If the median and the range of four numbers $$\{x, y, 2x + y, x - y\}$$, where $$0 < y < x < 2y$$, are 10 and 28 respectively, then the mean of the numbers is :

We are given four numbers: $$\{x, y, 2x + y, x - y\}$$ with the condition $$0 < y < x < 2y$$. The median is 10 and the range is 28. We need to find the mean of these numbers.

First, we arrange the numbers in ascending order. Given $$0 < y < x < 2y$$, we compare the numbers:

• Since $$y < x$$, we have $$x - y > 0$$.
• Also, $$x < 2y$$ implies $$x - y < 2y - y = y$$, so $$x - y < y$$.
• Now, $$y < x$$ is given.
• For $$2x + y$$, since $$x > y$$, we have $$2x + y > 2y + y = 3y$$. Also, $$x < 2y$$ implies $$2x + y < 4y + y = 5y$$. Moreover, $$2x + y > x$$ because $$2x + y - x = x + y > 0$$ (as both are positive). Similarly, $$2x + y > y$$ and $$2x + y > x - y$$.

Thus, the order is: $$x - y < y < x < 2x + y$$. The numbers in ascending order are: $$x - y$$, $$y$$, $$x$$, $$2x + y$$.

Since there are four numbers, the median is the average of the second and third numbers. The second number is $$y$$ and the third is $$x$$, so:

$$\text{Median} = \frac{y + x}{2} = 10$$

Multiplying both sides by 2:

$$x + y = 20 \quad \text{(Equation 1)}$$

The range is the difference between the largest and smallest numbers. The largest is $$2x + y$$ and the smallest is $$x - y$$, so:

$$\text{Range} = (2x + y) - (x - y) = 2x + y - x + y = x + 2y$$

Given that the range is 28:

$$x + 2y = 28 \quad \text{(Equation 2)}$$

We now have a system of equations:

$$\begin{cases} x + y = 20 \\ x + 2y = 28 \end{cases}$$

Subtract Equation 1 from Equation 2:

$$(x + 2y) - (x + y) = 28 - 20$$

Simplify:

$$x + 2y - x - y = 8$$

$$y = 8$$

Substitute $$y = 8$$ into Equation 1:

$$x + 8 = 20$$

$$x = 20 - 8$$

$$x = 12$$

Check the condition $$0 < y < x < 2y$$: $$y = 8$$, $$x = 12$$, $$2y = 16$$, and $$0 < 8 < 12 < 16$$ holds true.

Now, find the four numbers:

• $$x = 12$$
• $$y = 8$$
• $$2x + y = 2 \times 12 + 8 = 24 + 8 = 32$$
• $$x - y = 12 - 8 = 4$$

The numbers are 4, 8, 12, 32.

The mean is the sum of the numbers divided by 4:

$$\text{Mean} = \frac{4 + 8 + 12 + 32}{4} = \frac{56}{4} = 14$$

Verify the median and range:

• Ascending order: 4, 8, 12, 32. Median = $$\frac{8 + 12}{2} = \frac{20}{2} = 10$$, correct.
• Range = $$32 - 4 = 28$$, correct.

Hence, the mean is 14, which corresponds to Option D.

So, the answer is $$14$$.