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:
Solutions · 1
Official solution
Since 0<y<x<2y: y>2x⟹x−y<2x. So ordering is x−y<y<x<2x+y.
Median =2y+x=10⟹x+y=20 ...(i)
Range =(2x+y)−(x−y)=x+2y=28 ...(ii)
Solving: x=12,y=8.
\[\text{Mean} = \frac{(x-y)+y+x+(2x+y)}{4} = \frac{4x+y}{4} = 12+\frac84=14\]