Question

The mean of 15 numbers is 27. If each number is multiplied by 4, what will be the mean of the new numbers?

Answer 1

Hint: In this question, we need to determine the mean of the new numbers such that the mean of 15 numbers is 27 and each number is multiplied by 4. For this, we will first find the numbers and then multiply these numbers by 4 and then find the new mean.

Complete step-by-step answer:
Given the mean of the 15 numbers is \[ = 27\]
Let us assume 15 numbers to be \[ = {x_1},{x_2},{x_3}...........{x_n}\]
Now as we know that the mean of given numbers is the ratio of the sum of given numbers by the total numbers whose mean is to be found and since we have to find the mean of 15 numbers here so we will divide the sum of these numbers by 15, hence we can write
  \[
  Mean = \dfrac{{{x_1} + {x_2} + {x_3} + ......... + {x_n}}}{{15}} \\
   \Rightarrow \dfrac{{{x_1} + {x_2} + {x_3} + ......... + {x_n}}}{{15}} = 27 \;
 \]
So the value of the sum of these 15 numbers will be equal to \[{x_1} + {x_2} + {x_3} + ......... + {x_n} = 27 \times 15 = 405\]
Now it is said that each of these is multiplied by 4, hence we can write the new numbers as \[ = \left( {{x_1} \times 4} \right),\left( {{x_2} \times 4} \right)\left( {,{x_3} \times 4} \right)...........\left( {{x_n} \times 4} \right)\]
Hence their mean will be equal to
 \[Mean = \dfrac{{\left( {{x_1} \times 4} \right) + \left( {{x_2} \times 4} \right) + \left( {{x_3} \times 4} \right) + ......... + \left( {{x_n} \times 4} \right)}}{{15}}\]
Hence by taking 4 as a common we can further write this as
 \[Mean = \dfrac{{4\left\{ {{x_1} + {x_2} + {x_3} + ......... + {x_n}} \right\}}}{{15}}\]
Now since the sum of theses 15 numbers is \[{x_1} + {x_2} + {x_3} + ......... + {x_n} = 405\] , hence we can write the mean as
 \[
  Mean = \dfrac{{4\left\{ {405} \right\}}}{{15}} \\
   = 108 \;
 \]
Therefore the mean of the new 15 numbers \[ = 108\]
So, the correct answer is “108”.

Note: The mean of some numbers is the ratio of the sum of the numbers by the total numbers whose mean is to be found. Students must note that if they increase or decrease the values of the numbers whose mean is to be calculated then their mean will also change, in this question when each number was multiplied by 4 then the mean also increased by 4 times.