1. **State the problem:** Simplify the expression $$\frac{1}{4} \times -\frac{3}{5} + \frac{2}{5} \div -\frac{1}{2}$$. 2. **Recall the rules:** - Multiplying fractions: multiply numerators and denominators. - Dividing fractions: multiply by the reciprocal. - Addition and subtraction require common denominators. 3. **Multiply:** $$\frac{1}{4} \times -\frac{3}{5} = -\frac{3}{20}$$ 4. **Divide:** $$\frac{2}{5} \div -\frac{1}{2} = \frac{2}{5} \times -\frac{2}{1} = -\frac{4}{5}$$ 5. **Add the results:** $$-\frac{3}{20} + -\frac{4}{5} = -\frac{3}{20} - \frac{4}{5}$$ 6. **Find common denominator:** The common denominator of 20 and 5 is 20. 7. **Rewrite fractions:** $$-\frac{3}{20} - \frac{4}{5} = -\frac{3}{20} - \frac{4 \times 4}{5 \times 4} = -\frac{3}{20} - \frac{16}{20}$$ 8. **Add numerators:** $$-\frac{3 + 16}{20} = -\frac{19}{20}$$ **Final answer:** $$-\frac{19}{20}$$