First, let's find the values of \( \sqrt{8} \) and \( \sqrt{9} \):
\[ \sqrt{8} = 2\sqrt{2} \approx 2.828 \]
\[ \sqrt{9} = 3 \]
Thus, we are looking for a rational number \( \frac{a}{7} \) such that \( 2.828 < \frac{a}{7} < 3 \).
To isolate \( a \), we will multiply all parts of the inequality by 7:
\[ 2.828 \times 7 < a < 3 \times 7 \]
Calculating the left side:
\[ 2.828 \times 7 \approx 19.796 \]
And the right side:
\[ 3 \times 7 = 21 \]
This gives us the inequality:
\[ 19.796 < a < 21 \]
The integer values for \( a \) that fall between 19.796 and 21 are 20. Thus, we can choose:
\[ a = 20 \]
Now, expressing it as an improper fraction:
\[ \frac{20}{7} \]
To confirm that this satisfies our original inequality:
Calculating \( \frac{20}{7} \):
\[ \frac{20}{7} \approx 2.857 \]
Now we check:
\[ 2.828 < 2.857 < 3 \]
This holds true, so the rational number with a denominator of 7 that is between \( \sqrt{8} \) and \( \sqrt{9} \) is:
\[ \boxed{\frac{20}{7}} \]
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