Calculating the Average of Combined Numbers

This problem involves finding the average of a combined set of numbers when the averages of two separate sets are known. The key is to first find the sum of the numbers in each set and then combine them.

Understanding the Concept of Average

The average (or mean) of a set of numbers is calculated by dividing the sum of all the numbers by the total count of the numbers.

Mathematically, Average = $\frac{\text{Sum of numbers}}{\text{Number of numbers}}$

From this, we can find the sum of numbers if we know the average and the count: Sum of numbers = Average $\times$ Number of numbers.

Finding the Sum of the First Set of Numbers

We are given that the average of p numbers is q².

• Number of numbers in the first set = p
• Average of the first set = q²

Using the formula, the sum of these p numbers is:

Sum of p numbers = Average $\times$ Number of numbers = q² $\times$ p = pq²

Finding the Sum of the Second Set of Numbers

We are given that the average of q numbers is p².

• Number of numbers in the second set = q
• Average of the second set = p²

Using the formula, the sum of these q numbers is:

Sum of q numbers = Average $\times$ Number of numbers = p² $\times$ q = p²q

Calculating the Total Sum and Total Count

To find the average of the combined (p + q) numbers, we need the total sum of all the numbers and the total count of the numbers.

• Total sum of (p + q) numbers = Sum of p numbers + Sum of q numbers
• Total sum = pq² + p²q
• Total number of numbers = Number of p numbers + Number of q numbers
• Total count = p + q

Calculating the Average of (p + q) Numbers

Now we can find the average of the combined set using the total sum and total count.

Average of (p + q) numbers = $\frac{\text{Total sum}}{\text{Total count}}$

Average = $\frac{\text{pq² + p²q}}{\text{p + q}}$

Simplifying the Expression

We can simplify the expression for the average by factoring the numerator.

Numerator: pq² + p²q

Notice that both terms have a common factor of pq.

Factoring out pq, we get: pq(q + p)

So, the average expression becomes:

Average = $\frac{\text{pq(q + p)}}{\text{(p + q)}}$

Assuming that p + q is not equal to zero, we can cancel out the term (p + q) from the numerator and the denominator.

Average = pq

Comparing with Options

The calculated average of (p + q) numbers is pq. Let's look at the given options:

• Option 1: q/p
• Option 2: p + q
• Option 3: pq
• Option 4: p - q

Our calculated average matches Option 3.

Revision Table: Key Concepts for Average Problems

Here's a quick summary of the key concepts used in this problem:

Additional Information: Weighted Average

This problem can also be thought of in terms of a weighted average. When you combine sets with different numbers of elements and different averages, the overall average is a weighted average of the individual averages, where the weights are the number of elements in each set.

For two sets, one with count $n_1$ and average $\text{Avg}_1$, and another with count $n_2$ and average $\text{Avg}_2$, the combined average is:

Combined Average = $\frac{\text{n}_1 \times \text{Avg}_1 + \text{n}_2 \times \text{Avg}_2}{\text{n}_1 + \text{n}_2}$

In our problem:

• $n_1 = p$, $\text{Avg}_1 = q^2$
• $n_2 = q$, $\text{Avg}_2 = p^2$

Using the weighted average formula:

Combined Average = $\frac{\text{p} \times \text{q²} + \text{q} \times \text{p²}}{\text{p + q}}$

This is the same expression we obtained earlier, which simplifies to pq.

This shows that the method of finding total sum and total count is equivalent to using the weighted average formula for combining sets.