1. **State the problem:** Simplify the expression $$\frac{11}{2} \times \left( \frac{2}{5} + \frac{1}{2} - \frac{2}{3} \right) + \frac{3}{4} \div \frac{3}{5}.$$\n\n2. **Simplify inside the parentheses:** Find a common denominator for $$\frac{2}{5}, \frac{1}{2}, \text{ and } \frac{2}{3}.$$ The common denominator is 30.\n\n$$\frac{2}{5} = \frac{12}{30}, \quad \frac{1}{2} = \frac{15}{30}, \quad \frac{2}{3} = \frac{20}{30}.$$\n\nSo, inside the parentheses: $$\frac{12}{30} + \frac{15}{30} - \frac{20}{30} = \frac{12 + 15 - 20}{30} = \frac{7}{30}.$$\n\n3. **Multiply $$\frac{11}{2}$$ by $$\frac{7}{30}$$:**\n\n$$\frac{11}{2} \times \frac{7}{30} = \frac{11 \times 7}{2 \times 30} = \frac{77}{60}.$$\n\n4. **Simplify the division $$\frac{3}{4} \div \frac{3}{5}$$:**\n\nDivision of fractions means multiplying by the reciprocal:\n\n$$\frac{3}{4} \div \frac{3}{5} = \frac{3}{4} \times \frac{5}{3}.$$\n\nCancel the common factor 3:\n\n$$\frac{\cancel{3}}{4} \times \frac{5}{\cancel{3}} = \frac{1}{4} \times 5 = \frac{5}{4}.$$\n\n5. **Add the two results:**\n\n$$\frac{77}{60} + \frac{5}{4}.$$\n\nConvert $$\frac{5}{4}$$ to have denominator 60:\n\n$$\frac{5}{4} = \frac{5 \times 15}{4 \times 15} = \frac{75}{60}.$$\n\nSo,\n\n$$\frac{77}{60} + \frac{75}{60} = \frac{77 + 75}{60} = \frac{152}{60}.$$\n\n6. **Simplify $$\frac{152}{60}$$:**\n\nDivide numerator and denominator by 4:\n\n$$\frac{\cancel{152}^{38}}{\cancel{60}^{15}} = \frac{38}{15}.$$\n\n**Final answer:** $$\frac{38}{15}.$$