If the average of $$ p $$ numbers is $$ q^2 $$ and that of $$ q $$ numbers is $$ p^2 $$, then the average of $$ (p + q) $$ numbers is:

Question

Accepted Answer

Let the total sum of the $$ p $$ numbers be $$ pq^2 $$, since their average is $$ q^2 $$. Similarly, let the total sum of the $$ q $$ numbers be $$ qp^2 $$, as their average is $$ p^2 $$.

#### Step 1: Total Sum of All Numbers
The total sum of the $$ (p + q) $$ numbers is:  
$$
pq^2 + qp^2
$$

#### Step 2: Average of All Numbers
The average of the $$ (p + q) $$ numbers is given by:  
$$
	ext{Average} = \frac{	ext{Total Sum}}{	ext{Total Count}} = \frac{pq^2 + qp^2}{p + q}
$$

#### Step 3: Simplify the Expression
Factorize the numerator:  
$$
pq^2 + qp^2 = pq(q + p)
$$  
Thus,  
$$
	ext{Average} = \frac{pq(q + p)}{p + q} = pq
$$

Answer

Answer

Answer