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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:

$- \frac{2}{3} is the same as$ $\frac{- 2}{3} and \frac{2}{- 3}$

$\frac{- 2}{- 3} simplifies to \frac{2}{3}$

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.

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

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

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

$\begin{array}{l} \frac{9}{5} = \frac{9 \times 3}{5 \times 3} = \frac{27}{15} \\ - \frac{4}{3} = - \frac{4 \times 5}{3 \times 5} = - \frac{20}{15} \end{array}$

So,

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

Since the denominators are the same, add the numerators.

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

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.

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

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:

$\frac{- 25}{30} or \frac{- 5}{6}$