Question 246003
The club has x members now, so the average cost = 200,000/x.
If they add 5 members, the average cost = 200,000/(x+5) - 2000.
So, we can express this as:
{{{
200000/x = 200000/(x+5) + 2000
}}}
.
{{{
200000/x - 200000/(x+5) - 2000 = 0
}}}
.
Multiply through by {{{x(x+5)}}} to eliminate the denominators.
.
{{{
200000(x+5) - 200000x - 2000(x)(x+5) = 0
}}}
.
{{{
200000(x+5) - 200000x - 2000x^2 - 10000x = 0
}}}
.
{{{
200000x + 1000000 - 200000x - 2000x^2 - 10000x = 0
}}}
.
Divide by 2000
{{{
100x + 500 - 100x - x^2 - 5x = 0
}}}
.
The 100x cancel, so collect terms and rearrange
{{{
-x^2 - 5x + 500 = 0
}}}
.
Multiply by -1
{{{
x^2 + 5x - 500 = 0
}}}
.
Factor
{{{
-(x+25)(x-20) = 0
}}}
.
So x can = either -25 or 20.  -25 members does not make sense, so x = 20 and x+5 = 25.
.
Checking our answer.
average cost with 20 members
{{{200000/20 = 10000
}}}
.
average cost with 25 members
{{{200000/25 = 8000
}}}
.
So the difference in average cost is 2000, as the problem requires.