Question

If the median and the range of four numbers $\{x,y,2x+y,x-y\}$ where $0<x<y$ are $10$ and $32$ respectively, then the mean of the numbers is

Answer 1

Given $x<y\Rightarrow x-y<0$
Now, arranging given number's in increasing order
$x-y<x<y<2x+y$
$\therefore$Range $=(2x+y)-(x-y)=32$
$\Rightarrow x+2y=32\cdots \left(1\right)$
and median $=\frac{y+x}{2}=10$
$\Rightarrow x+y=20\cdots \left(2\right)$
solving equation (1) and (2)
$x=8,y=12$
$\therefore$ mean $=\frac{x-y+y+x+2x+y}{4}=x+\frac{y}{4}$
$\therefore$ mean $=11$