1. **Stating the problem:** We want to learn how to convert recurring decimals into fractions and vice versa. 2. **Converting recurring decimals to fractions:** - Suppose the recurring decimal is $x = 0.\overline{a}$ where $a$ is the repeating part. - Multiply $x$ by $10^n$ where $n$ is the length of the repeating block to shift the decimal point right after the repeating part. - Subtract the original $x$ from this new number to eliminate the repeating part. - Solve for $x$ to get the fraction. **Example:** Convert $0.\overline{3}$ to a fraction. - Let $x = 0.333...$ - Multiply by 10: $10x = 3.333...$ - Subtract: $10x - x = 3.333... - 0.333... = 3$ - So, $9x = 3 \Rightarrow x = \frac{3}{9} = \frac{1}{3}$. 3. **Converting fractions to recurring decimals:** - Divide numerator by denominator. - If division ends, decimal is terminating. - If division repeats, the repeating remainder sequence forms the recurring decimal. **Example:** Convert $\frac{1}{3}$ to decimal. - Divide 1 by 3: 0 remainder 1. - Multiply remainder by 10: 10 divided by 3 is 3 remainder 1. - Remainder repeats, so decimal is $0.\overline{3}$. This method works for any recurring decimal or fraction conversion.