The correct answer is D. $\mathbf{4 .}$

Here is a step-by-step breakdown of the solution.

Step 1: Define the variables and known values.
- Let $\(\mathbf{A}\)$ be the price per pound of apples.
- Let $\(\mathbf{B}\)$ be the price per pound of bananas.
- The quantity of apples purchased is $\(\mathbf{4}\)$ pounds.
- The quantity of bananas purchased is $\(\mathbf{3}\)$ pounds (since Anna bought one more pound of apples than bananas).

Step 2: Set up the equations.
The problem states that Anna spent the same amount of money on both fruits, so we can set up the following equation:
Cost of Apples = Cost of Bananas
$($ \(Price of Apples\) $) \(\times\)($ \(Quantity of Apples\) $)=($ \(Price of Bananas\) $) \(\times\)($ \(Quantity of Banana\)

$$
\(A \times 4=B \times 3\)
$$


We also know that the price of bananas is \(\$ 1\) more per pound than the price of apples:

$$
\(B=A+1\)
$$

Step 3: Substitute and solve.
Substitute the second equation ( $\(B=A+1\)$ ) into the first equation:

$$
\(A \times 4=(A+1) \times 3\)
$$


Now, solve for A:

$$
\(\begin{gathered}
4 A=3 A+3 \\
4 A-3 A=3 \\
A=3
\end{gathered}\)
$$


So, the price of apples is \(\$ 3\) per pound.

Step 4: Find the price of bananas.
Using the equation $\(B=A+1\)$, we can find the price of bananas:

$$
\(\begin{gathered}
B=3+1 \\
B=4
\end{gathered}\)
$$


The price per pound of bananas is \(\$ 4\).