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:** The number is $-42 - 0.1\overline{6}$. 4. **Let $x = 0.1\overline{6}$:** Multiply by 10 to shift decimal: $10x = 1.6\overline{6}$ 5. **Multiply by 10 again to isolate repeating part:** $100x = 16.6\overline{6}$ 6. **Subtract to eliminate repeating decimal:** $$100x - 10x = 16.6\overline{6} - 1.6\overline{6}$$ $$90x = 15$$ 7. **Solve for $x$:** $$x = \frac{15}{90} = \frac{1}{6}$$ 8. **Combine integer and fractional parts:** $$-42.1\overline{6} = -42 - \frac{1}{6} = -\frac{252}{6} - \frac{1}{6} = -\frac{253}{6}$$ 9. **Final answer:** $$-42.1\overline{6} = -\frac{253}{6} = -42 \frac{1}{6}$$