Question

Perform the indicated operation. \(-9 \frac{3}{5}+\left(-3 \frac{4}{15}\right)\)

Short answer

Answer: \(\frac{-167}{15}\)

Step-by-step solution

Convert to improper fractions

To convert the mixed numbers into improper fractions, we'll multiply the whole number by the denominator and then add the numerator. For \(-9 \frac{3}{5}\), this gives: \(-9 \times 5 + 3 = -45 + 3 = -42\). So, the improper fraction is \(\frac{-42}{5}\). Next, for \(-3 \frac{4}{15}\), we have: \(-3 \times 15 + 4 = -45 + 4 = -41\). Thus, the improper fraction is \(\frac{-41}{15}\). Now we have our two improper fractions: \( \frac{-42}{5} + \frac{-41}{15}\).

Find a common denominator

We need to find a common denominator that both 5 and 15 can divide into. The least common denominator (LCD) is the least common multiple (LCM) of the two denominators. lcm(5, 15) = 15, so our common denominator will be 15.

Add the fractions

To add the two fractions using the common denominator, we need to adjust the numerators of each fraction as follows: \( \frac{-42}{5} \times \frac{3}{3} = \frac{-126}{15}\) (multiply both the numerator and denominator by 3 to get a denominator of 15) Now we can add the two fractions: \(\frac{-126}{15}+\frac{-41}{15}=\frac{-126-41}{15}=\frac{-167}{15}\)

Simplify the fraction

Currently, the result is \(\frac{-167}{15}\). Since there are no common factors for the numerator and denominator other than 1, this fraction is already in its simplest form. So, the final answer is: \(\frac{-167}{15}\).