To solve the problem, we need to find the mean of the four numbers \( \{x, y, 2x + y, x - y\} \) given that the median is 10 and the range is 28, with the conditions \( 0 < y < x < 2y \).
### Step 1: Arrange the numbers in ascending order
Given the conditions \( 0 < y < x < 2y \), we can arrange the numbers as follows:
- \( x - y \) (smallest)
- \( y \)
- \( x \)
- \( 2x + y \) (largest)
So, the ordered set is \( \{x - y, y, x, 2x + y\} \).
### Step 2: Find the median
The median of four numbers is the average of the two middle numbers. Here, the two middle numbers are \( y \) and \( x \). Thus, the median is given by:
\[
\text{Median} = \frac{y + x}{2} = 10
\]
From this, we can derive:
\[
y + x = 20 \quad \text{(Equation 1)}
\]
### Step 3: Find the range
The range is calculated as the difference between the maximum and minimum values. Thus, we have:
\[
\text{Range} = (2x + y) - (x - y) = 2x + y - x + y = x + 2y
\]
Given that the range is 28, we have:
\[
x + 2y = 28 \quad \text{(Equation 2)}
\]
### Step 4: Solve the system of equations
Now we have two equations:
1. \( x + y = 20 \)
2. \( x + 2y = 28 \)
We can solve these equations simultaneously. Subtract Equation 1 from Equation 2:
\[
(x + 2y) - (x + y) = 28 - 20
\]
This simplifies to:
\[
y = 8
\]
Now, substitute \( y = 8 \) back into Equation 1:
\[
x + 8 = 20
\]
Thus, we find:
\[
x = 12
\]
### Step 5: Calculate the mean
The mean of the numbers \( x, y, 2x + y, x - y \) is given by:
\[
\text{Mean} = \frac{x + y + (2x + y) + (x - y)}{4}
\]
Substituting the values of \( x \) and \( y \):
\[
\text{Mean} = \frac{12 + 8 + (2 \cdot 12 + 8) + (12 - 8)}{4}
\]
Calculating each term:
- \( 2x + y = 2 \cdot 12 + 8 = 24 + 8 = 32 \)
- \( x - y = 12 - 8 = 4 \)
Now substituting back:
\[
\text{Mean} = \frac{12 + 8 + 32 + 4}{4} = \frac{56}{4} = 14
\]
### Final Answer
The mean of the numbers is \( \boxed{14} \).
