Answer
$34$ students
Work Step by Step
Initial number of students: $x$
Initial cost per student: $c$
We have:
$xc=1700$
$(x+6)(c-7.5)=1700$
From the first equation $c=\frac{1700}{x}$. We substitute it in the second equation:
$(x+6)\left(\frac{1700}{x}-7.5\right)=1700$
We solve the equation:
$(x+6)(1700-7.5x)=1700x$
$1700x-7.5x^2+10200-45x=1700x$
$7.5x^2+45x-10200=0$
Divide by $15$:
$0.5x^2+3x- 680=0$
$x_{1,2}=\frac{-3\pm\sqrt{9+1360}}{1}=-3\pm 37$
$x_1=-3-37=-40$
$x_2=-3+37=34$
As $x>0$, the only solution is $x=34$.
