How long will it take an investment to double at 10% annual interest?

Question

assume that all interest rates are compound continously

Answer 1

If compounding continuously, you use the equation A = Pe^rt where A is the amount you want to end with, P is starting amount, e is Euler's number, r is the rate (in decimal form), and t is the time.

 

The problem told you the rate is 10%, so r = .10

Even though you don't have an actual number for where you start or end, that's no problem since we want to double it.  Whatever P is, that makes A = 2P (double what you started with).  Our equation should look like this:

 

2P = Pe^.10t

2 = e^.10t (divided both sides by P, which essentially cancels them out)

 

Next, we have to change this into a logarithmic form.  If you're unsure how to do this, then take the natural log of both sides (since we're dealing with 'e').

 

ln 2 = ln e^.10t

 

ln 2 = .10t ln e (power rule of logs say the exponents in logs can be brought in front of the log)

 

ln 2 = .10t   (ln e = 1, so that part goes away)

 

(ln 2) / .10 = t  (just divided both sides by .10)

 

t = 6.93 years