Find the sum and then classify it.

To evaluate \( \frac{5}{6} + \sqrt{91} \), we first need to determine the value of \( \sqrt{91} \):

1. Calculate \( \sqrt{91} \):
- The approximate value of \( \sqrt{91} \) is roughly \( 9.539 \) (since \( 9.539 \times 9.539 \approx 91 \)).

2. Now, add \( \frac{5}{6} \):
- Convert \( \frac{5}{6} \) to a decimal, which is approximately \( 0.8333 \).

3. Adding these values together:
\[
0.8333 + 9.539 \approx 10.3723
\]

Now let's classify \( 10.3723 \):

- The sum \( 10.3723 \) is a decimal and is not a fraction of two integers, meaning it could be classified as irrational.
- However, it is also approximated, so the exact nature of \( \sqrt{91} \) being irrational means that the sum \( \frac{5}{6} + \sqrt{91} \) is also irrational.

Given these conclusions, the most accurate representation from the options provided would be:

**10.3727253 . .., irrational**

This reflects the infinite nature of the decimal expansion from the contributions of the irrational number \( \sqrt{91} \).