Question

The average age of boys in a class of 30 is 15 years. If 10 more boys
join the class, the average of the whole class is reduced by 1 year. What
is the average age of new comers?​

Answer 1

$\huge \mathfrak \color{purple}{solution}$

The formula to calculate average of a distribution is given as:

$\boxed{\textbf{Average} = \dfrac{\textbf{Sum of all observations}}{\textbf{Total no. of observations}}}$

According to the question,

Average age of 30 boys in a class is 15 years. Writing it in terms of the above formula we get:

$\begin{gathered}\implies 15 = \dfrac{\text{Sum of all observations}}{30}\\\\\\\implies \text{Sum of all observations} = 30 \times 15\\\\\implies \boxed{\text{Sum of all observations} = 450}\end{gathered}$

Now, it is given that 10 more boys join the class. Hence total number of students increase from 30 to 40.

It is also given that Average age becomes 14 years. Hence we need to find the average age of newcomers.

Let the sum of ages of the new boys be 'x'.

Therefore the sum of all observations after newcomers enter is given as:

⇒ Sum = 450 + x

Substituting in the Average formula we get:

$\begin{gathered}\implies 14 = \dfrac{450+x}{40}\\\\\\\text{Cross multiplying we get:}\\\\\implies 450 + x = 14 \times 40\\\\\implies 450 + x = 560\\\\\implies x = 560 - 450\\\\\implies \boxed{x = 110}\end{gathered}$

Hence the sum of ages of newcomers is 110.

Finding the average age of newcomers we get:

$\begin{gathered}\implies \text{Average} = \dfrac{110}{10}\\\\\implies \boxed{ \bf{\textbf{Average} = 11\:years}}\end{gathered}$

Hence 11 years is the required answer.