Given:

• The average of $P$ numbers is $Q^2$.
• The average of $Q$ numbers is $P^2$.

We previously found that the average of $P + Q$ numbers is: $\frac{P \times Q^2 + Q \times P^2}{P + Q}$

We need to match this result with one of the provided options. Let's examine each option to see if it matches our result.

1. $P - Q$
2. $PQ$
3. $P + Q$
4. $2PQ$

Let's simplify our expression and check each option:

$\frac{P \times Q^2 + Q \times P^2}{P + Q}$

Factor out a common term from the numerator:

$\frac{PQ(P + Q)}{P + Q}$

Since $P + Q$ is not zero, we can cancel $P + Q$ in the numerator and denominator:

$\frac{PQ(P + Q)}{P + Q} = PQ$

So, the average of $P + Q$ numbers is $PQ$.

Thus, the correct answer is:

2. $PQ$

Would you like further details or have any questions?

Here are 5 questions you might want to ask:

1. How do you factor common terms in algebraic expressions?
2. What are the properties of arithmetic operations in averages?
3. Can you provide more examples of simplifying complex expressions?
4. How do you verify solutions to multiple-choice math problems?
5. What are some tips for solving algebra problems involving averages?

Tip: Always check your final expression against all given options in multiple-choice questions to ensure you have the correct solution.