1. **State the problem:** Simplify the expression $$\left(\frac{8}{9} - \frac{2}{3} + \frac{5}{8}\right) \div \frac{1}{3}$$. 2. **Find a common denominator for the terms inside the parentheses:** The denominators are 9, 3, and 8. The least common denominator (LCD) is 72. 3. **Convert each fraction to have denominator 72:** $$\frac{8}{9} = \frac{8 \times 8}{9 \times 8} = \frac{64}{72}$$ $$\frac{2}{3} = \frac{2 \times 24}{3 \times 24} = \frac{48}{72}$$ $$\frac{5}{8} = \frac{5 \times 9}{8 \times 9} = \frac{45}{72}$$ 4. **Perform the operations inside the parentheses:** $$\frac{64}{72} - \frac{48}{72} + \frac{45}{72} = \frac{64 - 48 + 45}{72} = \frac{61}{72}$$ 5. **Divide by $$\frac{1}{3}$$:** Dividing by a fraction is the same as multiplying by its reciprocal. $$\frac{61}{72} \div \frac{1}{3} = \frac{61}{72} \times \frac{3}{1} = \frac{61 \times 3}{72} = \frac{183}{72}$$ 6. **Simplify the fraction $$\frac{183}{72}$$:** The greatest common divisor (GCD) of 183 and 72 is 3. $$\frac{\cancel{3}61}{\cancel{3}24} = \frac{61}{24}$$ 7. **Final answer:** $$\frac{61}{24}$$ or as a mixed number $$2 \frac{13}{24}$$.