$0.4\overline{23}$ is equivalent to the fraction

Concept & Formula To convert a mixed recurring decimal to a vulgar fraction, subtract the non-repeating part from the entire number to determine the numerator. For the denominator, write as many $9$s as there are repeating digits, followed by as many $0$s as there are non-repeating digits after the decimal point. $$ \text{Fraction} = \frac{\text{Complete Number} - \text{Non-repeating part}}{\text{9s (repeating count) } \text{0s (non-repeating count)}} $$ Step-by-Step Solution Given the mixed recurring decimal: $0.4\overline{23}$ Identify the components after the decimal: The complete sequence is $423$. The non-repeating part is $4$. The repeating part is $23$ (two digits long). Calculate the Numerator: Subtract the non-repeating part from the entire sequence. $$ 423 - 4 = 419 $$ Calculate the Denominator: We have two repeating digits ($2$ and $3$), so we use two $9$s. We have one non-repeating digit ($4$), so we use one $0$. This gives a denominator of $990$. Form the final fraction: $$ \frac{419}{990} $$ Exam Strategy & Shortcut Analyze the denominator first. The decimal $0.4\overline{23}$ has two repeating digits and one non-repeating digit. The denominator MUST be $990$. This instantly eliminates options (a) and (b). Then, quickly do the numerator subtraction $423 - 4 = 419$ to select option (d). Common Pitfall A very common mistake is simply writing the entire number over $990$ (e.g., getting $\frac{423}{990}$ and simplifying incorrectly) or adding the non-repeating part instead of subtracting it. Always remember the rule: "Subtract the stagnant part from the whole". Final Answer Therefore, the correct answer is $\frac{419}{990}$.