To find the average of $P + Q$ numbers when given the averages of $P$ and $Q$ numbers, we can use the properties of averages and sums.

Given:

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

First, let's find the total sum of the $P$ numbers: $\text{Sum of } P \text{ numbers} = P \times Q^2$

Next, let's find the total sum of the $Q$ numbers: $\text{Sum of } Q \text{ numbers} = Q \times P^2$

Now, we need to find the total sum of all $P + Q$ numbers combined: $\text{Total sum} = P \times Q^2 + Q \times P^2$

The average of the $P + Q$ numbers is the total sum divided by the total number of numbers: $\text{Average of } (P + Q) \text{ numbers} = \frac{P \times Q^2 + Q \times P^2}{P + Q}$

This can be simplified further if needed, but the expression itself is the required solution.

Therefore, the average of $P + Q$ numbers is: $\frac{P \times Q^2 + Q \times P^2}{P + Q}$

Would you like further details or have any questions?

Here are 5 questions you might want to ask:

1. How do you calculate the average of a combined group of numbers?
2. What are the properties of averages and sums?
3. How can you simplify complex algebraic expressions?
4. Can you provide more examples involving averages and sums?
5. What are some common mistakes to avoid when dealing with averages?

Tip: When solving problems involving averages, always remember that the average is the sum of all the values divided by the number of values. This principle can help you solve more complex problems step by step.