Adding and Subtracting Fractions with Negatives

Once you've learned how to add and subtract positive fractions , you can extend the method to include negative fractions.

Note that:

$\frac{-2\text{\hspace{0.17em}}\text{\hspace{0.17em}}}{\text{\hspace{0.17em}}\text{\hspace{0.17em}}3}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\frac{\text{\hspace{0.17em}}2}{-3\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}}$

When you are adding or subtracting a negative fraction, you usually want to consider the numerator as negative. The method is just the same, except now you may need to add negative or positive numerators.

Example 1:

Find the sum.

$\frac{9}{5}+\left(-\frac{4}{3}\right)$

The LCM of $5$ and $3$ is $15$ .

To add the fractions with unlike denominators, rename the fractions with a common denominator.

$\begin{array}{l}\frac{9}{5}=\frac{9\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3}{5\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3}=\frac{27}{15}\\ -\frac{4}{3}=-\frac{4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}5}{3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}5}=-\frac{20}{15}\end{array}$

So,

$\frac{9}{5}+\left(-\frac{4}{3}\right)=\frac{27}{15}+\left(-\frac{20}{15}\right)$

Since the denominators are the same, add the numerators.

$\begin{array}{l}=\frac{27+\left(-20\right)}{15}\\ =\frac{7}{15}\end{array}$

Example 2:

Find the difference.

$-\frac{7}{10}-\frac{2}{15}$

The LCM of $10$ and $15$ is $30$ .

To subtract the fractions with unlike denominators, rename the fractions with a common denominator.

$\begin{array}{l}-\frac{7}{10}=-\frac{7}{10}×\frac{3}{3}=-\frac{21}{30}\\ \frac{2}{15}=\frac{2}{15}×\frac{2}{2}=\frac{4}{30}\end{array}$

So,

$-\frac{7}{10}-\frac{2}{15}=-\frac{21}{30}-\frac{4}{30}$

Since the denominators are the same, subtract the numerators.

$-\frac{21}{30}-\frac{4}{30}=\frac{-21-4}{30}$

Simplify. We get: