Let's solve the given expression step by step.
The given fractions are:
[tex]\[ \frac{-3}{5} + \frac{3}{10} - \frac{1}{5} + \frac{3}{15} \][/tex]
Step 1: Find a common denominator.
The denominators here are 5, 10, 5, and 15. We'll find the least common denominator (LCD) for these values. The least common multiple of 5, 10, and 15 is 30. Thus, our common denominator is 30.
Step 2: Convert each fraction to have the common denominator of 30.
1. For \(\frac{-3}{5}\):
[tex]\[ \frac{-3}{5} = \frac{-3 \times 6}{5 \times 6} = \frac{-18}{30} \][/tex]
2. For \(\frac{3}{10}\):
[tex]\[ \frac{3}{10} = \frac{3 \times 3}{10 \times 3} = \frac{9}{30} \][/tex]
3. For \(\frac{-1}{5}\):
[tex]\[ \frac{-1}{5} = \frac{-1 \times 6}{5 \times 6} = \frac{-6}{30} \][/tex]
4. For \(\frac{3}{15}\):
[tex]\[ \frac{3}{15} = \frac{3 \times 2}{15 \times 2} = \frac{6}{30} \][/tex]
Now, we have:
[tex]\[ \frac{-18}{30} + \frac{9}{30} + \frac{-6}{30} + \frac{6}{30} \][/tex]
Step 3: Sum the numerators over the common denominator.
Combine the fractions:
[tex]\[ \frac{-18 + 9 - 6 + 6}{30} \][/tex]
Calculate the numerator:
[tex]\[ -18 + 9 - 6 + 6 = -18 + 9 = -9 \][/tex]
[tex]\[ -9 - 6 = -15 \][/tex]
[tex]\[ -15 + 6 = -9 \][/tex]
So, the sum of the numerators is -9.
Step 4: Write the result as a simplified fraction.
[tex]\[ \frac{-9}{30} \][/tex]
We can simplify this fraction by dividing the numerator and the denominator by their greatest common divisor, which is 3:
[tex]\[ \frac{-9 \div 3}{30 \div 3} = \frac{-3}{10} \][/tex]
Therefore, the simplified result of the given expression is:
[tex]\[ \boxed{-\frac{1}{30}} \][/tex]
