Rage Math X^(x/x) / years, just check it you'll understand.

Vraces:

*Need serious help

In a crack smoking neighborhood the people have been over 20 years reduced to half of what they originally were. What yearly reducing in percent is this?

[I already have the answer, but I can't figure out w*f to do to solve it(I'm getting the wrong answer) cause I'm an idiot]
If you don't think you can solve it don't write s**t that'll confuse me further, I'm confused enough as it is.

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Hi Vraces,
I sent you 2 private messages way back at the beginning of Dec, I know that you haven't read them because they are still sitting in my outbox
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Now for your problem, (this is lead up to your problem)
Just say that there were 1000 rabbits in a field and the population is decreasing by an average of 5% each year. 5% as a decimal is 0.05
so, at the end of each year there would be (100%-5%=95%) which is (1-0.05) of the early January population.
So at the end of the first year there would be 1000*(1-0.05)
At the end of the second year there would be 1000*(1-0.05) (1-0.05) = 1000*(1-0.05) ² [this is the number of rabbits at the beginning of the second year * (1-0.05)]
.....
At the end of the 20th year there would be about 1000*(1-0.05) ²⁰ rabbits.

Are you with me so far?
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Now lets look at your problem
In the beginning there were P crack users.
After 20 years there were P/2 crack users. (P/2 is the same as 0.5P)
Let the average yearly reduction rate be r (not r%, this is a decimal, so if it was 5%, r=0.05)

We have

P/2 = P(1-r) ²⁰
0.5P = P(1-r) ²⁰
divide both sides by P
0.5 = (1-r) ²⁰
Raise both sidesby the power of 1/20
0.5^(1/20) = 1-r
r = 1 - 0.5^(1/20)
r = 0.034 (correct to 3 dp)
so the number of crack users is reducing by a average yearly rate of 3.4%
Is this answer the same as the one given?
do you understand?

PS You are not an idiot and a little less bad language on the forum would be good.

Thanks! That was it, now I get it.
For some reason I had excluded the '1' in the equation.
0.5^0.05 = 0.9659363289248456
/ 20 = Wrong answer

1 -0.5^0.05 = 0.0340636710751544 = right answer

That's excellent!