1. **State the problem:** Convert the repeating decimal $-42.1\overline{6}$ to a fraction. 2. **Understand the notation:** The bar over 6 means the digit 6 repeats infinitely: $-42.1666\ldots$ 3. **Separate the integer and decimal parts:** $$-42.1\overline{6} = -42 - 0.1\overline{6}$$ 4. **Focus on converting $0.1\overline{6}$ to a fraction:** Let $x = 0.1\overline{6} = 0.1666\ldots$ 5. Multiply $x$ by 10 to shift one decimal place: $$10x = 1.6666\ldots$$ 6. Multiply $x$ by 100 to shift two decimal places: $$100x = 16.6666\ldots$$ 7. Subtract the two equations: $$100x - 10x = 16.6666\ldots - 1.6666\ldots$$ $$90x = 15$$ 8. Solve for $x$: $$x = \frac{15}{90}$$ 9. Simplify the fraction by dividing numerator and denominator by 15: $$x = \frac{\cancel{15}^1}{\cancel{90}^6} = \frac{1}{6}$$ 10. So, $$0.1\overline{6} = \frac{1}{6}$$ 11. Now combine with the integer part: $$-42.1\overline{6} = -42 - \frac{1}{6} = -\left(42 + \frac{1}{6}\right) = -\frac{252}{6} - \frac{1}{6} = -\frac{253}{6}$$ **Final answer:** $$-42.1\overline{6} = -\frac{253}{6}$$