Expert: Paul Klarreich - 11/4/2011

Question
If an account that earns interest compounded continuously takes 12 years to double in value, how long will it take to triple in value?

Answer
Questioner: Kriste  
Country: Tennessee, United States
Category: Calculus
Private: No
Subject: Logarithmic Functions
Question: If an account that earns interest compounded continuously takes 12 years to double in value, how long will it take to triple in value?


CIF:  A = P(1 + i)^n, where

i = interest per interval
n = number of intervals

Assume Annual rate = r, and d = divisor, as in d = 4 means quarterly.

Since lim (x -> inf) (1 + 1/x)^x = e, we will massage the C.I.F. into that form:

A = P(1 + r/d)^d  << the(discrete) C.I.F.

If r/d = 1/x,  x = d/r,  d = rx

A = P(1 + 1/x)^rx

A = P((1 + 1/x)^x)^r

Now take lim(x->inf):

A = P e^r   << the CONTINUOUS C.I.F.  for one year.

and

A = p (e^r)^y,  for  y years.

A = P e^(ry)

Now your problem:

In 12 years:

2P = P e^12r

2 =  e^12r

12r = ln 2
     ln 2
r = --------  <<  computed interest rate.
      12

Now you want to triple, you say, in y  years:

3P = P e^(ry)

Now simplify, put in your computed value of r, and solve for y.

You can finish up.