The average of x numbers Is y and average of y numbers is x Then the average of all the numbers taken together is

Options

A. `(x+y)/(2xy)`
B. `(2xy)/(x+y)`
C. `(x^(2) + y^(2))/(x+y)`
D. `(xy)/(x+y)`

Answer

To solve the problem step by step, we will use the concept of averages and the relationship between the sum of numbers and their average.

### Step 1: Understand the given information
We know that:
- The average of \( x \) numbers is \( y \).
- The average of \( y \) numbers is \( x \).

### Step 2: Write the equations for the sums
From the definition of average, we can express the sums of the numbers:
1. The sum of \( x \) numbers (let's call it \( S_x \)) can be calculated as:
\[ S_x = x \times y \]
2. The sum of \( y \) numbers (let's call it \( S_y \)) can be calculated as:
\[ S_y = y \times x \]

### Step 3: Calculate the total sum of all numbers
Now, we need to find the total sum of all the numbers taken together:
\[ S_{total} = S_x + S_y = (x \times y) + (y \times x) = 2xy \]

### Step 4: Calculate the total number of terms
The total number of terms when combining both sets of numbers is:
\[ N_{total} = x + y \]

### Step 5: Calculate the average of all numbers
Now, we can find the average of all the numbers taken together using the formula for average:
\[ \text{Average} = \frac{S_{total}}{N_{total}} = \frac{2xy}{x + y} \]

### Final Answer
Thus, the average of all the numbers taken together is:
\[ \frac{2xy}{x + y} \]

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