Question

The mean of 16 numbers is 54. If each number is divided by 9, what will be the new mean?

Answer 1

Hint: First assume the 16 numbers. After that find the sum of the numbers by substituting the value of the mean and total count of numbers by mean formula, $\dfrac{{\sum {{x_i}} }}{n}$. Then divide each number by 9. After that find the sum of new numbers which are ${\dfrac{1}{9}^{th}}$ of the original number. Then, find the mean of those numbers.

Complete step-by-step answer:
Let the 16 numbers be ${a_1},{a_2}, \ldots ,{a_{16}}$.
We know that the general formula to find the mean value is,
Mean $ = \dfrac{{\sum {{x_i}} }}{n}$
Substitute the values in the above formula,
$ \Rightarrow 54 = \dfrac{{{a_1} + {a_2} + \ldots + {a_{16}}}}{{16}}$
Cross-multiply the terms,
$ \Rightarrow {a_1} + {a_2} + \ldots + {a_{16}} = 864$..............….. (1)
Now, divide each number by 9,
$ \Rightarrow \dfrac{{{a_1}}}{9},\dfrac{{{a_2}}}{9}, \ldots ,\dfrac{{{a_{16}}}}{9}$
Now find the sum of the numbers,
$ \Rightarrow \dfrac{{{a_1}}}{9} + \dfrac{{{a_2}}}{9} + \ldots + \dfrac{{{a_{16}}}}{9}$
Take $\dfrac{1}{9}$ common from each number,
\[ \Rightarrow \dfrac{1}{9}\left( {{a_1} + {a_2} + \ldots + {a_{16}}} \right)\]
Substitute the value from equation (1),
$ \Rightarrow \dfrac{1}{9} \times 864$
Cancel out the common factors,
$ \Rightarrow \dfrac{{{a_1}}}{9} + \dfrac{{{a_2}}}{9} + \ldots + \dfrac{{{a_{16}}}}{9} = 96$
Now, find the mean of the new numbers by the mean formula,
$ \Rightarrow $Mean $ = \dfrac{{\dfrac{{{a_1}}}{9} + \dfrac{{{a_2}}}{9} + \ldots + \dfrac{{{a_{16}}}}{9}}}{{16}}$
Substitute the values,
$ \Rightarrow $Mean $ = \dfrac{{96}}{{16}}$
Divide numerator by the denominator,
$\therefore $Mean $ = 16$

Hence, the new mean is 16.

Note: Arithmetic mean represents a number that is obtained by dividing the sum of the elements of a set by the number of values in the set. So, you can use the layman term Average, or be a little fancier and use the word “Arithmetic mean“. Arithmetic means utilizing two basic mathematical operations, addition and division to find a central value for a set of values.
Whenever we face such types of problems the key point that we need to recall is that Mean or Arithmetic mean is the average of given numbers. It is found by calculating the sum of the given numbers and dividing it by total numbers