1. **State the problem:** Simplify the expression $$-\frac{6}{5} \times \frac{1}{4} + \frac{4}{5} \div \frac{5}{6}$$. 2. **Recall the rules:** - Multiplication and division of fractions: $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$ and $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. - Addition of fractions requires a common denominator. 3. **Calculate each part:** - Multiply $$-\frac{6}{5} \times \frac{1}{4} = -\frac{6 \times 1}{5 \times 4} = -\frac{6}{20}$$. - Simplify $$-\frac{6}{20}$$ by dividing numerator and denominator by 2: $$-\frac{\cancel{6}^{3}}{\cancel{20}^{10}} = -\frac{3}{10}$$. 4. **Divide $$\frac{4}{5} \div \frac{5}{6}$$:** - Rewrite division as multiplication by reciprocal: $$\frac{4}{5} \times \frac{6}{5} = \frac{4 \times 6}{5 \times 5} = \frac{24}{25}$$. 5. **Add the two results:** $$-\frac{3}{10} + \frac{24}{25}$$. 6. **Find common denominator:** - The least common denominator of 10 and 25 is 50. - Convert fractions: $$-\frac{3}{10} = -\frac{3 \times 5}{10 \times 5} = -\frac{15}{50}$$ $$\frac{24}{25} = \frac{24 \times 2}{25 \times 2} = \frac{48}{50}$$ 7. **Add the fractions:** $$-\frac{15}{50} + \frac{48}{50} = \frac{-15 + 48}{50} = \frac{33}{50}$$. **Final answer:** $$\frac{33}{50}$$.