Question

If $p,q,r$ and $s$ are four different positive integers selected from $1$ to $200,$ then find the highest possible value of
$\frac{(p+q)+(r+s)}{(p+q)+(r-s)}$

Answer 1

$I=\frac{(p+q)+(r+s)}{(p+q)+(r-s)}0<p,q,r,s\le 200$

$\Rightarrow \frac{(p+q+r)+s}{(p+q+r)-s}$

$\Rightarrow \frac{(p+q+r-s)+2s}{(p+q+r-s)}$

$I\Rightarrow 1+\frac{2s}{(p+q+r-s)}$

Fir $I$ to be maximum

$(p+q+r)-s=1$ and $s$ should be maximum

$\therefore if(p+q+r)=200$

And $s=199$

$I=1+\frac{2\left(199\right)}{(200-199)}=1+2\left(199\right)$

$I\Rightarrow 1+398=399$

$\therefore$ maximum value of $I=399$