1. The problem is to convert the repeating decimal $89.6\overline{6}$ into a fraction. 2. Let $x = 89.6\overline{6}$, which means $x = 89.6666\ldots$ with 6 repeating. 3. To isolate the repeating part, multiply $x$ by 10 to shift the decimal one place: $$10x = 896.6666\ldots$$ 4. Subtract the original $x$ from this equation to eliminate the repeating decimal: $$10x - x = 896.6666\ldots - 89.6666\ldots$$ 5. This simplifies to: $$9x = 807$$ 6. Solve for $x$: $$x = \frac{807}{9}$$ 7. Simplify the fraction by dividing numerator and denominator by 9: $$x = \frac{807 \div 9}{9 \div 9} = \frac{89.666\ldots}{1}$$ but since $807 \div 9 = 89.666\ldots$, we need to express $x$ as a mixed number. 8. Alternatively, rewrite $x$ as $89 + 0.6\overline{6}$. 9. Convert $0.6\overline{6}$ to a fraction. Let $y = 0.6\overline{6}$. 10. Multiply $y$ by 10: $$10y = 6.6\overline{6}$$ 11. Subtract $y$ from $10y$: $$10y - y = 6.6\overline{6} - 0.6\overline{6} = 6$$ 12. So, $$9y = 6 \implies y = \frac{6}{9} = \frac{2}{3}$$ 13. Therefore, $x = 89 + \frac{2}{3} = 89 \frac{2}{3}$. 14. The fraction form of $89.6\overline{6}$ is $89 \frac{2}{3}$. 15. Among the given options, $89 \frac{2}{3}$ is correct.