1. **State the problem:** Simplify the expression $$\frac{5}{4} \times \frac{2}{3} \div \frac{1}{5} + \frac{2}{5} \div \frac{1}{10} \times \frac{3}{4} + \frac{1}{2} \times \frac{4}{3} \div \frac{1}{6}$$. 2. **Recall the rule:** Dividing by a fraction is the same as multiplying by its reciprocal. So, $$a \div \frac{b}{c} = a \times \frac{c}{b}$$. 3. **Rewrite the expression using multiplication by reciprocals:** $$\frac{5}{4} \times \frac{2}{3} \times \frac{5}{1} + \frac{2}{5} \times \frac{10}{1} \times \frac{3}{4} + \frac{1}{2} \times \frac{4}{3} \times \frac{6}{1}$$ 4. **Calculate each term separately:** - First term: $$\frac{5}{4} \times \frac{2}{3} \times \frac{5}{1} = \frac{5 \times 2 \times 5}{4 \times 3 \times 1} = \frac{50}{12}$$ - Second term: $$\frac{2}{5} \times \frac{10}{1} \times \frac{3}{4} = \frac{2 \times 10 \times 3}{5 \times 1 \times 4} = \frac{60}{20}$$ - Third term: $$\frac{1}{2} \times \frac{4}{3} \times \frac{6}{1} = \frac{1 \times 4 \times 6}{2 \times 3 \times 1} = \frac{24}{6}$$ 5. **Simplify each fraction:** - First term: $$\frac{50}{12} = \frac{\cancel{2} \times 25}{\cancel{2} \times 6} = \frac{25}{6}$$ - Second term: $$\frac{60}{20} = \frac{\cancel{20} \times 3}{\cancel{20}} = 3$$ - Third term: $$\frac{24}{6} = \frac{\cancel{6} \times 4}{\cancel{6}} = 4$$ 6. **Add the simplified terms:** $$\frac{25}{6} + 3 + 4 = \frac{25}{6} + \frac{18}{6} + \frac{24}{6} = \frac{25 + 18 + 24}{6} = \frac{67}{6}$$ 7. **Final answer:** $$\boxed{\frac{67}{6}}$$