Question

If the mean of a set of observation \[{x_1},{x_2},...{x_{10}}\] is $20$ then find the mean of ${x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40$ .
(A) $34$
(B) $42$
(C) $38$
(D) $40$

Answer 1

Hint: Use the definition of mean, i.e. $Mean = \dfrac{{{\text{Sum of the observations}}}}{{{\text{Number of observations}}}}$ on the given information and find the sum of all the numbers. Then again use the formula for mean to write mean of ${x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40$ and substitute the value of the sum of the numbers \[{x_1},{x_2},...{x_{10}}\] . This will give us the required mean.

Complete step-by-step answer:
In this problem, the mean of ten observations \[{x_1},{x_2},{x_3},{x_4},{x_5},{x_6},{x_7},{x_8},{x_9},{x_{10}}\] is given as $20$. Using this information we need to find the mean of the observations ${x_1} + 4,{x_2} + 8,{x_3} + 12,{x_4} + 16,{x_5} + 20,{x_6} + 24,{x_7} + 28,{x_8} + 32,{x_9} + 36,{x_{10}} + 40$ .
Before moving towards the solution we should understand the concept of the mean. A mean is the simple mathematical average of a set of two or more numbers. The mean is a statistical indicator that can be used to gauge the performance of a company’s stock price over days, months, or years, a company through its earnings over several years, a firm by assessing its fundamentals such as price-to-earnings ratio, free cash flow, and liabilities on the balance sheet, and a portfolio by estimating its average returns over a certain period.
$ \Rightarrow Mean = \dfrac{{{\text{Sum of the observations}}}}{{{\text{Number of observations}}}}$
According to the given information, by using the above relation, we get:
$ \Rightarrow Mean\left( {{x_1},{x_2},{x_3},{x_4},{x_5},{x_6},{x_7},{x_8},{x_9},{x_{10}}} \right) = 20 = \dfrac{{{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8} + {x_9} + {x_{10}}}}{{10}}$
Now the above equation can be rearranged to find the value of the sum of all the numbers:
\[ \Rightarrow \left( {{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8} + {x_9} + {x_{10}}} \right) = 20 \times 10 = 200\] ………..(i)
So, now we can use the definition of mean to calculate the mean of numbers ${x_1} + 4,{x_2} + 8,{x_3} + 12,{x_4} + 16,{x_5} + 20,{x_6} + 24,{x_7} + 28,{x_8} + 32,{x_9} + 36{\text{ and }}{x_{10}} + 40$
$ \Rightarrow Mean\left( {{x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40} \right) = $
$ = \dfrac{{\left( {{x_1} + 4 + {x_2} + 8 + {x_3} + 12 + {x_4} + 16 + {x_5} + 20 + {x_6} + 24 + {x_7} + 28 + {x_8} + 32 + {x_9} + 36 + {x_{10}} + 40} \right)}}{{10}}$
Here we can further solve the numerator of the above expression by separating the terms \[{x_1},{x_2},...{x_{10}}\] and $4,8,12,....40$ , we get:
$ = \dfrac{{\left( {{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8} + {x_9} + {x_{10}}} \right) + \left( {4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40} \right)}}{{10}}$
But we already have the value of the sum of \[{x_1},{x_2},...{x_{10}}\] from relation (i), we get:
$ \Rightarrow Mean\left( {{x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40} \right) = \dfrac{{\left( {200} \right) + \left( {4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40} \right)}}{{10}}$
Now, this can be further evaluated as follows:
$ \Rightarrow Mean\left( {{x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40} \right) = \dfrac{{\left( {200} \right) + \left( {220} \right)}}{{10}} = \dfrac{{420}}{{10}} = 42$
Therefore, the mean of ${x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40$ will be $42$ .
Hence, the option (B) is the correct answer.

Note: In questions like this, the use of the definition of mean always helps in solving the problem. This problem can also be solved using an alternative approach by first writing the mean for ${x_1} + 4,{x_2} + 8,{x_3} + 12,...{x_{10}} + 40$ and then split it into two fractions as:
$ = \dfrac{{\left( {{x_1} + {x_2} + {x_3} + {x_4} + {x_5} + {x_6} + {x_7} + {x_8} + {x_9} + {x_{10}}} \right)}}{{10}} + \dfrac{{\left( {4 + 8 + 12 + 16 + 20 + 24 + 28 + 32 + 36 + 40} \right)}}{{10}}$
The first fraction is the mean of \[{x_1},{x_2},...{x_{10}}\] which is given in the question.