# Thread: exponential growth, doubling

1. ## exponential growth, doubling

Hello I'm a new user on this forum and I've difficulties with this question in my handbook regarding exponential growth.
I'll post the question below and how i would solve it(seems wrong according to my handbook)

In 1990, the chinese population was 3 times as high as the population of europe.
The chinese population rises each year with 2% ,while the population of europe goes down with 0.5%.
When will the chinese population be 4 times as high as the population of europe.

I'd say: 400 * 1.02t = 100 * 0.995t
400/100 = 0.995t/1.02t
log (4) = t*log (0.995) - t*log(1.02)
log (4) = t*(log (0.995) - log(1.02))
log (4)/(log 0.995 - log 1.02) = t = -55.864

This seems to be wrong. According to my handbook the answer is after 11.6 years.

The answer in my handbook is a lot more logical than mine, and I really want to know what I did wrong.

2. ## Re: exponential growth, doubling

In 1990, $C_0=3E_0$

In year 1990+t (where t is a number of years):

$E_t = 0.995^tE_0$
$C_t = 1.02^tC_0$

You want to know when will $C_t = 4E_t$. This is the same as saying $\dfrac{C_t}{E_t} = 4$

$C_t = 1.02^tC_0 = 1.02^t\cdot 3E_0 = 3\left(\dfrac{1.02}{0.995}\right)^t E_t$

Dividing both sides by $E_t$ gives:

$\dfrac{C_t}{E_t} = 3\left( \dfrac{1.02}{0.995} \right)^t$

Setting equal to 4 and solving for $t$ gives:

$t = \dfrac{\ln\left(\dfrac{4}{3}\right)}{\ln\left( \dfrac{1.02}{0.995} \right)} \approx 11.5930$

3. ## Re: exponential growth, doubling

3*(1.02)^n = 4*(.995)^n
n = 11.593 ...per Sir Slip

4. ## Re: exponential growth, doubling

Thanks for the fast reply!
I would never come up with the first answer, but i understand it.

But i do not get why DenisB says "3*(1.02)^n= 4*(0.995)^n"
Now it seems, the population of europe will be 3/4th of the chinese instead of 1/4th. (my thought process might be pretty stupid, i'm sorry)
I'd say "1*(1.02)^n = 4*(0.955)^n

5. ## Re: exponential growth, doubling

There's a "1" in there:

3*(1.02)^n = 4*1*(.995)^n

Hokay?!

6. ## Re: exponential growth, doubling

Does not matter what we start with, as long as it's 3:1
Can be looked at as 2 savings acoounts, one
earning -1/2% (like a Canadian bank!) and
one earnimg 2%:
Code:
YR        EUROPE             CHINA
0        1000.00           3000.00
1   5.00  995.00    60.00  3060.00
2   4.97  990.03    61.20  3121.20
...
11  4.76  946.36    73.14  3730.12
11.593 ?
12  4.73  941.62    74.60  3804.72
Got it?