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The arithmetic mean of $x,y$ and $z$ is $80$, and that of $x,y,z,u$ and $v$ is $75$, where $u=\left (x+y \right)/2$ and $v=\left (y+z \right)/2$. If $x\geq z$, then the minimum possible value of $x$ is ____________
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Given that,

$ \frac{x+y+z}{3} = 80 $

$ \Rightarrow x+y+z = 240 \quad \longrightarrow (1) $

And, $\frac{x+y+z+u+v}{5} = 75 $

$ \Rightarrow x+y+z+u+v = 375 \quad \longrightarrow {2} $

Also, $ u = (x+y)/2 \; , \; v = (y+z)/2 $

Put the value of $u$ and $v$ in the equation $(2),$ we get

$ x+y+z+\frac{x+y}{2}+\frac{y+z}{2} = 375 $

$ \Rightarrow \frac{2(x+y+z) + x+y+z+y}{2} = 375 $

$ \Rightarrow 2(240) + 240 + y = 750 $

$ \Rightarrow 720 + y = 750 $

$ \Rightarrow \boxed{y = 30} $

Put the value of $y$ in equation $(1),$ we get

$ x+y+z = 240 $

$ \Rightarrow x+30+z = 240 $

$ \Rightarrow x+z = 240 – 30 $

$ \Rightarrow x+z = 210 \quad \longrightarrow (3) $

Since, $ x \geqslant z, \; x $ takes the minimum possible value, when $ x = z.$

From equation $(3),$

$ x+z = 210 $

$ \Rightarrow x+x = 210 $

$ \Rightarrow 2x = 210 $

$ \Rightarrow \boxed{x = 105} $

$\therefore$ The minimum possible value of $x$ is $105.$

Correct Answer $:105$

$\textbf{PS:}$ The arithmetic mean is the sum of all the numbers in a data set divided by the quantity of numbers in that set.

The arithmetic mean $\overline{x}$ of a collection of $n$ numbers $(\text{from}\; a_1$​ through $a_n​)$ is given by the formula:

$$\overline{x}=\displaystyle \frac{1}{n}\sum_{i=1}^n a_i = \frac{a_1+a_2+a_3+\dots + a_n}{n}.\ _\square$$

Reference: https://brilliant.org/wiki/arithmetic-mean/

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