The correct answer is: in.
The preposition “in” is used to indicate a manner or way of doing something. In this case, the speaker is saying that the woman explained the solution in a patient manner. This means that she was calm and understanding, and took the time to explain things clearly.
The other options are not correct because they do not indicate a manner or way of doing something. The preposition “of” is used to indicate possession or association. The preposition “at” is used to indicate a location or time. The preposition “on” is used to indicate a surface or position.
Here are some examples of how the preposition “in” can be used to indicate a manner or way of doing something:
- She spoke in a loud voice.
- He acted in a friendly manner.
- They worked in a careful manner.