Question 201205
Let x=# of people in the club (before 5 more join)



Before they get 5 people, we have the equation:



Cost Per Individual:  {{{c=200000/x}}}  (since the cost is divided among "x" people)



When 5 more people join, the equation becomes



{{{c-2000=200000/(x+5)}}} (simply replace "x" with "x+5" and subtract 2000 from "c" to form the new equation)



So we have the two equations



{{{c=200000/x}}} Equation 1



{{{c-2000=200000/(x+5)}}} Equation 2




{{{c-2000=200000/(x+5)}}} Start with Equation 2



{{{200000/x-2000=200000/(x+5)}}} Plug in {{{c=150000/x}}} (which is Equation 1)



{{{cross(x)(x+5)(200000/cross(x))-x(x+5)(2000)=x*cross((x+5))(200000/cross((x+5)))}}} Multiply EVERY term by the LCD {{{x(x+5)}}} to clear out the fractions.



{{{200000(x+5)-2000x(x+5)=200000x}}} Multiply and simplify.



{{{200000x+1000000-2000x^2-10000x=200000x}}} Distribute



{{{200000x+1000000-2000x^2-10000x-200000x=0}}} Subtract 200000x from both sides.



{{{-2000x^2-10000x+1000000=0}}} Combine like terms.



Notice that the quadratic {{{-2000x^2-10000x+1000000}}} is in the form of {{{Ax^2+Bx+C}}} where {{{A=-2000}}}, {{{B=-10000}}}, and {{{C=1000000}}}



Let's use the quadratic formula to solve for "x":



{{{x = (-B +- sqrt( B^2-4AC ))/(2A)}}} Start with the quadratic formula



{{{x = (-(-10000) +- sqrt( (-10000)^2-4(-2000)(1000000) ))/(2(-2000))}}} Plug in  {{{A=-2000}}}, {{{B=-10000}}}, and {{{C=1000000}}}



{{{x = (10000 +- sqrt( (-10000)^2-4(-2000)(1000000) ))/(2(-2000))}}} Negate {{{-10000}}} to get {{{10000}}}. 



{{{x = (10000 +- sqrt( 100000000-4(-2000)(1000000) ))/(2(-2000))}}} Square {{{-10000}}} to get {{{100000000}}}. 



{{{x = (10000 +- sqrt( 100000000--8000000000 ))/(2(-2000))}}} Multiply {{{4(-2000)(1000000)}}} to get {{{-8000000000}}}



{{{x = (10000 +- sqrt( 100000000+8000000000 ))/(2(-2000))}}} Rewrite {{{sqrt(100000000--8000000000)}}} as {{{sqrt(100000000+8000000000)}}}



{{{x = (10000 +- sqrt( 8100000000 ))/(2(-2000))}}} Add {{{100000000}}} to {{{8000000000}}} to get {{{8100000000}}}



{{{x = (10000 +- sqrt( 8100000000 ))/(-4000)}}} Multiply {{{2}}} and {{{-2000}}} to get {{{-4000}}}. 



{{{x = (10000 +- 90000)/(-4000)}}} Take the square root of {{{8100000000}}} to get {{{90000}}}. 



{{{x = (10000 + 90000)/(-4000)}}} or {{{x = (10000 - 90000)/(-4000)}}} Break up the expression. 



{{{x = (100000)/(-4000)}}} or {{{x =  (-80000)/(-4000)}}} Combine like terms. 



{{{x = -25}}} or {{{x = 20}}} Simplify. 



So the <i>possible</i> answers are {{{x = -25}}} or {{{x = 20}}} 

  
  
However, a negative number of people doesn't make any sense. So we must ignore the solution {{{x=-25}}}




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Answer:



So the solution is {{{x = 20}}} which means that there are 20 members currently in the club (before the 5 people join).