1. **State the problem:** Simplify the expression $$\frac{7}{3} - \frac{1}{3} \times \frac{7}{2} + \frac{15}{8} \times \frac{4}{5} - 3 \div \left( \frac{3}{4} \times \frac{3}{2} \right)$$. 2. **Recall the order of operations:** Perform multiplication and division before addition and subtraction. 3. **Calculate each part step-by-step:** - Multiply $$\frac{1}{3} \times \frac{7}{2} = \frac{7}{6}$$. - Multiply $$\frac{15}{8} \times \frac{4}{5} = \frac{15 \times 4}{8 \times 5} = \frac{60}{40} = \frac{3}{2}$$. - Multiply inside the division denominator: $$\frac{3}{4} \times \frac{3}{2} = \frac{9}{8}$$. - Divide 3 by $$\frac{9}{8}$$: $$3 \div \frac{9}{8} = 3 \times \frac{8}{9} = \frac{24}{9} = \frac{8}{3}$$. 4. **Rewrite the expression with these results:** $$\frac{7}{3} - \frac{7}{6} + \frac{3}{2} - \frac{8}{3}$$ 5. **Find a common denominator to combine all terms:** The common denominator is 6. - $$\frac{7}{3} = \frac{14}{6}$$ - $$\frac{7}{6}$$ stays the same - $$\frac{3}{2} = \frac{9}{6}$$ - $$\frac{8}{3} = \frac{16}{6}$$ 6. **Combine all terms:** $$\frac{14}{6} - \frac{7}{6} + \frac{9}{6} - \frac{16}{6} = \frac{14 - 7 + 9 - 16}{6} = \frac{0}{6} = 0$$ **Final answer:** $$0$$