You can skip the algebra once you understand the concept. We are just looking at a ratio. So it can be less than 5, or much greater than 5, so long as the ratio of 2/3 is maintained.

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\(\text { For positive numbers } \mathrm{p} \text { and } \mathrm{q}, \frac{p-q}{p+q}=\frac{2}{3}\)

Quantity A	Quantity B
p+q	5

A) Quantity A is greater.
B) Quantity B is greater.
C) The two quantities are equal.
D) The relationship cannot be determined from the information given.

Given:

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

with positive $p$ and $q$.
We want to compare:
- Quantity A $\(=p+q\)$
- Quantity $\(\mathrm{B}=5\)$

Step 1: Express $p$ in terms of $q$
From the equation:

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

cross-multiply:

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


Step 2: Find $\(p+q\)$
Since $\(p=5 q\)$,

$$
\(p+q=5 q+q=6 q\)
$$

Step 3: Compare $\(6 q\)$ to 5
- $\(q>0\)$, so $6 q$ can be any positive number.
- There is no restriction on the magnitude of $q$.
- For example:
- If $\(q=1\)$, then $\(p+q=6\)$, which is greater than 5 .
- If $\(q=0\).5$, then $\(p+q=3\)$, which is less than 5 .

Conclusion:
- Quantity A could be greater than, less than, or equal to Quantity B depending on the value of $q$.
- Therefore, the relationship cannot be determined from the information given.