math question?

Question

A commonly asked question is, “How long will it take to double my money?” At 8% interest rate and continuous compounding, what is the answer? Round your answer to the hundredth's place. i got 8.66 years

Answer 1

Your answer is correct!

The continuous compounding formula is:

A = a [e^(rt)],

where A is the final amount after compounding, a is the initial investment, e is the base of the natural logarithmic system, r is the annual interest rate and t is the elapsed time in years.

Since the problem stipulates that the initial investment be doubled, then A/a = 2. Then we can rewrite the above equation like so:

2 = e^(rt).

Now we can take the natural logarithm of both sides of this equation to find t.

ln 2 = (rt) ln e, and since ln e = 1, this last equation becomes:

ln 2 = rt.

To find t, we simply divide ln 2 by the annual interest rate:

ln 2/r = t
ln 2/.08 ~ t
8.6643 ~ t.

So, it will actually take slightly more than 8.66 years to double, but the difference is really inconsequential. 0.0043 years is only equivalent to slightly more than 1.5 days.

Some of the other responders did not seem to take notice of the fact that this calls for continuous compounding, not straight compounding. Continuous compounding increases the value of an investment more quickly, thus it takes less time to reach a given value than it would with straight compounding. That's why the use of the special formula above is necessitated. If this problem had been done using straight compounding, then the answer would have been about 9.0065 years.

Answer 2

It will not be double at rate of 8% in 8.66 years. It will take 9 years to double at this rate of interest.

Answer 3

use the rule of 72. Just divide 72 by 8

Answer 4

hi,
my answer is ~9 years (9.0064...).

here is how I got to it:
1. every year your assets grows by another 8%.
2. that mean that (year n)=(year n-1)*1.08
3. the generic form is:
"assets after n years" = "first year" * 1.08^n (n power 1.08 or wise versa - I am not sure how you say it)
4. let's call your initial assets "a" and then we want to calculate
how many years it should take for "a" to become "2a" so:
2a = a*1.08^n
5. (since a is not 0 we can divide by a)
2=1.08^n
6. (this is quite simple, you do log 1.08 for both sides):
log 1.08 (2) = n ==> n = ~9

to calculate log1.08 with a calculator you can use:
log 1.08 (2) = log 10 (2) / log 10 (1.08)

to verify my calculation:
1.08^9 = 1.999 ~= 2