1. **State the problem:** Simplify the expression $$\frac{5}{6} \div 2 \frac{1}{3} + \frac{1}{2}$$. 2. **Convert mixed number to improper fraction:** $$2 \frac{1}{3} = \frac{2 \times 3 + 1}{3} = \frac{7}{3}$$ 3. **Rewrite the division as multiplication by reciprocal:** $$\frac{5}{6} \div \frac{7}{3} = \frac{5}{6} \times \frac{3}{7}$$ 4. **Multiply the fractions:** $$\frac{5 \times 3}{6 \times 7} = \frac{15}{42}$$ 5. **Simplify the fraction:** $$\frac{15}{42} = \frac{5}{14}$$ because 15 and 42 share a common factor of 3. 6. **Add $$\frac{1}{2}$$ to $$\frac{5}{14}$$:** Find a common denominator: $$\frac{1}{2} = \frac{7}{14}$$ Then add: $$\frac{5}{14} + \frac{7}{14} = \frac{12}{14}$$ 7. **Simplify the sum:** $$\frac{12}{14} = \frac{6}{7}$$ after dividing numerator and denominator by 2. **Final answer:** $$\frac{6}{7}$$