First, we determine Richard's reading rate. According to the problem, Richard can read \( \frac{1}{4} \) of a book in \( \frac{4}{5} \) of an hour.
To find the reading rate in terms of books per hour, we set up the equation as follows:
- Determine how much of the book Richard can read in one hour. If he reads \( \frac{1}{4} \) of a book in \( \frac{4}{5} \) hours, we can calculate how much he would read in \( 1 \) hour by scaling up: \[ \text{Reading rate} = \frac{\frac{1}{4}}{\frac{4}{5}} = \frac{1}{4} \times \frac{5}{4} = \frac{5}{16} \text{ books per hour} \]
Now, we need to see how much he can read in one hour. Since his reading rate is \( \frac{5}{16} \) books per hour, we conclude that he reads:
\[ \frac{5}{16} \text{ of a book in one hour} \]
Thus, the answer is:
\[ \boxed{\frac{5}{16}} \]
This corresponds to option B.