No, since there are actually five questions, answering them with “The answer is simple: both” would not only be grammatically incorrect but also confusing. This statement provided in the passage is therefore correct: “In fact, the answers are simple: both.”
I can understand your confusion though, for even that correct answer is rather semantically slippery. It’s hard to pin down mentally to which “both” refers. For clarity, I would word that statement as follows:
“In fact, the answer to each question is simple: both.”
Another clear way to word that statement:
“In fact, the answers to all five questions are simple: both.”
The first alternative construction, which treats “answer” as a singular subject, makes it clear that the answer “both” applies to each of the five questions; the second alternative construction, which treats “answers” as a plural subject, makes it clear that the answer “both” applies to all five questions.