In this lesson, we shall be learning how to add fractions with different denominators. There are many methods available, some of which involve various diagrams or tables. In this lesson, we shall stick to some standard methods.

## How to Add Fractions with Different Denominators

Before we begin, let's just have a quick recap of the vocabulary associated with fractions.

numerator - the top number of the fraction

denominator - the bottom number of the fraction

vinculum - the line separating the numerator and denominator

## Let's consider the following question:

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

In order to add the fractions, you need to have the same denominators. One of the quickest methods for doing this is to:

multiply the numerator and denominator of the first fraction by the denominator of the second fraction

multiply the numerator and denominator of the second fraction by the denominator of the first fraction

$$

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

$$

So, we now have:

$$

=\frac{14}{35} + \frac{15}{35}\\

$$

Now that we have common denominators, we can add the 2 fractions together:

$$

=\frac{29}{35}

$$

### More Examples of Adding Fractions with Different Denominators

**EXAMPLE 2**

$$\begin{align}

&\frac{3}{5}+\frac{1}{8}\\\\

&=\frac{8}{8}\times\frac{3}{5}+\frac{5}{5}\times\frac{1}{8}\\\\

&=\frac{24}{40}+\frac{5}{40}\\\\

&=\frac{29}{40}

\end{align}

$$

**EXAMPLE 3 **

$$\begin{align}

&\frac{2}{9}+\frac{2}{5}\\\\

&=\frac{5}{5}\times\frac{2}{9}+\frac{9}{9}\times\frac{2}{5}\\\\

&=\frac{25}{45}+\frac{18}{45}\\\\

&=\frac{43}{45}

\end{align}

$$

### Common Errors when Adding Fractions

The main error I see is that students add the denominators together as well as the numerators. In the question above that would have given an incorrect answer:

$$

\frac{14}{35}+\frac{15}{35}\ne\frac{29}{70}

$$

## How to Avoid Errors when Adding Fractions with Different Denominators

One of the methods that can avoid the error above is often called the Bow Tie Method. The idea is that you draw a large vinculum and then cross-multiply the numerators and denominators like this:

$$\begin{align}

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

&=\frac{2\times5+3\times7}{7\times5}\\\\

&=\frac{10+21}{35}\\\\

&=\frac{31}{35}

\end{align}

$$

I commonly skip the second step as follows:

$$\begin{align}

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

&=\frac{10+21}{35}\\\\

&=\frac{31}{35}

\end{align}

$$

## Do you want more addition of fractions questions?

The addition and subtractions of fractions worksheet pack below contains 1300 questions with answers. Attempting some questions on a regular basis is a great way to strengthen your skills.