Solving for "t" to find equivelant compounding period.

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JaredRoss

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I have been stuck on this problem for quite awhile because my algebra skills are a bit lagging. It goes as follows:

Real GDP per Capita in the US is currently $56,000 and grows at approximately 1.5% each year.
Real GDP per Capita in China is currently $8,000 and grows at approximately 6.5% each year.
If these growth rates continue, Real GDP per Capita for each country will be equal in how many years?

I am trying to solve for t in the following equation: 56,000(1+.015)^t = 8,000(1+.065)^t

I have tried multiple variations to solve for t, but I am currently stumped. How would one solve for t in this equation?
 
Hello, and welcome to FMH! :)

I would first divide through by 8000 to get:

[MATH]7\cdot1.015^t=1.065^t[/MATH]
Then:

[MATH]7=\left(\frac{213}{203}\right)^t[/MATH]
Can you finish?
 
JaredRoss said:
I have been stuck on this problem for quite awhile because my algebra skills are a bit lagging. It goes as follows:

Real GDP per Capita in the US is currently $56,000 and grows at approximately 1.5% each year.
Real GDP per Capita in China is currently $8,000 and grows at approximately 6.5% each year.
If these growth rates continue, Real GDP per Capita for each country will be equal in how many years?

I am trying to solve for t in the following equation: 56,000(1+.015)^t = 8,000(1+.065)^t

I have tried multiple variations to solve for t, but I am currently stumped. How would one solve for t in this equation?
Click to expand...
You will need to use "Log".

56,000(1+.015)^t = 8,000(1+.065)^t

7 = (1+.065)^t/(1+.015)^t = [(1+.065)/(1+.015)]^t

7 = [(1+.065)/(1+.015)]^t

Simplify and take "Log"

If you are still stuck - write back showing your work.

[Mark beat me to it]
 
Subhotosh Khan said:
You will need to use "Log".

56,000(1+.015)^t = 8,000(1+.065)^t

7 = (1+.065)^t/(1+.015)^t = [(1+.065)/(1+.015)]^t

7 = [(1+.065)/(1+.015)]^t

Simplify and take "Log"

If you are still stuck - write back showing your work.

[Mark beat me to it]
Click to expand...


I understand up until the point of, "Simplify and take 'Log'". Can you please explain this?
 
Or to deal with this kind of problem more generally


[MATH]56000 * 1. 015^t = 8000 * 1.065^2 \implies 7 * 1.015^t = 1.0165^2 \implies log(7 * 1.015^t) = log(1.065^t).[/MATH]
Now what?
 
I believe I was finally able to derive the correct answer with: log(7) = log(1.065/1.015)^t ==> .8451 = .0209T ==> T=40.43

Am I correct?
 
I get (after converting the equation I posted from exponential to logarithmic form):

[MATH]t=\log_{\frac{213}{203}}(7)\approx40.4671337842832[/MATH]
 
JaredRoss said:
I believe I was finally able to derive the correct answer with: log(7) = log(1.065/1.015)^t ==> .8451 = .0209T ==> T=40.43

Am I correct?
Click to expand...
T=40.43 ..................... what's the unit of T? seconds? Centuries?....
 
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