Question 132413
In the beginning, there were x students, when five didn't show up, there were x - 5 students.


In the beginning, the cost per student was {{{500/x}}} but when five students didn't show up, the cost was {{{500/(x-5)}}} and this cost was $5 more than the original cost per student, so:


{{{500/x=500/(x-5)-5}}}


Need an LCD of {{{x-5}}} on the right:
{{{500/x=(500-5(x-5))/(x-5)}}}


Simplify the right hand numerator:
{{{500/x=(525-5x)/(x-5)}}}


Cross-multiply:
{{{500(x-5)=(525-5x)x}}}


Expand:
{{{500x-2500=525x-5x^2}}}


Collect terms and put into standard form:
{{{5x^2-25x-2500=0}}}


Divide by 5:
{{{x^2-5x-500=0}}}


{{{-25*20=-500}}}, {{{-25+20=-5}}}, so:
{{{(x-25)(x+20)=0}}}


{{{x=25}}} or {{{x=-20}}}.   A negative number for a number of students doesn't make much sense in this context, so exclude -20 as an extraneous root.  There were originally 25 students, so 25 - 5 = 20 actually attended.


Check:
{{{500/25=20}}}, {{{500/20=25}}}, and {{{20 + 5 = 25}}}