Compound Interest

**BeSweeet** · October 5th 2009, 01:05 PM

I need to find the initial investment and time to double. It is compounded continuously. I'm given the annual % rate and the amount after 6 years. So:

Initial Investment: $?????
Annual % Rate: 5%
Time to Double: ???? yr
Amount After 6 Years: $2000.00

Not sure where to begin on this one.

**masters** · October 5th 2009, 01:39 PM

Originally Posted by BeSweeet

I need to find the initial investment and time to double. I'm given the annual % rate and the amount after 6 years. So:

Initial Investment: $?????
Annual % Rate: 5%
Time to Double: ???? yr
Amount After 6 Years: $2000.00

Not sure where to begin on this one.

How is the investment compounded? Annually, Semi-annually, quarterly????

**BeSweeet** · October 5th 2009, 01:41 PM

Continuously.

**masters** · October 5th 2009, 02:09 PM

Originally Posted by BeSweeet

I need to find the initial investment and time to double. It is compounded continuously. I'm given the annual % rate and the amount after 6 years. So:

Initial Investment: $?????
Annual % Rate: 5%
Time to Double: ???? yr
Amount After 6 Years: $2000.00

Not sure where to begin on this one.

Hi BeSweet,

Your formula for amount accrued with interest compounded continuously is

$A=Pe^{rt}$

$2000=Pe^{(.05)(6)}$

$2000=Pe^{.3}$

$P=\frac{2000}{e^{.3}}\approx 1481.64$

This will give you the initial investment. Now the time to double that investment......

$2= e^{rt}$

$2=e^{(.05)t}$

$\ln 2 = \ln e^{.05t}$

$\ln 2 = .05t$

$t=\frac{\ln 2}{.05} \approx 13.86$ years.

**BeSweeet** · October 5th 2009, 04:21 PM

Wow, thanks a bunch! See, this is how I feel I learn better. I like it when someone posts the steps, which lets me analyze them and figure out how that step came about. No offense to anyone else who has ever helped me.

One question: in the second part, where did P go from Pe^rt?

**HallsofIvy** · October 6th 2009, 04:23 AM

"Time to double" meant t such that $Pe^{rt}= 2P$ . Divide both sides by P.

And "feeling that I learn better" when someone shows you all the steps is often an illusion. Yes, everything makes sense but can you do it on your own? The only way to really learn mathematics is to do it yourself!

**masters** · October 6th 2009, 04:24 AM

Originally Posted by BeSweeet

Wow, thanks a bunch! See, this is how I feel I learn better. I like it when someone posts the steps, which lets me analyze them and figure out how that step came about. No offense to anyone else who has ever helped me.

One question: in the second part, where did P go from Pe^rt?

Since you were looking for the time when your original investment would double, then A divided by P would be 2. That's where the P went.

$A=Pe^{rt}$

$\frac{A}{P}=e^{rt}$

$2=e^{rt}$

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