Question

The average of \[{x_1},{x_2},{x_3},{x_4}\] is $16$. Half the sum of \[{x_2},{x_3},{x_4}\] is $23$. What is the value of \[{x_1}\]?
${\text{(A) 17}}$
${\text{(B) 18}}$
${\text{(C) 19}}$
${\text{(D) 20}}$

Answer 1

Hint: We will first try to make two equations using the data given to us and then find the value of the remaining unknown value.
We will make use of the mean formula to solve the given question.
${\text{Mean = }}\dfrac{{{\text{sum of terms}}}}{{{\text{number of terms}}}}$

Complete step-by-step solution:
It is given that the question stated as the mean of the four terms \[{x_1},{x_2},{x_3},{x_4}\] is $16$
So we can write the total number of terms in the distribution is $4$.
Now, using the formula of mean this statement can be written mathematically as:
$ \Rightarrow 16 = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4}}}{4} \to (1)$
Also, half the sum of \[{x_2},{x_3},{x_4}\] is $23$, mathematically it can be written as:
$ \Rightarrow 23 = \dfrac{{{x_2} + {x_3} + {x_4}}}{2}$
On cross multiplying we get:
\[ \Rightarrow {x_2} + {x_3} + {x_4} = 23 \times 2\]
On simplifying we get:
\[ \Rightarrow {x_2} + {x_3} + {x_4} = 46 \to (2)\]
Now we will substitute equation $(2)$ in $(1)$,
On substituting we get:
$ \Rightarrow 16 = \dfrac{{{x_1} + 46}}{4}$
Now on cross multiplying we get:
$ \Rightarrow 16 \times 4 = {x_1} + 46$
On simplifying we get:
$ \Rightarrow 64 = {x_1} + 46$
Now taking $46$across the $ = $ sign it becomes negative therefore, we get:
$ \Rightarrow {x_1} = 64 - 46$
On subtracting we get:
${x_1} = 18$, which is the required answer.

Therefore, the correct option is ${\text{(B)}}$ which is $18$.

Note: There is a property of mean; the mean in the distribution is not always one actual value from the distribution itself because of the division which is done to calculate it.
Mean can only be used if the distribution is numerical, if the distribution is non-numerical than median is most commonly used.
A number when shifted across the $ = $ sign, if it is positive it will become negative and if it is negative it will become positive. Same rule is followed for multiplication and division across the $ = $ sign.
The formula of mean should be remembered to solve these types of questions. In the question it is written “average”, which means mean. Average is a word used for mean in layman terms.