{4}+\frac{1}{2}=\frac{1}{4}+\frac{2}{4}=\frac{1+2}{4}=\frac{3}{4}

Common Denominators and Multiplication by 1=a/a

The addition of 2 fractions involves the processes of finding a common denominator, preferably the Least Common Denominator, of both fractions and equivalent fraction conversion in order for us to properly add them.

Finding the common denominator in the last example was relatively easy because 4 is a multiple of 2.

Here we will outline the actual technique for finding a common denominator.

Example

Add \frac{2}{3} and \frac{3}{4}

To perform this problem we must multiply each fraction by an equivalent form of the number 1 that will make each fraction have a common denominator.

What is the first number that both denominators 3 and 4 divide into evenly?

In this case, it is just 3\times 4=12

How do we make each fraction into an equivalent form that has a denominator of 12?

We multiply each fraction by 1 so to speak

\frac{4}{4}\times\frac{2}{3}=\frac{8}{12}

and

\frac{3}{3}\times\frac{3}{4}=\frac{9}{12}

Notice that \frac{3}{3}=1 and \frac{4}{4}=1.

Now we've put ourselves in a position to add these fractions

\frac{2}{3}+\frac{3}{4}=(\frac{4}{4})(\frac{2}{3})+(\frac{3}{4})(\frac{3}{3})=\frac{8}{12}+\frac{9}{12}=\frac{8+9}{12}=\frac{17}{12}

We multiplied both fractions by a form of 1 in order to find a common denominator: 1=\frac{a}{a} for the first and 1=\frac{b}{b} for the second.

Example

Add

\frac{5}{6}+\frac{3}{8}

Find the first number that both 6 and 8 divide into evenly: \frac{24}{6}=4 and \frac{24}{8}=3

Multiply each fraction by an equivalent form of 1 that makes its denominator 24: 1=\frac{4}{4} for \frac{5}{6} and 1=\frac{3}{3} for \frac{3}{8}

\frac{5}{6}+\frac{3}{8}=(\frac{4}{4})(\frac{5}{6})+(\frac{3}{8})(\frac{3}{3})=\frac{20}{24}+\frac{9}{24}=\frac{20+9}{4}=\frac{29}{24}

Example

Add

\frac{2}{5}+\frac{1}{6}

\frac{2}{5}+\frac{1}{6}=(\frac{6}{6})(\frac{2}{5})+(\frac{1}{6})(\frac{5}{5})=\frac{12}{30}+\frac{5}{30}=\frac{17}{30}

Addition of 3 or more Fractions

The addition of 3 or more fractions is performed the same way but we now have 3 or more denominators to account for in finding a common denominator.

Example

Add

\frac{7}{6}+\frac{4}{3}+\frac{5}{4}+\frac{3}{2}

Find a common denominator for 2, 3, 4 and 6. It is 12.

\frac{7}{6}+\frac{4}{3}+\frac{5}{4}+\frac{3}{2}=

(\frac{2}{2})(\frac{7}{6})+(\frac{4}{4})(\frac{4}{3})+(\frac{5}{4})(\frac{3}{3})+(\frac{3}{2})(\frac{6}{6})=

\frac{14}{12}+\frac{16}{12}+\frac{15}{12}+\frac{18}{12}=\frac{63}{12}=\frac{21}{4}

Quick and Easy Formula

\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}

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Emma

I am passionate about travelling and currently live and work in Paris. I like to spend my time reading, gardening, running, learning languages and exploring new places.