1. **State the problem:** Simplify the expression $$\frac{\left(\frac{3}{2} \cdot \frac{1}{4} + \frac{2}{3} - \frac{1}{4}\right) \cdot 6}{\left(\frac{3}{2} - \frac{1}{2}\right)^{-1} : \left(\frac{3}{2} + \frac{1}{4} - \frac{1}{6}\right)}$$ 2. **Simplify the numerator:** Calculate the product: $$\frac{3}{2} \cdot \frac{1}{4} = \frac{3}{8}$$ Sum inside numerator parentheses: $$\frac{3}{8} + \frac{2}{3} - \frac{1}{4}$$ Find common denominator 24: $$\frac{9}{24} + \frac{16}{24} - \frac{6}{24} = \frac{9 + 16 - 6}{24} = \frac{19}{24}$$ Multiply by 6: $$\frac{19}{24} \cdot 6 = \frac{19 \cdot 6}{24} = \frac{114}{24}$$ Simplify fraction: $$\frac{\cancel{114}^{19} \cdot \cancel{6}^1}{\cancel{24}^{4} \cdot \cancel{6}^1} = \frac{19}{4}$$ 3. **Simplify the denominator:** First part: $$\frac{3}{2} - \frac{1}{2} = \frac{2}{2} = 1$$ Raise to power -1: $$1^{-1} = 1$$ Second part: $$\frac{3}{2} + \frac{1}{4} - \frac{1}{6}$$ Find common denominator 12: $$\frac{18}{12} + \frac{3}{12} - \frac{2}{12} = \frac{19}{12}$$ Division in denominator means: $$1 : \frac{19}{12} = 1 \cdot \frac{12}{19} = \frac{12}{19}$$ 4. **Combine numerator and denominator:** $$\frac{\frac{19}{4}}{\frac{12}{19}} = \frac{19}{4} \cdot \frac{19}{12} = \frac{361}{48}$$ 5. **Final answer:** $$\boxed{\frac{361}{48}}$$