a. Which rational numbers are between \( \frac{1}{5} \) and \( 0 . \overline{20} \) ? Select all that apply. A. 0.0020 B. 0.204 C. 0.2013 D. 0.2010 E. 0.2019 F. 0.2023

We first write the two boundary values in decimal form. 1. Convert \( \frac{1}{5} \) to a decimal: \[ \frac{1}{5} = 0.2 \] 2. Interpret \( 0.\overline{20} \) as the repeating decimal: Let \[ x = 0.\overline{20} = 0.202020\ldots \] Multiplying both sides by 100: \[ 100x = 20.202020\ldots \] Subtracting the original equation: \[ 100x - x = 20.202020\ldots - 0.202020\ldots \quad \Longrightarrow \quad 99x = 20 \] Thus: \[ x = \frac{20}{99} \approx 0.202020\ldots \] We are to find which of the following decimals lie strictly between \(0.2\) and approximately \(0.202020\). Examine each option: - **A. \(0.0020\)** \(0.0020 < 0.2\). Not between \(0.2\) and \(0.202020\). - **B. \(0.204\)** \(0.204 > 0.202020\). Not between \(0.2\) and \(0.202020\). - **C. \(0.2013\)** \(0.2 < 0.2013 < 0.202020\). Lies between. - **D. \(0.2010\)** \(0.2 < 0.2010 < 0.202020\). Lies between. - **E. \(0.2019\)** \(0.2 < 0.2019 < 0.202020\). Lies between. - **F. \(0.2023\)** \(0.2023 > 0.202020\). Not between. Thus, the decimals that are between \( \frac{1}{5} \) and \( 0.\overline{20} \) are the ones given in Options **C, D,** and **E**.