a. Find three rational numbers between $$ \frac{1}{5} $$ and $$ 0 . \overline{20} $$. b. Find three rational numbers between $$ \frac{1}{3} $$ and $$ 0 . \overline{34} $$. $$ \begin{array}{ll} ext { a. Which rational numbers are between } \frac{1}{5} ext { and } 0 . \overline{20} ext { ? Select all that apply. } \ \square ext { A. } 0.0020 & \square ext { B. } 0.204 \ \square ext { C. } 0.2013 \ \square ext { E. } 0.2019 & \square ext { D. } 0.2010 \ ext { b. Which rational numbers are between } \frac{1}{3} ext { and } 0 . \overline{347} ext { ? Select all that apply. } \ \square ext { A. } 0.3303 & \square ext { B. } 0.2023 \ \square ext { C. } 0.3368 & \square ext { D. } 0.7902 \ \square ext { E. } 0.0034 & \square ext { F. } 0.3371\end{array} $$

Question

a. Find three rational numbers between $$ \frac{1}{5} $$ and $$ 0 . \overline{20} $$. b. Find three rational numbers between $$ \frac{1}{3} $$ and $$ 0 . \overline{34} $$. $$ \begin{array}{ll}	ext { a. Which rational numbers are between } \frac{1}{5} 	ext { and } 0 . \overline{20} 	ext { ? Select all that apply. } \ \square 	ext { A. } 0.0020 & \square 	ext { B. } 0.204 \ \square 	ext { C. } 0.2013 \ \square 	ext { E. } 0.2019 & \square 	ext { D. } 0.2010 \ 	ext { b. Which rational numbers are between } \frac{1}{3} 	ext { and } 0 . \overline{347} 	ext { ? Select all that apply. } \ \square 	ext { A. } 0.3303 & \square 	ext { B. } 0.2023 \ \square 	ext { C. } 0.3368 & \square 	ext { D. } 0.7902 \ \square 	ext { E. } 0.0034 & \square 	ext { F. } 0.3371\end{array} $$

UpStudy AI · Accepted Answer

**Part (a)**

1. First, note that
   $$
   \frac{1}{5} = 0.2.
   $$
2. Next, consider the repeating decimal:
   $$
   0.\overline{20} = 0.202020\ldots
   $$
3. We are to choose numbers strictly between $$0.2$$ and $$0.202020\ldots$$. Now, examine the options:

- **Option A: $$0.0020$$**  
     This is much less than $$0.2$$, so it does not lie in the interval.
     
   - **Option B: $$0.204$$**  
     This number is greater than $$0.202020\ldots$$, so it is not between the endpoints.
     
   - **Option C: $$0.2013$$**  
     Since $$0.2 < 0.2013 < 0.202020\ldots$$, this number qualifies.
     
   - **Option D: $$0.2010$$**  
     Likewise, $$0.2 < 0.2010 < 0.202020\ldots$$; it lies in the interval.
     
   - **Option E: $$0.2019$$**  
     Also, $$0.2 < 0.2019 < 0.202020\ldots$$, so this option is acceptable.

4. Therefore, the three rational numbers between $$\frac{1}{5}$$ and $$0.\overline{20}$$ among the choices are the numbers in **Options C, D, and E**.

---

**Part (b)**

1. First, note that
   $$
   \frac{1}{3} \approx 0.3333\ldots
   $$
2. Next, for the repeating decimal:
   $$
   0.\overline{347} = 0.347347347\ldots
   $$
3. We seek numbers strictly between approximately $$0.3333$$ and $$0.347347\ldots$$. Now, check the options:

- **Option A: $$0.3303$$**  
     This is less than $$0.3333$$, so it is not in the required interval.
     
   - **Option B: $$0.2023$$**  
     This is significantly lower than $$0.3333$$.
     
   - **Option C: $$0.3368$$**  
     Since $$0.3333 < 0.3368 < 0.347347\ldots$$, this number qualifies.
     
   - **Option D: $$0.7902$$**  
     This is far greater than $$0.347347\ldots$$.
     
   - **Option E: $$0.0034$$**  
     This is much less than $$0.3333$$.
     
   - **Option F: $$0.3371$$**  
     We observe that $$0.3333 < 0.3371 < 0.347347\ldots$$, so this number also qualifies.

4. Thus, of the given choices, the numbers that lie between $$\frac{1}{3}$$ and $$0.\overline{347}$$ are those in **Options C and F**.

---

**Summary of Answers:**

- For part (a): Options **C (0.2013)**, **D (0.2010)**, and **E (0.2019)**.
- For part (b): Options **C (0.3368)** and **F (0.3371)**.
**Part (a):** - **Option C:** 0.2013 - **Option D:** 0.2010 - **Option E:** 0.2019 **Part (b):** - **Option C:** 0.3368 - **Option F:** 0.3371

a. Find three rational numbers between and .
b. Find three rational numbers between and .

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