1. Stating the problem: Simplify the expression $$\frac{1}{4} \div \left(3^2 \times \frac{1}{9}\right) + \frac{3}{2} \div \frac{2}{3}$$. 2. Recall the rules: Division by a fraction is the same as multiplication by its reciprocal. Also, powers and multiplication should be handled carefully. 3. Simplify inside the parentheses first: $$3^2 = 9$$ So, $$9 \times \frac{1}{9} = \frac{9}{1} \times \frac{1}{9} = \cancel{9} \times \frac{1}{\cancel{9}} = 1$$ 4. Now the expression becomes: $$\frac{1}{4} \div 1 + \frac{3}{2} \div \frac{2}{3}$$ 5. Division by 1 leaves the number unchanged: $$\frac{1}{4} + \frac{3}{2} \div \frac{2}{3}$$ 6. Convert division by a fraction to multiplication by its reciprocal: $$\frac{3}{2} \times \frac{3}{2}$$ 7. Multiply the fractions: $$\frac{3 \times 3}{2 \times 2} = \frac{9}{4}$$ 8. Now add the two fractions: $$\frac{1}{4} + \frac{9}{4} = \frac{1+9}{4} = \frac{10}{4}$$ 9. Simplify the fraction: $$\frac{10}{4} = \frac{\cancel{10}}{\cancel{4}} = \frac{5}{2}$$ Final answer: $$\frac{5}{2}$$