Question

The average of six numbers is $x$ and the average of three of these is $y$. If the average of the remaining three is $z$, then :

• A. $x=y+z$
• B. $2x=y+z$
• C. $x=2y+2z$
• D. None of these

Answer 1

The correct option is B $2x=y+z$

Let the six numbers be $a_1,a_2,a_3,a_4,a_5,a_6$

Given that the average of six numbers is $x$

Therefore, $x=\frac{a_1+a_2+a_3+a_4+a_5+a_6}{6}$

$⟹a_1+a_2+a_3+a_4+a_5+a_6=6x$ ------(1)

Given that the average of three of the numbers is $y$

Let the three numbers from the six be $a_1,a_2,a_3$

Therefore, $y=\frac{a_1+a_2+a_3}{3}$

$⟹a_1+a_2+a_3=3y$ ------(2)

Given that the average of remaining three numbers is $z$

Therefore, $z=\frac{a_4+a_5+a_6}{3}$

$⟹a_4+a_5+a_6=3z$ ------(3)

substituting (2) and (3) in (1) we get

$3y+3z=6x$

$⟹y+z=2x$