Question

The average age of x persons is 60 years. If
two persons of 52 years and 68 years leave the
group and two new persons of y years and 72 years
join the group, then the average age of the group
increases by 1 year. If x is a perfect square and
54 < y < 64, find the value of y.​

Answer 1

I don't know

Later I know I will give you answer

Answer 2

Average

The average age of x persons is 60.

Hence total age of x persons will be $60\times x=60 x$

As per information, it is given that,

Two persons of age $52\ years \ and \ 68 \ years$ leave the group and two new persons of age $y \ years \ and \ 72 \ years$.  joined which increases the average with 1 years.

Which can be represented as,

$\frac{x\times 60-(52+68)+72+y}{x}=(60+1)$

$\implies 60x-120+72+y=61x\\\\\implies y-48=x$

As it is given that,

$54<y<64$ and also x is a perfect square.

Check the value of x for the value of y  between 55 and 63 such that x becomes a perfect square.

We  find that x will only be a perfect square between $(55-48)=7 \ to \ (63-48)=15$

Hence checking the possible perfect square between 7 and 15 is only 9.

Hence, the value of x as 9 is only possible when,

$y-48=9\\\implies y=48+9=57$

Hence, possible value of y is 57 years.