\[ \text { QUESTION } 2 \] Write the repeating decimal number \( 2.7 \mathbf{1} \delta \) as an improper fraction Write the repeating decimal number \( 10, \overline{213} \) as an improper fraction

Solve the equation by following steps: - step0: Solve for \(x\): \(x=10+0.\overline{213}\) - step1: Simplify: \(x=\frac{3401}{333}\) Solve the equation \( x=2.7+0.0\overline{1} \). Solve the equation by following steps: - step0: Solve for \(x\): \(x=2.7+0\overline{1}\) - step1: Simplify: \(x=\frac{122}{45}\) Let's convert the repeating decimal numbers into improper fractions step by step. ### 1. Converting \( 2.7\overline{1} \) to an improper fraction Let \( x = 2.7\overline{1} \). To express this as a fraction, we can separate the non-repeating and repeating parts: 1. **Separate the decimal**: \[ x = 2.7 + 0.0\overline{1} \] 2. **Convert \( 2.7 \) to a fraction**: \[ 2.7 = \frac{27}{10} \] 3. **Convert \( 0.0\overline{1} \)**: Let \( y = 0.0\overline{1} \). \[ 10y = 0.1\overline{1} \] Subtracting the original equation from this: \[ 10y - y = 0.1\overline{1} - 0.0\overline{1} \implies 9y = 0.1 \implies y = \frac{1}{90} \] 4. **Combine the fractions**: \[ x = \frac{27}{10} + \frac{1}{90} \] To add these fractions, find a common denominator (which is 90): \[ x = \frac{27 \times 9}{90} + \frac{1}{90} = \frac{243 + 1}{90} = \frac{244}{90} \] Simplifying gives: \[ x = \frac{122}{45} \] ### 2. Converting \( 10.\overline{213} \) to an improper fraction Let \( x = 10.\overline{213} \). 1. **Separate the decimal**: \[ x = 10 + 0.\overline{213} \] 2. **Convert \( 0.\overline{213} \)**: Let \( y = 0.\overline{213} \). \[ 1000y = 213.\overline{213} \] Subtracting the original equation from this: \[ 1000y - y = 213.\overline{213} - 0.\overline{213} \implies 999y = 213 \implies y = \frac{213}{999} \] Simplifying gives: \[ y = \frac{71}{333} \] 3. **Combine the fractions**: \[ x = 10 + \frac{71}{333} = \frac{10 \times 333 + 71}{333} = \frac{3330 + 71}{333} = \frac{3401}{333} \] ### Final Results - The improper fraction for \( 2.7\overline{1} \) is \( \frac{122}{45} \). - The improper fraction for \( 10.\overline{213} \) is \( \frac{3401}{333} \).