1. **State the problem:** Simplify the expression $$\frac{1}{8} + \left( \frac{3}{5} + \frac{5}{6} - \frac{1}{3} \right)$$. 2. **Use the order of operations:** First simplify inside the parentheses. 3. **Find a common denominator for the fractions inside the parentheses:** The denominators are 5, 6, and 3. The least common denominator (LCD) is 30. 4. **Convert each fraction inside the parentheses to have denominator 30:** $$\frac{3}{5} = \frac{3 \times 6}{5 \times 6} = \frac{18}{30}$$ $$\frac{5}{6} = \frac{5 \times 5}{6 \times 5} = \frac{25}{30}$$ $$\frac{1}{3} = \frac{1 \times 10}{3 \times 10} = \frac{10}{30}$$ 5. **Perform the addition and subtraction inside the parentheses:** $$\frac{18}{30} + \frac{25}{30} - \frac{10}{30} = \frac{18 + 25 - 10}{30} = \frac{33}{30}$$ 6. **Simplify the fraction inside the parentheses:** $$\frac{33}{30} = \frac{11}{10}$$ 7. **Now add $$\frac{1}{8}$$ to $$\frac{11}{10}$$:** 8. **Find the LCD of 8 and 10, which is 40:** $$\frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40}$$ $$\frac{11}{10} = \frac{11 \times 4}{10 \times 4} = \frac{44}{40}$$ 9. **Add the fractions:** $$\frac{5}{40} + \frac{44}{40} = \frac{5 + 44}{40} = \frac{49}{40}$$ 10. **Final answer:** $$\frac{49}{40}$$ or as a mixed number $$1 \frac{9}{40}$$. This is the simplified result of the original expression.