Question

What is a rational number with a denominator of 7 that is between √7 and √8 . Write your answer as an improper fraction.

Answer 1

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

Calculating these values:

\[
\sqrt{7} \approx 2.6457513110645906
\]
\[
\sqrt{8} = 2\sqrt{2} \approx 2.8284271247461903
\]

Next, we note that we want to find a rational number of the form \( \frac{a}{7} \) for some integer \( a \) that falls between \( \sqrt{7} \) and \( \sqrt{8} \). This means we need to solve the inequalities:

\[
\sqrt{7} < \frac{a}{7} < \sqrt{8}
\]

Multiplying all parts of this inequality by 7 gives:

\[
7\sqrt{7} < a < 7\sqrt{8}
\]

Calculating the lower and upper bounds:

\[
7\sqrt{7} \approx 7 \times 2.6457513110645906 \approx 18.520258177452134
\]
\[
7\sqrt{8} = 7 \times 2.8284271247461903 \approx 19.79898987322333
\]

Thus, we have:

\[
18.520258177452134 < a < 19.79898987322333
\]

Since \( a \) must be an integer, the only integer \( a \) that satisfies this inequality is \( 19 \).

Therefore, the rational number we seek is:

\[
\frac{19}{7}
\]

To verify, we check if \( \frac{19}{7} \) lies between \( \sqrt{7} \) and \( \sqrt{8} \):

Calculating \( \frac{19}{7} \):

\[
\frac{19}{7} \approx 2.7142857142857144
\]

Now we check the inequalities:

\[
\sqrt{7} \approx 2.6457513110645906 < 2.7142857142857144 < 2.8284271247461903 \approx \sqrt{8}
\]

Thus, \( \frac{19}{7} \) is indeed between \( \sqrt{7} \) and \( \sqrt{8} \).

The answer is:

\[
\boxed{\frac{19}{7}}
\]