Question

The average of six numbers is $x$ and the average of three of these is $y$. The average of the remaining three is $z$, then
$A.)$ $x = y + z$
$B.)$ $2x = y + z$
$C.)$ $x = 2y + 2z$
$D.)$ $x = y2z$

Answer 1

First of all we need to calculate the total of these six numbers. Also, we have to calculate the total of three numbers as well as the total of the remaining three numbers. Then we will try to put it into a general equation form.

Formula used: Average of ${x_1},{x_2},{x_3},......,{x_n} = \dfrac{{{x_1} + {x_2} + {x_3} + ..... + {x_n}}}{n}$ .

Complete step by step solution:
Here, in the question, it is given that the average of six numbers is $x$.
So, the total of these numbers is = $$$$(6 \times x) = 6x.
Now, the average of three of these numbers is $y$.
So, total of these numbers is $=$ $(3 \times y) = 3y.$
Now, the average of the rest of the three numbers is = $$$$z.
So, total of these numbers is = $$$$(3 \times z) = 3z.
Let us consider, ${n_1},{n_2},{n_3},{n_4},{n_5},{n_6}$ are the six numbers whose average is $x$ and average of ${n_1},{n_2},{n_3}$ is $y$ and average of ${n_4},{n_5},{n_6}$ is$z$.
So, from the above information, we can derive the following equations:
${n_1} + {n_2} + {n_3} + {n_4} + {n_5} + {n_6} = 6x.$
Also, $({n_1} + {n_2} + {n_3}) = 3y.$
And, $({n_4} + {n_5} + {n_6}) = 3z.$
So, we can write it as,
${n_1} + {n_2} + {n_3} + {n_4} + {n_5} + {n_6} = ({n_1} + {n_2} + {n_3}) + ({n_4} + {n_5} + {n_6}) = (3y + 3z).$
So, according to the question,
$\Rightarrow 6x = (3y + 3z).$
Taking the common value as RHS we get,
$\Rightarrow 6x = 3(y + z).$
Divide both the sides by three.
$\Rightarrow \dfrac{{6x}}{3} = (y + z).$
Let us divide the term and we get
$\Rightarrow 2x = (y + z).$
$\therefore$ Hence the correct answer is $2x = (y + z)$

So, Option B is the correct choice.

Note: Total of numbers can be found by multiplying the average of numbers by the number of numbers in the data set.
We can also make the total of all the numbers and then subtract the total of the any three numbers. Then we can tally this total with the total of the rest of the numbers in the set.
Also, by using the options we can check whether the given options are correct or not.

Total of the numbers	Total of the first sub part	Total of the second sub part
$6 \times x$	$3 \times y$	$3 \times z$

Then tally both the sides as the total of two sub parts is equal to the original sum.