Express $0.29\overline{56}$ in the form $\frac{p}{q}$ (vulgar fraction)

Concept & Formula To convert a mixed recurring decimal into a fraction, subtract the non-repeating digits from the entire sequence to find the numerator. For the denominator, append 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. $$ \text{Fraction} = \frac{\text{Complete Number} - \text{Non-repeating Part}}{\text{9s (for repeating digits)} \text{0s (for non-repeating digits)}} $$ Step-by-Step Solution Given the decimal: $0.29\overline{56}$ 1. Identify the components after the decimal point: - Complete sequence: $2956$ - Non-repeating digits: $29$ (two digits) - Repeating digits: $56$ (two digits under the bar) 2. Compute the numerator: $$ 2956 - 29 = 2927 $$ 3. Compute the denominator: - Two repeating digits $\rightarrow$ two $9$s ($99$) - Two non-repeating digits $\rightarrow$ two $0$s ($00$) - Denominator = $9900$ 4. Combine to form the fraction: $$ \frac{2927}{9900} $$ Exam Strategy & Shortcut Look at the bar position instantly. Since there are two repeating digits ($56$) and two non-repeating digits ($29$), the denominator must be $9900$. This immediately eliminates options (a) and (b). A quick subtraction $2956 - 29 = 2927$ confirms option (c) directly. Common Pitfall Students often place the entire number over $10000$ (Option b) by treating it as a standard terminating decimal, or they forget to subtract the non-repeating part ($29$) from the numerator. Remember, the bar changes the entire denominator logic. Final Answer Therefore, the correct answer is $\frac{2927}{9900}$.