Find two rational numbers in the form `(p)/(q)` between `0.343443444344443….and 0.363663666366663….`

Options

A. `7/20` and `9/25`
B. `6/20` and `8/25`
C. `9/20` and `3/25`
D. `5/20` and `6/25`

Explanation

To find two rational numbers in the form \( \frac{p}{q} \) between the two given decimal numbers \( 0.343443444344443... \) and \( 0.363663666366663... \), we will follow these steps:

### Step 1: Identify the Decimal Numbers
The two numbers we are considering are:
- \( x = 0.343443444344443... \)
- \( y = 0.363663666366663... \)

### Step 2: Convert the Decimal Numbers to Fractions
To find rational numbers between these two decimals, we first convert them to fractions.

1. **For \( x = 0.343443444344443... \)**:
- This number can be approximated as \( 0.3434 \) (considering the repeating part).
- Converting \( 0.3434 \) to a fraction:
\[ 0.3434 = \frac{3434}{10000} = \frac{1717}{5000} \quad \text{(after simplification)} \]

2. **For \( y = 0.363663666366663... \)**:
- This number can be approximated as \( 0.3636 \).
- Converting \( 0.3636 \) to a fraction:
\[ 0.3636 = \frac{3636}{10000} = \frac{909}{2500} \quad \text{(after simplification)} \]

### Step 3: Find Rational Numbers Between \( x \) and \( y \)
Now that we have the approximate fractions:
- \( x \approx \frac{1717}{5000} \)
- \( y \approx \frac{909}{2500} \)

We can find rational numbers between these two fractions.

1. **First Rational Number**:
- Choose a number like \( 0.35 \):
\[ 0.35 = \frac{35}{100} = \frac{7}{20} \quad \text{(after simplification)} \]
- Check if \( \frac{7}{20} \) is between \( \frac{1717}{5000} \) and \( \frac{909}{2500} \):
\[ \frac{7}{20} = \frac{1750}{5000} \quad \text{(converting to a common denominator)} \]
- Since \( \frac{1717}{5000} < \frac{1750}{5000} < \frac{909}{2500} \), \( \frac{7}{20} \) is a valid rational number.

2. **Second Rational Number**:
- Choose another number like \( 0.36 \):
\[ 0.36 = \frac{36}{100} = \frac{9}{25} \quad \text{(after simplification)} \]
- Check if \( \frac{9}{25} \) is between \( \frac{1717}{5000} \) and \( \frac{909}{2500} \):
\[ \frac{9}{25} = \frac{18}{50} = \frac{3600}{10000} \quad \text{(converting to a common denominator)} \]
- Since \( \frac{1717}{5000} < \frac{3600}{10000} < \frac{909}{2500} \), \( \frac{9}{25} \) is also a valid rational number.

### Final Answer
The two rational numbers between \( 0.343443444344443... \) and \( 0.363663666366663... \) are:
1. \( \frac{7}{20} \)
2. \( \frac{9}{25} \)