Question

The average age of a group of eight members is the same as it was $3$ years ago when a young member is substituted for an old member. The incoming member is younger to the outgoing member by
$A) 11$ Years
$B) 24$ Years
$C) 28$ Years
$D) 16$ Years

Answer 1

In order to solve this type of problems related to age, we need to follow the following steps:
$a)$ Express what we don’t know as a variable.
$b)$ Create an equation based on the information provided.
$c)$ Solve the unknown variable.
$d)$ Substitute our answer back into the equitation to see if the left side of the equation equals the right side of the equation and hence, we will get our final result.

Complete step-by-step solution:
Let the average age of the eight members be $\overline x$.
Let the ages be ${a_1},{a_2},........{a_8}$
Therefore, we can write the average of eight members, $\overline x = \dfrac{{{a_1} + {a_2} + ....... + {a_8}}}{8}$
By doing cross multiply, we get
$\Rightarrow {a_1} + {a_2} + ....... + {a_8} = 8\overline x$
We are just taking the ${a_1}$ variable to the right hand side, we can write it as,
$\Rightarrow {a_2} + ........ + {a_8} = 8\overline x - {a_1}............(1)$
Let us consider the old member age is ${a_1}$ and the young member age is ${a_9}$.
Also it stated as in the question those three years ago, the young member is substituted for old member.
Again it is stated as, average is same after substitution.
Therefore, $\overline x = \dfrac{{\left( {{x_9} - 3} \right) + ({a_2} - 3) + ........ + ({a_8} - 3)}}{8}$
On solving, we can have,
$\Rightarrow {a_9} + {a_2} + ........ + {a_8} - 3(8) = 8\overline x$
By multiplying the numbers, we get
$\Rightarrow {a_9} + {a_2} + ....... + {a_8} - 24 = 8\overline x .....(2)$
Substituting $(1)$ in $(2)$ we get,
$\Rightarrow {a_9} + 8\overline x - {a_1} - 24 = 8\overline x$
We can cancel the same terms on each side and take $- 24$ to the right hand side.
Hence we get
$\Rightarrow {a_9} - {a_1} = 24$
Therefore, the incoming members are $24$ years younger to the outgoing member.

Thus the correct option is $(B)$ that is $24$.

Note: In order to solve age related problems, we need to know some important points to remember which are as follows:
$i)$ If the present age is $y$ then $n$ times the present age $ny.$
$ii)$ If the present age is $x$, then the age $n$ years later we can write $x + n$