Question

The mean of a certain number of observations is $35$. What is the new value of the mean if each observation is increased by $7$?
A) $35$
B) $40$
C) $42$
D) None of these

Answer 1

In this question, we are given a mean of a certain number of observations and we have been asked the new mean if all the observations are increased by a certain number. At first, assume the number of observations and use it to find the sum of observations. Then add the total increase in the observations to the original sum. Divide the new sum by the assumed number of observations. It will give you a new mean.

Formula used: Mean = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Total observations}}}}$

Complete step-by-step solution:
We are given a mean of certain number observations and we have been asked for the new mean if each observation is increased by $7$.
Let us assume the number of observations to be $x$ and put all the values in the formula- Mean = $\dfrac{{{\text{Sum of observations}}}}{{{\text{Total observations}}}}$.
Mean = $35$, Total observations = $x$.
Putting in the formula,
$\Rightarrow 35 = \dfrac{{{\text{Sum}}}}{x}$
$\Rightarrow {\text{Sum = }}35x$
Now, since all the observations have been increased by $7$, the sum will increase by $7x$.
$\Rightarrow {\text{Sum = }}35x + 7x$
$\Rightarrow {\text{New Sum = 42}}x$
Putting in the formula to find new mean,
$\Rightarrow {\text{New Mean = }}\dfrac{{42x}}{x} = 42$

Therefore, the mean has now changed from 35 to 42.

Note: Instead of doing the entire solution, it should be understood that if all the observations are increased or decreased by a certain value, then the mean will be affected in the similar way. It is because the mean considers the average value and on average, every observation has increased or decreased.