Question

Two years ago, the mean age of $40$ people was $11$ years. Now a person has left the group and the mean age has changed to $12$ years. Find the age of the person who left the group.

Answer 1

Hint: First, we will calculate the total age of the people in both the scenarios and then subtract them to get the age of the person who left the group.
Finally we get the required answer.

Formula used: ${\text{Mean = }}\dfrac{{{\text{sum of terms}}}}{{{\text{number of terms}}}}$

Complete step-by-step solution:
It is given that the question stated as the mean age of $40$ people is $11$
Now, we used the formula of mean,
On substituted the values of mean and the number of people we get:
$ \Rightarrow 11 = \dfrac{{{\text{total age}}}}{{{\text{40}}}}$
On cross multiplying we get:
${{\text{total age}} = {11 \times 40}}$
On multiplying we get:
${\text{total age = 440}}$
Therefore, the total age of all the $40$ people is $440$.
Now this was $2$ years ago, the new total age should go up $2$ times for all the $40$ people in the group therefore the new total age is:
${\text{total age}} = (2 \times 40) + 440$
On multiplying the bracket term, we get:
${\text{total age}} = 80 + 440$
On adding both the terms is:
${\text{total age}} = 520$
Therefore, the new total age after $2$ years is $520$.
Now from the question stated as that after one person left the group, the mean is$12$.
Since $1$ person left the group the remaining numbers of people in the group are $40 - 1 = 39$ people.
Therefore, the mean age of $39$ people is $12$ on using the formula of mean and substituting the values we get:
$12 = \dfrac{{{\text{total age}}}}{{39}}$
On cross multiplying we get:
${{\text{total age}} = {12 \times 39}}$
On multiplying we get:
${\text{total age = 468}}$
Therefore, the total age of all the $39$ people is $468$.
Now we have to know that the total age of $40$ people and the total age of $39$ people, all in the same year
Therefore, the age of the person who left the group is:
$ \Rightarrow 520 - 468$
On simplifying we get:
$ \Rightarrow 52$

$\therefore $ The age of the person who left the group is \[52\]

Note: Mean is called average in layman terms and it is always the total of a value of a property in a distribution divided by the total number of terms in that distribution.
A common place to make mistakes is Cross multiplying, always cross multiplying the denominator of the fraction in R.H.S with the numerator of the fraction in the L.H.S.