Question

The average age of three friends is 23. Even if the age of the \[{4^{th}}\] friend is added, the average remains 23. What is the age of the \[{4^{th}}\] friend?
A) $21$ years
B) $23$ years
C) $32$ years
D) $34$ years

Answer 1

Hint: In this question, we have been told that the average age of three friends is the same as the average age of four friends. Then we are asked the age of the fourth friend. Use an average basic formula for this. First, find the sum of the ages of the three friends. Use it to find the sum of the ages of four friends. Then subtract them to find the age of the fourth friend.

Complete step-by-step solution:
We have been told that the average age of three friends is the same as the average age of four friends and we have been asked the age of the \[{4^{th}}\] friend. Let us use the basic formula of averages to find the age of the \[{4^{th}}\] friend.
$\bar X = \dfrac{{{\text{Sum of observations}}}}{{{\text{Total observations}}}}$
We know that the average age of three friends is 23. Let us put this in the formula-
$ \Rightarrow 23 = \dfrac{{{\text{Sum}}}}{3}$
$ \Rightarrow {\text{Sum = }}23 \times 3 = 69$
Therefore, the sum of the ages of three friends is $69$.
Now, let the age of \[{4^{th}}\] friend be $x$. We also know that the average age of four friends is also 23. Using this information,
$ \Rightarrow 23 = \dfrac{{69 + x}}{4}$
Shifting to find the value of x,
$ \Rightarrow 23 \times 4 = 69 + x$
Simplifying we get,
$ \Rightarrow 92 - 69 = x$
Subtracting we get,
$ \Rightarrow x = 23$

$\therefore $ The age of \[{4^{th}}\] friend is $23$.

Note: If this question has been asked in an MCQ and you are not required to show the solution, then you can find the answer by thinking in this way-
It is given that the average age of three friends is 23. It means that on an average basis, every friend is 23. When the \[{4^{th}}\] friend enters, the average still remains 23. If the three friends are of 23 each, and the average age of four friends is also 23, this can only happen if the \[{4^{th}}\] friend is also 23.