1. **Stating the problem:** Simplify and verify the expression: $$\frac{11}{4} - \frac{3}{2} - 1 : \left\{ -\frac{17}{3} \cdot \left[ \frac{1}{6} : \left( \frac{5}{3} + 4 \right) \right] + \frac{36}{24} + \left(-\frac{1}{5} - \frac{1}{10}\right) \cdot \left(-\frac{1}{6} - \frac{1}{4}\right) \right\} : \left[ \frac{3}{5} \cdot \left(-\frac{2}{7}\right) \right] = -1$$ 2. **Recall important rules:** - Division of fractions: $a : b = a \times \frac{1}{b}$ - Addition/subtraction of fractions requires common denominators. - Multiplication and division are performed from left to right. 3. **Step-by-step simplification:** - Calculate inside parentheses first: $$\frac{5}{3} + 4 = \frac{5}{3} + \frac{12}{3} = \frac{17}{3}$$ - Compute the division inside the brackets: $$\frac{1}{6} : \frac{17}{3} = \frac{1}{6} \times \frac{3}{17} = \frac{3}{102} = \frac{1}{34}$$ - Multiply by $-\frac{17}{3}$: $$-\frac{17}{3} \times \frac{1}{34} = -\frac{17}{3} \times \frac{1}{34} = -\frac{17}{102} = -\frac{1}{6}$$ - Simplify $\frac{36}{24}$: $$\frac{36}{24} = \frac{3}{2}$$ - Calculate the products in the last term: $$-\frac{1}{5} - \frac{1}{10} = -\frac{2}{10} - \frac{1}{10} = -\frac{3}{10}$$ $$-\frac{1}{6} - \frac{1}{4} = -\frac{2}{12} - \frac{3}{12} = -\frac{5}{12}$$ - Multiply these two results: $$-\frac{3}{10} \times -\frac{5}{12} = \frac{15}{120} = \frac{1}{8}$$ - Sum inside the braces: $$-\frac{1}{6} + \frac{3}{2} + \frac{1}{8}$$ Find common denominator 24: $$-\frac{4}{24} + \frac{36}{24} + \frac{3}{24} = \frac{35}{24}$$ - Calculate the expression in the denominator brackets: $$\frac{3}{5} \times -\frac{2}{7} = -\frac{6}{35}$$ - Calculate the numerator before the braces: $$\frac{11}{4} - \frac{3}{2} - 1 = \frac{11}{4} - \frac{6}{4} - \frac{4}{4} = \frac{11 - 6 - 4}{4} = \frac{1}{4}$$ 4. **Putting it all together:** $$\frac{1}{4} : \frac{35}{24} : \left(-\frac{6}{35}\right) = \frac{1}{4} \times \frac{24}{35} \times \frac{35}{-6}$$ - Cancel $35$ numerator and denominator: $$= \frac{1}{4} \times \frac{24}{\cancel{35}} \times \frac{\cancel{35}}{-6} = \frac{1}{4} \times \frac{24}{-6}$$ - Simplify $\frac{24}{-6} = -4$: $$= \frac{1}{4} \times (-4) = -1$$ 5. **Final answer:** $$-1$$ This matches the given result, confirming the expression is correct.