Question

What is a rational number with a denominator of 7 that is between 7–√ and 8–√ . Write your answer as an improper fraction.(1 point) $$

Answer 1

To find a rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{7} \), we first need to calculate \( 7 - \sqrt{7} \) and \( 8 - \sqrt{7} \).

1. **Calculate approximations**:
- The square root of 7 is approximately \( \sqrt{7} \approx 2.6457513110645906 \).
- Thus:
\[
7 - \sqrt{7} \approx 7 - 2.6457513110645906 \approx 4.354248688935409
\]
\[
8 - \sqrt{7} \approx 8 - 2.6457513110645906 \approx 5.354248688935409
\]

2. **Set up the inequality**:
We want a rational number \( \frac{a}{7} \) such that:
\[
4.354248688935409 < \frac{a}{7} < 5.354248688935409
\]

3. **Multiply through by 7**:
Multiplying the entire inequality by 7 results in:
\[
7 \times 4.354248688935409 < a < 7 \times 5.354248688935409
\]
Which simplifies to:
\[
30.480740822548863 < a < 37.48074082254886
\]

4. **Identify integer values for \( a \)**:
The integer values for \( a \) must be between 31 and 37:
- Possible integer values for \( a \) are \( 31, 32, 33, 34, 35, 36, 37 \).

5. **Select a value for \( a \)**:
Any of these values will yield a rational number with the desired properties, but we can select \( a = 34 \) for simplicity.

Thus, the rational number with a denominator of 7 that lies between \( 7 - \sqrt{7} \) and \( 8 - \sqrt{7} \) is:

\[
\frac{34}{7}
\]