Question 129060
Let x be the number of occupied units.  Then 80 - x is the number of unoccupied units.


The rental fee is 200 plus 20 times the number of unoccupied units, so:


{{{200+20(80-x)}}}, which simplifies to:


{{{200+1600-20x}}}
{{{1800-20x}}}


This is the amount charged per apartment, and there are x occupied apartments, so the total revenue is:


{{{x(1800-20x)}}} and that is equal to $20,020.


So:


{{{x(1800-20x)=20020}}}


{{{1800x-20x^2-20020=0}}}


Divide by -20:
{{{-90x+x^2+1001=0}}}


{{{x^2-90x+1001=0}}}


You can factor this (the prime factors of 1001 are 7, 11, and 13), or use the quadratic formula, whichever you prefer, but the roots come out to be {{{x=77}}} and {{{x=13}}},


So, if the rental fee is 200 + 20(3) = 260 (3 unoccupied units), then the total revenue is 77 * 260 = 20020.


But if the rental fee is 200 + 20(67) = 1540, then the total revenue is 13 * 1540, also = 20020.


The second root presumes that the relationship of rent to unoccupied units is linear across the entire range of possibilities, which I suspect is not the case in real life.