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Q.
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 :
Statistics
Solution:
Since 0 < y < x < 2y
$\therefore \, y > \frac{x}{2} \Rightarrow \, x - y < \frac{x}{2} $
$\therefore \, x -y < y < x < 2x + y$
Hence median $ = \frac{y + x}{2} = 10 $
$\Rightarrow \, x + y = 20$ ...(i)
And range = (2x + y) - (x - y) = x + 2y
But range = 28
$\therefore $ x + 2y = 28 ...(ii)
From equations (i) and (ii),
x = 12, y = 8
$\therefore $ Mean $ = \frac{\left(x-y\right) +y +x + \left(2x+y\right)}{4} = \frac{4x+y}{4}$
$ = x + \frac{y}{4} = 12 + \frac{8}{4} =14 $