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 of fractions: $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$. - Division of fractions: $$\frac{a}{b} \div \frac{c}{d} = \frac{a}{b} \times \frac{d}{c}$$. - Addition requires a common denominator. 3. **Calculate the multiplication part:** $$\frac{6}{5} \times \frac{1}{4} = \frac{6 \times 1}{5 \times 4} = \frac{6}{20}$$ Simplify by canceling common factor 2: $$\frac{\cancel{6}^{3}}{\cancel{20}^{10}} = \frac{3}{10}$$ 4. **Calculate the division part:** $$\frac{4}{5} \div \frac{5}{6} = \frac{4}{5} \times \frac{6}{5} = \frac{4 \times 6}{5 \times 5} = \frac{24}{25}$$ 5. **Add the two results:** $$\frac{3}{10} + \frac{24}{25}$$ Find common denominator: 50 $$\frac{3}{10} = \frac{3 \times 5}{10 \times 5} = \frac{15}{50}$$ $$\frac{24}{25} = \frac{24 \times 2}{25 \times 2} = \frac{48}{50}$$ 6. **Add numerators:** $$\frac{15}{50} + \frac{48}{50} = \frac{15 + 48}{50} = \frac{63}{50}$$ 7. **Final answer:** $$\frac{63}{50}$$ or as a mixed number $$1 \frac{13}{50}$$.