Question

The average age of 30 boys in class is 15 year. One boy aged 20 years left the class, but two boys come in his place whose ages differ by 5 years. If the average age of all the boys in the class still remains 15 years, the age of the younger new comer is
A.20 years
B.15 years
C.10 years
D.8 years

Answer 1

Hint: First we will find the total age of all the initial boys. Then after subtracting the age of the 20-years old boy who left, we will add the age of two boys which will be variable \[x\] and \[x + 5\], where the younger boy will be aged \[x\] years. Now we will have the total age and the average age will still be 15 years. We can use this average to form an equation, because we know that the new number of boys will be 31, and we can solve accordingly to get the value of \[x\] that is the age of the younger boy.

Complete step-by-step answer:
Initially there were 30 boys in the class, with an average age of 15 years, so their total age will be given by \[30 \times 15 = 450\] years. Now a boy aged 20-years leaves the class, leaving behind 29 boys with their total age as \[450 - 20 = 430\].
Now, two new boys enter the class, whose ages differ by 5 years. Let us assume that the younger boy has age \[x\] years and the elder one has age \[x + 5\] years.
So, now we have a class with 31 boys and their total age is given as \[430 + (x) + (x + 5) = 435 + 2x\]. Thus, the total age of 31 boys will be \[435 + 2x\] years. Also it is given in the question that the average age of the new class is still 15. Using this we can form an equation for average age of the class as
\[\dfrac{{435 + 2x}}{{31}} = 15\]. We will solve this equation and get the value of \[x\] that is the age of the younger boy.
The equation will be solved as
\[
  \dfrac{{435 + 2x}}{{31}} = 15 \\
\Rightarrow 435 + 2x = 15 \times 31 \\
\Rightarrow 435 + 2x = 465 \\
\Rightarrow 2x = 465 - 435 \\
\Rightarrow 2x = 30 \\
\Rightarrow x = 15 \\
\]
Thus, the age of the younger boy will be 15 years old and the age of the elder boy will be 20 years old.
Hence, option (B) is the correct option.

Note: In this type of question, it is best to form an equation using a single variable. Also if the average of data and the count of the data is given then it is better to find the total sum of data by the formula of average, as we can easily add or subtract the total sum but averages may confuse us.