If ${\text{x + y = 9}}$, ${\text{y + z = 7}}$ and ${\text{z + x = 5}}$ then-
A) ${\text{x + y + z = 10}}$
B) Arithmetic mean of x, y, z is $3.5$
C) median of x, y, z is $3.5$
D)${\text{x + y + z = 10}}.5$

VerifiedVerified
150.3k+ views
Hint: We can find the arithmetic mean by using formula, Arithmetic mean=$\dfrac{{{\text{sum of the variables}}}}{{{\text{total number of variables}}}}$.
On adding the given three equations we can find the sum of the variables x, y, z.

Complete step-by-step answer:
We are given, ${\text{x + y = 9}}$ --- (i)
${\text{y + z = 7}}$--- (ii)
${\text{z + x = 5}}$ --- (iii)
To find the sum of all the variables, add eq. (i), (ii) and (iii).
$ \Rightarrow {\text{x + y + y + z + z + x = 9 + 7 + 5}}$
On adding the given values we get,
$ \Rightarrow 2{\text{x + 2y + 2z = 21}}$
On taking $2$ common in the equation, we get
$ \Rightarrow 2\left( {{\text{x + y + z}}} \right){\text{ = 21}}$
On transferring $2$ on the right side, we get
$ \Rightarrow {\text{x + y + z = }}\dfrac{{{\text{21}}}}{2}$ $ = 10.5$ --- (iv)
So option D is correct.
Now we know the sum of the variables and we know there are three variables. We know that,
$ \Rightarrow $ Arithmetic mean=$\dfrac{{{\text{sum of the variables}}}}{{{\text{total number of variables}}}}$
$ \Rightarrow $ Arithmetic mean=$\dfrac{{{\text{x + y + z}}}}{3}$
On substituting the values from eq. (iv), we get
$ \Rightarrow $ Arithmetic mean=$\dfrac{{\dfrac{{21}}{2}}}{3}$
On solving further we get-
$ \Rightarrow $ Arithmetic mean=$\dfrac{{21}}{2} \times \dfrac{1}{3} = \dfrac{{21}}{6}$
On division, we get-
$ \Rightarrow $ Arithmetic mean=$3.5$
So option B is correct.
On substituting values of eq. (i) in eq. (iv), we get-
$ \Rightarrow x + 7 = 10.5$
On solving we get-
$ \Rightarrow {\text{x}} = 10.5 - 7 = 3.5$
On substituting value of x in eq. (iii), we get
$ \Rightarrow {\text{z = 5 - 3}}{\text{.5 = 1}}{\text{.5}}$
On substituting the value of z in eq. (i) we get
$ \Rightarrow {\text{y}} + 3.5 = 9$
On solving we get-
$ \Rightarrow {\text{y = 9 - 3}}{\text{.5 = 5}}{\text{.5}}$
So the values of x, y and z are $3.5,{\text{5}}{\text{.5,1}}{\text{.5}}$ respectively
And we know the median is the middle value in the given numbers. The middle value is $5.5$
$ \Rightarrow $ Median=$5.5$
So option C is not correct.
Hence the correct options are B and D.

Note: We can also find the median using formula-
Median=${\left[ {\dfrac{{\left( {{\text{n + 1}}} \right)}}{2}} \right]^{th}}$ term for odd number of observations.
So on substituting the values we get,
Median =$\dfrac{{3 + 1}}{2} = \dfrac{4}{2} = {2^{nd}}$ term
Second term is $5.5$ so the median is $5.5$.