Fractions have lots of different names, and renaming is often needed to add or subtract fractions.

Here's how you add $\displaystyle\,\frac{1}{2}\,$ and $\displaystyle\,\frac{1}{3}\,$:

\begin{align} \cssId{s6}{\frac12 + \frac13} &\cssId{s7}{\ \ =\ \ \frac12\cdot\frac33 + \frac13\cdot \frac22}\cr\cr &\cssId{s8}{\ \ =\ \ \frac36 + \frac26}\cr\cr &\cssId{s9}{\ \ =\ \ \frac56} \end{align}

Notice that $\,\frac{3}{6}\,$ is just another name for $\,\frac{1}{2}\,$, and $\,\frac{2}{6}\,$ is just another name for $\,\frac{1}{3}\,$. However, these are the names that are needed for addition!

The next few paragraphs discuss the ideas in this example.

## Adding Fractions with a Common Denominator

When fractions have the same denominator (called a common denominator), then it's easy to add them:

For all real numbers $\,a\,$ and $\,b\,$, and for $\,c\ne 0\,$:
$$\frac{a}{c} + \frac{b}{c} = \frac{a+b}{c}$$
That is, to add two fractions with the same denominator, just add the numerators, and keep the denominator the same.

When fractions to be added don't have a common denominator, you need to find one,

In general, you'll want to use the least common multiple of the individual denominators as the new denominator. This is called the least common denominator.

Subtraction works the same way, because subtraction is just a special kind of addition.

## Examples

\displaystyle \begin{align} \cssId{s25}{\frac13 + \frac25} &\cssId{s26}{\ \ =\ \ \frac13\cdot\frac55 + \frac25\cdot \frac33}\cr\cr &\cssId{s27}{\ \ =\ \ \frac5{15} + \frac{6}{15}} \cssId{s28}{\ \ =\ \ \frac{11}{15}} \end{align}

Note that the least common multiple of $\,3\,$ and $\,5\,$ is $\,15\,$.

\displaystyle \begin{align} \cssId{s30}{\frac16 + \frac12} &\cssId{s31}{\ \ =\ \ \frac16 + \frac12\cdot \frac33}\cr\cr &\cssId{s32}{\ \ =\ \ \frac1{6} + \frac{3}{6}} \cssId{s33}{\ \ =\ \ \frac{4}{6}} \cssId{s34}{\ \ =\ \ \frac{2}{3}} \end{align}

Note that the least common multiple of $\,6\,$ and $\,2\,$ is $\,6\,$. In the exercises below, you may input your answer in either form, simplified or not.

\displaystyle \begin{align} \cssId{s37}{\frac13 - \frac25} &\cssId{s38}{\ \ =\ \ \frac13\cdot\frac55 - \frac25\cdot \frac33}\cr\cr &\cssId{s39}{\ \ =\ \ \frac5{15} - \frac{6}{15}} \cssId{s40}{ \ \ =\ \ -\frac{1}{15}} \end{align}

To subtract two fractions with the same denominator, just subtract the numerators, and keep the denominator the same:

\begin{align} \cssId{s42}{\frac{a}{c} - \frac{b}{c}} &\cssId{s43}{\ \ =\ \ \frac{a}{c} + \left(-\frac{b}{c}\right)}\cr\cr &\cssId{s44}{\ \ =\ \ \frac{a}{c} + \left(\frac{-b}{c}\right)}\cr\cr &\cssId{s45}{\ \ =\ \ \frac{a + (-b)}{c}}\cr\cr &\cssId{s46}{\ \ =\ \ \frac{a - b}{c}} \end{align}

## Practice

Use the least common denominator. You must input your answer as a diagonal fraction (like ‘2/5’), since you can't input horizontal fractions. You'll probably want to use paper-and-pencil to solve each problem; key in your answer when you're done.