Question

Perform the indicated operation. Write all results in lowest terms. $$-\frac{2}{5}+\left(-\frac{1}{4}\right)$$

Short answer

Answer: The sum of the given fractions is $\frac{-13}{20}$.

Step-by-step solution

Identify the common denominator

The given fractions have denominators 4 and 5. The least common denominator (LCD) or the smallest number that both denominators can divide into is found by finding the least common multiple (LCM) of the two denominators. LCM(4, 5) = 20. So, our common denominator is 20.

Rewrite the fractions with the common denominator

To rewrite these fractions with the common denominator (20), we need to multiply both the numerator and the denominator of each fraction by an appropriate value such that their denominator becomes 20. For the first fraction, multiply the numerator and the denominator by 4. For the second fraction, multiply the numerator and the denominator by 5. $$-\frac{2}{5}\cdot\frac{4}{4}+\frac{-1}{4}\cdot\frac{5}{5}=-\frac{8}{20}+\left(-\frac{5}{20}\right)$$

Add the numerators

Now that both fractions have the same denominator (20), we can add the numerators and place the result over the common denominator. $$-\frac{8}{20}+\left(-\frac{5}{20}\right)=\frac{-8-5}{20}$$

Simplify the fraction

Calculate the value in the numerator and simplify it, if possible. $$\frac{-8-5}{20}=\frac{-13}{20}$$ Since the numerator and the denominator have no common factors other than 1, the given fraction is already in its lowest terms. So, the final answer is: $$-\frac{2}{5}+\left(-\frac{1}{4}\right)=\frac{-13}{20}$$