Question

If the arithmetic mean of the numbers ${x_1},{x_2},{x_3},...,{x_n}$ is $\bar x$, then the arithmetic mean of the numbers $a{x_1} + b,a{x_2} + b,a{x_3} + b,...,a{x_n} + b$, where $a$ and $b$ are two constants, would be
A. $\bar x$
B. $na\bar x + nb$
C. $a\bar x$
D. $a\bar x + b$

Answer 1

Hint: Here arithmetic mean of some numbers is given and arithmetic mean of some numbers related to the given numbers is asked. To solve this type of problem, you need to use the formula of finding the arithmetic mean of numbers. Basically, the arithmetic mean is nothing but the average of some given numbers. It is obtained by calculating the sum of the numbers divided by the total number of given data.

Formula Used:
Arithmetic mean of the numbers ${x_1},{x_2},{x_3},...,{x_n}$ is defined by $\bar x = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n}$, where $n$ is total number of given data.

Complete step by step solution:
The given numbers are ${x_1},{x_2},{x_3},...,{x_n}$
Arithmetic mean of the numbers is the sum of the numbers divided by total number of given data i.e.
$\bar x = \dfrac{{{x_1} + {x_2} + {x_3} + ... + {x_n}}}{n}$
$ \Rightarrow {x_1} + {x_2} + {x_3} + ... + {x_n} = n\bar x - - - - - \left( i \right)$
We have to find the arithmetic mean of the numbers $a{x_1} + b,a{x_2} + b,a{x_3} + b,...,a{x_n} + b$
Sum of the numbers is $\left( {a{x_1} + b} \right) + \left( {a{x_2} + b} \right) + \left( {a{x_3} + b} \right) + ... + \left( {a{x_n} + b} \right)$
Arrange the terms.
$ = a\left( {{x_1} + {x_2} + {x_3} + ... + {x_n}} \right) + \left( {b + b + b + ... + b} \right)$
There are total $n$ numbers.
So, $b + b + b + ... + b = nb$
and substitute ${x_1} + {x_2} + {x_3} + ... + {x_n} = n\bar x$ from equation $\left( i \right)$
Thus the sum becomes $an\bar x + nb$
Taking $n$ as common, we get the sum $n\left( {a\bar x + b} \right)$
Dividing the sum by $n$, we get
Arithmetic mean of the numbers $ = \dfrac{{n\left( {a\bar x + b} \right)}}{n} = a\bar x + b$

Option ‘D’ is correct

Note: Arithmetic mean is simply average of the given numbers. If we add or subtract a number from each of the given numbers then, the mean of the resulting numbers will be equal to the mean of the given numbers with added or subtracted the number which we previously added or subtracted.