1. Problem a) Simplify $\frac{3}{4} \times 6 - \frac{5}{6} \times 4$ and then divide by 2. 2. Use multiplication of fractions and subtraction: $$\frac{3}{4} \times 6 = \frac{3 \times 6}{4} = \frac{18}{4}$$ $$\frac{5}{6} \times 4 = \frac{5 \times 4}{6} = \frac{20}{6}$$ 3. Find common denominator for subtraction: $$\frac{18}{4} = \frac{18 \times 6}{4 \times 6} = \frac{108}{24}$$ $$\frac{20}{6} = \frac{20 \times 4}{6 \times 4} = \frac{80}{24}$$ 4. Subtract: $$\frac{108}{24} - \frac{80}{24} = \frac{108 - 80}{24} = \frac{28}{24}$$ 5. Simplify $\frac{28}{24}$ by dividing numerator and denominator by 4: $$\frac{\cancel{28}^7}{\cancel{24}^6} = \frac{7}{6}$$ 6. Now divide $\frac{7}{6}$ by 2: $$\frac{7}{6} \div 2 = \frac{7}{6} \times \frac{1}{2} = \frac{7}{12}$$ 7. Final answer for a): $\frac{7}{12}$. --- 1. Problem b) Simplify $\frac{3}{-5} + \frac{9}{3} - \frac{5}{-9} + \frac{2}{-15}$. 2. Rewrite with negative signs in numerator: $$-\frac{3}{5} + 3 + \frac{5}{9} - \frac{2}{15}$$ 3. Find common denominator for fractions: 45. Convert fractions: $$-\frac{3}{5} = -\frac{27}{45}$$ $$\frac{5}{9} = \frac{25}{45}$$ $$-\frac{2}{15} = -\frac{6}{45}$$ 4. Sum fractions: $$-\frac{27}{45} + \frac{25}{45} - \frac{6}{45} = \frac{-27 + 25 - 6}{45} = \frac{-8}{45}$$ 5. Add integer 3: $$3 + \left(-\frac{8}{45}\right) = \frac{135}{45} - \frac{8}{45} = \frac{127}{45}$$ 6. Final answer for b): $\frac{127}{45}$. --- 1. Problem c) Simplify $\left(-\frac{2}{5} + \frac{1}{-2}\right) \div \left(\frac{5}{-8} - \frac{1}{2}\right)$. 2. Rewrite negatives: $$-\frac{2}{5} - \frac{1}{2}$$ $$-\frac{5}{8} - \frac{1}{2}$$ 3. Find common denominators: For numerator: 10 $$-\frac{2}{5} = -\frac{4}{10}$$ $$-\frac{1}{2} = -\frac{5}{10}$$ Sum numerator: $$-\frac{4}{10} - \frac{5}{10} = -\frac{9}{10}$$ For denominator: 8 $$-\frac{5}{8} - \frac{1}{2} = -\frac{5}{8} - \frac{4}{8} = -\frac{9}{8}$$ 4. Division of fractions: $$\left(-\frac{9}{10}\right) \div \left(-\frac{9}{8}\right) = -\frac{9}{10} \times -\frac{8}{9}$$ 5. Simplify: $$\frac{\cancel{-9}}{10} \times \frac{8}{\cancel{-9}} = \frac{8}{10} = \frac{4}{5}$$ 6. Final answer for c): $\frac{4}{5}$. --- 1. Problem e) Simplify $-\frac{2}{3} \div -\frac{4}{5} \div -\frac{3}{4}$. 2. Division is left to right: First division: $$-\frac{2}{3} \div -\frac{4}{5} = -\frac{2}{3} \times -\frac{5}{4} = \frac{10}{12} = \frac{5}{6}$$ 3. Second division: $$\frac{5}{6} \div -\frac{3}{4} = \frac{5}{6} \times -\frac{4}{3} = -\frac{20}{18} = -\frac{10}{9}$$ 4. Final answer for e): $-\frac{10}{9}$. --- 1. Problem f) Simplify $\left(-\frac{3}{2}\right)^2 \times \left(-\frac{1}{3}\right)$. 2. Square the first fraction: $$\left(-\frac{3}{2}\right)^2 = \frac{9}{4}$$ 3. Multiply: $$\frac{9}{4} \times -\frac{1}{3} = -\frac{9}{12} = -\frac{3}{4}$$ 4. Final answer for f): $-\frac{3}{4}$.