1. **State the problem:** Simplify the expression $$\frac{1}{2} - \frac{1}{3} \times \frac{7}{2} + \frac{15}{8} \times \frac{4}{5} - 3 \div \frac{3}{4} \times \frac{3}{2}$$. 2. **Recall order of operations:** Multiplication and division are done before addition and subtraction, from left to right. 3. **Calculate each multiplication and division:** - $$\frac{1}{3} \times \frac{7}{2} = \frac{1 \times 7}{3 \times 2} = \frac{7}{6}$$ - $$\frac{15}{8} \times \frac{4}{5} = \frac{15 \times 4}{8 \times 5} = \frac{60}{40}$$ Simplify by canceling common factors: $$\frac{\cancel{60}}{\cancel{40}} = \frac{3}{2}$$ (since 60/20=3 and 40/20=2) - $$3 \div \frac{3}{4} = 3 \times \frac{4}{3} = \frac{3 \times 4}{3} = \frac{12}{3}$$ Simplify: $$\frac{\cancel{12}}{\cancel{3}} = 4$$ - Then multiply by $$\frac{3}{2}$$: $$4 \times \frac{3}{2} = \frac{4 \times 3}{2} = \frac{12}{2} = 6$$ 4. **Rewrite the expression with these results:** $$\frac{1}{2} - \frac{7}{6} + \frac{3}{2} - 6$$ 5. **Find a common denominator to combine fractions:** The denominators are 2 and 6; the least common denominator is 6. - Convert $$\frac{1}{2} = \frac{3}{6}$$ - $$\frac{7}{6}$$ stays the same - Convert $$\frac{3}{2} = \frac{9}{6}$$ 6. **Combine the fractions:** $$\frac{3}{6} - \frac{7}{6} + \frac{9}{6} = \frac{3 - 7 + 9}{6} = \frac{5}{6}$$ 7. **Subtract 6:** Convert 6 to fraction with denominator 6: $$6 = \frac{36}{6}$$ So, $$\frac{5}{6} - \frac{36}{6} = \frac{5 - 36}{6} = \frac{-31}{6}$$ **Final answer:** $$\boxed{\frac{-31}{6}}$$