According to the question, $P = (40\% \text{ of } A) + (65\% \text{ of } B)$

and $Q = (50\% \text{ of } A) + (50\% \text{ of } B)$, where $A > B$

Now, $P - Q = (40\% \text{ of } A) + (65\% \text{ of } B) - (50\% \text{ of } A) - (50\% \text{ of } B) = (15\% \text{ of } B) - (10\% \text{ of } A)$

We know that $A > B$, but not by how much.

Hence, all three cases are possible, as illustrated below:

1. If $B = 100$ and $A = 110$, then $P - Q = (15\% \text{ of } B) - (10\% \text{ of } A) = 15 - 11 = 4$. So, $P > Q$

2. If $B = 100$ and $A = 150$, then $P - Q = (15\% \text{ of } B) - (10\% \text{ of } A) = 15 - 15 = 0$. So, $P = Q$

3. If $B = 100$ and $A = 200$, then $P - Q = (15\% \text{ of } B) - (10\% \text{ of } A) = 15 - 20 = -5$. So, $P < Q$ Hence, nothing can be said about the relation between $P$ and $Q$.