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

Given $0<y<x<2y$

$x<2y\Rightarrow \frac{x}{2}<y\Rightarrow y>\frac{x}{2}$$\Rightarrow -y<\frac{-x}{2}$$\Rightarrow x-y<\frac{x}{2}$

$\therefore x-y<y<x<2x+y$

$$\begin{gathered} \therefore median=10 Range=28 \\ \Rightarrow \frac{x+y}{2}=10 2x+y-x+y=28 \\ \Rightarrow x+y=20 x+2y=28 \\ \underline{x+2y=28} x=28-2y \\ -y=-8\Rightarrow y=8 \text{\hspace{0.17em}x\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}}=28-16\Rightarrow 12 \end{gathered}$$

$mean=\frac{x-y+y+x+2x+y}{4}=\frac{4x+y}{4}=\frac{48+8}{4}$$\frac{56}{4}=14$