Question

Which one of these is a rational number that comes between √7 and √8?

5/6
19/7
3/4
6/3

Answer 1

To determine which of the given options is a rational number that comes between \( \sqrt{7} \) and \( \sqrt{8} \), we first need to find the approximate values of \( \sqrt{7} \) and \( \sqrt{8} \):

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

Now we will convert each rational number into decimal form and check if it falls between these two values.

1. **\( \frac{5}{6} \)**

\[
\frac{5}{6} \approx 0.8333
\]

2. **\( \frac{19}{7} \)**

\[
\frac{19}{7} \approx 2.714285714285714
\]

3. **\( \frac{3}{4} \)**

\[
\frac{3}{4} = 0.75
\]

4. **\( \frac{6}{3} \)**

\[
\frac{6}{3} = 2
\]

Now we compare these values with \( \sqrt{7} \) and \( \sqrt{8} \):

- \( \frac{5}{6} \approx 0.8333 \) (not between \( \sqrt{7} \) and \( \sqrt{8} \))
- \( \frac{19}{7} \approx 2.714285714285714 \) (this is between \( \sqrt{7} \) and \( \sqrt{8} \))
- \( \frac{3}{4} = 0.75 \) (not between \( \sqrt{7} \) and \( \sqrt{8} \))
- \( \frac{6}{3} = 2 \) (not between \( \sqrt{7} \) and \( \sqrt{8} \))

Thus, the rational number that comes between \( \sqrt{7} \) and \( \sqrt{8} \) is

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