1. **State the problem:** We need to add the fractions $\frac{3}{10}$ and $-\frac{1}{5}$. 2. **Formula and rules:** To add fractions, they must have a common denominator. The sum is given by $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$. 3. **Find a common denominator:** The denominators are 10 and 5. The least common denominator (LCD) is 10. 4. **Convert fractions:** $\frac{3}{10}$ stays the same. Convert $-\frac{1}{5}$ to $-\frac{2}{10}$ by multiplying numerator and denominator by 2. 5. **Add the fractions:** $\frac{3}{10} + (-\frac{2}{10}) = \frac{3 - 2}{10} = \frac{1}{10}$. 6. **Final answer:** The sum is $\frac{1}{10}$.