Given Information
1. $p$ and $q$ are positive numbers ( $\(p>0\)$ and $\(q>0\)$ ).
2. The ratio $\(\frac{p-q}{p+q}=\frac{2}{3}\)$.

Quantities to Compare
Quantity A: $\(p+q\)$

Quantity B: 5

Analysis
Step 1: Use the given equation to find the relationship between $p$ and $q$.

We cross-multiply the given equation:

$$
\(\begin{gathered}
\frac{p-q}{p+q}=\frac{2}{3} \\
3(p-q)=2(p+q)
\end{gathered}\)
$$


Distribute the numbers:

$$
\(3 p-3 q=2 p+2 q\)
$$


Isolate $p$ on one side and $q$ on the other:

$$
\(\begin{gathered}
3 p-2 p=2 q+3 q \\
p=5 q
\end{gathered}\)
$$

Step 2: Express Quantity A $\((p+q)\)$ in terms of a single variable.
Since $p=5 q$, we can substitute $5 q$ for $p$ in Quantity A:

Quantity $\(\mathrm{A}=p+q=(5 q)+q=6 q\)$

Step 3: Determine the value of Quantity A based on the constraints.
We know that $q$ is a positive number, so $q>0$.
- If $q$ were equal to 1 , then Quantity A would be $6(1)=6$.
- If $q$ were equal to 0.5 , then Quantity A would be $6(0.5)=3$.
- If $q$ were equal to $5 / 6 \approx 0.833$, then Quantity A would be $\(6(5 / 6)=5\)$.

Since $q$ can be any positive number, Quantity A ( $6 q$ ) can be any positive number greater than 0 .

Conclusion
We compare Quantity A (6q) to Quantity B (5):
- If we choose $q=1$, then $6 q=6$. Quantity A > Quantity B ( $6>5$ ).
- If we choose $q=0.5$, then $6 q=3$. Quantity B > Quantity A ( $3<5$ ).
- If we choose $q=5 / 6$, then $6 q=5$. Quantity A = Quantity B ( $5=5$ ).

Since the relationship between Quantity $A$ and Quantity $B$ changes depending on the choice of the positive number $q$, the relationship cannot be determined.

The correct choice is The relationship cannot be determined from the information given.