How do you write \( -0.3 \overline{8} \) as a fraction? \begin{tabular}{l}\( -\frac{7}{18} \) \\ \( \frac{7}{18} \) \\ \hline\end{tabular}

To convert \( -0.3 \overline{8} \) to a fraction, we first need to split it into two parts: the non-repeating and the repeating decimal. Let \( x = -0.3 \overline{8} \). Then, we can write the equation as \( x = -0.3 + (-0.0\overline{8}) \). The repeating part can be converted by multiplying by 10 and subtracting the original: Let \( y = -0.0\overline{8} \) Then \( 10y = -0.8\overline{8} \) Subtracting gives: \( 10y - y = -0.8\overline{8} - (-0.0\overline{8}) \Rightarrow 9y = -0.8 \) Solving for \( y \) gives \( y = -\frac{8}{90} = -\frac{4}{45} \). Now, adding \( -0.3 \) (which is \( -\frac{3}{10} \)), we have: \[ x = -\frac{3}{10} - \frac{4}{45} \] Finding a common denominator of 90, we convert each fraction: \[ x = -\frac{27}{90} - \frac{8}{90} = -\frac{35}{90} \] This simplifies to \( -\frac{7}{18} \). So the answer is \( -\frac{7}{18} \). Now for some extra knowledge! Did you know that the concept of repeating decimals has fascinated mathematicians for centuries? It connects beautifully with the discovery of rational numbers, which are simply ratios of two integers. In fact, any repeating or terminating decimal can be expressed as a fraction, showcasing the intrigue of mathematics and its numeric relationships! If you ever find yourself confused while converting repeating decimals to fractions, remember to always multiply to eliminate the repeat, then subtract! A common mistake is forgetting to adjust signs or miscalculating the common denominator. Keep a close eye on your fractions, and you'll turn those pesky decimals into neat rational numbers in no time!