The value of $4.1\overline{2}$ is

Concept & Formula For mixed recurring decimals with a whole number component, separate the integer part from the fractional component. Convert the decimal part using the standard mixed recurring profile: $$ 4.1\overline{2} = 4 + 0.1\overline{2} $$ Step-by-Step Solution Given expression: $4.1\overline{2}$ 1. Separate the whole number: $$ 4 + 0.1\overline{2} $$ 2. Convert $0.1\overline{2}$ into a fraction: - Numerator: Complete number minus non-repeating part $\rightarrow 12 - 1 = 11$ - Denominator: One repeating digit ($9$) and one non-repeating digit ($0$) $\rightarrow 90$ $$ 0.1\overline{2} = \frac{11}{90} $$ 3. Combine back into a mixed fraction: $$ 4\frac{11}{90} $$ Exam Strategy & Shortcut Quickly check the decimal component $0.1\overline{2}$. The denominator must contain one $9$ and one $0$ (yielding $90$). This immediately points directly to option (a) without needing to evaluate the rest of the options. Common Pitfall A common error is writing the fraction as $\frac{12}{99}$ or $\frac{11}{99}$ by ignoring the fact that only the digit $2$ repeats while the digit $1$ does not. Always differentiate between repeating and fixed decimal positions. Final Answer Therefore, the correct answer is $4\frac{11}{90}$.