Converting the following recurring decimals to common fractions in the simplest form: \( \begin{array}{lll}\text { (a) } 0, \dot{8} & \text { (b) } 0, \dot{6} \dot{3} & \text { (c) } 0, \dot{3} 1 \dot{2}\end{array} \)

Sure, let's convert each of the recurring decimals to their simplest fractional forms. ### (a) \( 0, \dot{8} \) (which represents \( 0.\overline{8} \)) **Step-by-Step Conversion:** 1. Let \( x = 0.\overline{8} \). 2. Multiply both sides by 10 to shift the decimal: \( 10x = 8.\overline{8} \). 3. Subtract the original equation from this result: \[ 10x - x = 8.\overline{8} - 0.\overline{8} \Rightarrow 9x = 8 \] 4. Solve for \( x \): \[ x = \frac{8}{9} \] **Simplest Form:** \( \frac{8}{9} \) --- ### (b) \( 0, \dot{6}\dot{3} \) (which represents \( 0.\overline{63} \)) **Step-by-Step Conversion:** 1. Let \( x = 0.\overline{63} \). 2. Multiply both sides by 100 to shift the decimal two places: \( 100x = 63.\overline{63} \). 3. Subtract the original equation from this result: \[ 100x - x = 63.\overline{63} - 0.\overline{63} \Rightarrow 99x = 63 \] 4. Solve for \( x \): \[ x = \frac{63}{99} \] 5. Simplify the fraction by dividing numerator and denominator by 9: \[ \frac{63}{99} = \frac{7}{11} \] **Simplest Form:** \( \frac{7}{11} \) --- ### (c) \( 0, \dot{3}1\dot{2} \) (which represents \( 0.3121212\ldots \) or \( 0.3\overline{12} \)) **Step-by-Step Conversion:** 1. Let \( x = 0.3\overline{12} \). 2. Break it into two parts: the non-repeating part \( 0.3 \) and the repeating part \( 0.\overline{12} \). - **Non-repeating Part:** \( 0.3 = \frac{3}{10} \) - **Repeating Part:** 1. Let \( y = 0.\overline{12} \). 2. Multiply by 100 to shift the decimal two places: \( 100y = 12.\overline{12} \). 3. Subtract the original equation: \( 100y - y = 12.\overline{12} - 0.\overline{12} \Rightarrow 99y = 12 \). 4. Solve for \( y \): \( y = \frac{12}{99} = \frac{4}{33} \). 5. Since this part starts after the first decimal place, divide by 10: \( \frac{4}{33} \times \frac{1}{10} = \frac{4}{330} \). 3. Combine both parts: \[ x = \frac{3}{10} + \frac{4}{330} = \frac{99}{330} + \frac{4}{330} = \frac{103}{330} \] **Simplest Form:** \( \frac{103}{330} \) --- **Final Answers:** - **(a)** \( \frac{8}{9} \) - **(b)** \( \frac{7}{11} \) - **(c)** \( \frac{103}{330} \)