When two fractions have the same denominator, adding them is simple. We simply add the numerators of each fraction together with the same denominator.

Example

Add frac 2 3 to frac 5 3 to get frac 2 + 5 3 to get frac 7 3.

## Fractions with Unlike Denominators Added

Let’s say we want to add frac1 to frac1 to get frac1+frac1

Because the denominators of the two fractions are not equal, we cannot simply add the numerators and denominators to obtain the answer.

Finding a common (same) denominator for both fractions that will allow us to add them correctly is the trick.

What number can we pick where 2 and 4 will split equally?

Since we always want to choose the smallest such number, our best option would be 4.

Any number that can be divided by both 2 and 4 is acceptable, such as 8, 12, or 16, and it will work. We simply save time by selecting the smallest one, as we will see in a moment.

Our task is to convert frac12 into an equivalent fraction with a 4 in the denominator because frac14 already has a 4 in the denominator.

With a 4 in the denominator, frac12 becomes frac24, which is the equivalent fraction.

We can now combine them by adding them together.

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

## Multiplication by 1 and Common Denominators equal a/a

A common denominator for both fractions—preferably the Least Common Denominator—and equivalent fraction conversion are required to add two fractions correctly.

The last example's common denominator was relatively simple to find because 4 is a multiple of 2.

Here, we'll go over how we can actually find a common denominator.

For instance, add "frac 2 3 " and "frac 3 "

To solve this problem, multiply each fraction by a form of 1 that has an equivalent value and will give all the fractions a common denominator.

What is the first integer into which the numerators 3 and 4 divide equally?

In this instance, it is simply 3 x 4 = 12.

How can we convert each fraction to its equivalent form with a 12 as the denominator?

So to speak, we multiply each fraction by 1.

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

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

Take note that frac 3 3 and frac 4 4 both equal 1.

Now that we've prepared ourselves, we can 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}

In order to find a common denominator, we multiplied both fractions by a form of 1: 1=fracaa for the first and 1=fracbb for the second.

Example \sAdd

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

The first number that 6 and 8 divide evenly into is: \frac{24}{6}=4 and frac 24 8 = 3.

Divide each fraction by the equivalent form of 1 to get the denominator, which is 24. For frac5-6, 1=frac4-4, and for frac3-8, 1=frac3-3.

\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} \s\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}

3 or more fractions added

The same procedure is used to add three or more fractions, but there are now three or more denominators to take into consideration when determining a common denominator.

Example \sAdd

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

2, 3, 4, and 6 all have a common denominator. It is 12.

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

(\frac{7}{6})+(\frac{4}{4})

(\frac{4}{3})+(\frac{5}{4})

(\frac{3}{3})+(\frac{3}{2})

(\frac{6}{6})

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

Quick and simple Formula \s\frac{a}{b}+\frac{c}{d}=\frac{ad+bc}{bd}