# Fractions with the Same Numerators Added

Fractions with the Same Numerators Added

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.

\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

\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}

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.

\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}