Question

The average age of a class of $20$ students is $12$ years, out of which one student whose age is $10$ years left the class and two new boys entered the class. The average of the class remains the same and the difference between the ages of new boys is $4$ years. What will be the age of the younger one?

Answer 1

In this question we have to find the age of the younger one. They give the average of the class with students and there are some changes in the number of students. We are going to solve this problem by using multiple variables in algebraic expressions. From the given, we have to get the required data and do some mathematical calculations on them. Then we get the required age of the younger one.

Formula used: Let the age of one student be ${\text{x}}$. Then, the age of ${\text{n}}$ students is ${\text{nx}}$.
Let the sum of two terms be expressed as ${\text{x}} + {\text{y}}$.
Let the difference of two terms be expressed as${\text{x}} - {\text{y}}$.

Complete step-by-step solution:
From the given, we have the average age of a class of $20$students is $12$ years. Then we get the total age of $20$ students from the given data.
Therefore, the total age of $20$ students is $20 \times 12 = 240$ years.
It is given that the student who is $10$ years old left the class.
So, we have to find the age of $19$ students by subtracting $10$ from the total age of $20$ students is $20 \times 12 = 240$ years.
Therefore, the total age of $19$ students $= 240 - 10 = 230$ years
By the given, then $2$ new students entered into the class, after the left of $10$ years old students but its average $\left( { = 12} \right)$ remains the same.
Now, we have to find the total age of $21$ students.
$\therefore$ The total age of $21$students is $21 \times 12 = 252$ years.
Here, to find the total age of $2$new students who entered in the class by subtracting the total age of $21$ students from the total age of $19$ students.
$\therefore$ The total age of $2$ new students who entered the class is $252 - 230 = 22$ years.
Let ${\text{x}}$ and ${\text{y}}$ be the two new students. Then the sum of the two students be $22$
$\Rightarrow {\text{x}} + {\text{y}} = 22$
According to the question, we have the difference between the ages of $2$ new boys is $4$years.
$\Rightarrow {\text{x}} - {\text{y}} = 4$
Now, adding the above two equations ${\text{x}} + {\text{y}} = 22$and ${\text{x}} - {\text{y}} = 4$. Then, we get
$\Rightarrow {\text{x}} + {\text{y}} + {\text{x}} - {\text{y}} = 22 + 4$
Add and subtract the terms,
$\Rightarrow 2{\text{x}} = 26$
Hence,
$\Rightarrow {\text{x}} = \dfrac{{26}}{2} = 13$
Substitute the ${\text{x}}$ value in ${\text{x}} - {\text{y}} = 4$. Then, we get the ${\text{y}}$ value.
$\Rightarrow 13 - {\text{y}} = 4$
Rearranging the terms,
$\Rightarrow - {\text{y}} = 4 - 13$
Simplifying we get,
$\Rightarrow - {\text{y}} = - 9$
Hence,
$\Rightarrow {\text{y}} = 9$

$\therefore$ The age of the younger one is $9$ years.

Note: We can solve linear equation in substituting method,
$\Rightarrow {\text{x}} + {\text{y}} = 22 - - - \left( 1 \right)$
$\Rightarrow {\text{x}} - {\text{y}} = 4 - - - \left( 2 \right)$
Let us consider the equation (2), rearranging the terms for x,
$\Rightarrow {\text{x}} = 4 + y - - - \left( 3 \right)$
Substitute the equation (3) in equation (1),
$\Rightarrow 4 + {\text{y}} + {\text{y}} = 22$
Add and subtract the terms,
$\Rightarrow 2{\text{y}} = 22 - 4$
Hence,
$\Rightarrow {\text{y}} = \dfrac{{18}}{2} = 9$
Substitute the y value into the equation (3),
$\Rightarrow {\text{x}} = 4 + 9 = 13$
Hence we got the required result.