If $8,500 is invested at 6% compounded continuously, how long will it take to double the investment?

Continuous compounding:

future value = present value * ert
where t=number of periods, and r=rate

future value/present value = 2
or
ert=2
e0.06t=2
take natural log on both sides,
0.06t = ln(2)
t=11.55 years.

The rule of 69
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In fact, you can apply the rule of 69 for continuous compounding:
the time (in years) to double is 69.31 divided by the annual rate in %.

To determine the time it will take for an investment to double with continuous compounding, you can use the formula for continuous compound interest:

A = P * e^(rt)

Where:
A = the final amount
P = the principal (initial investment)
e = Euler's number (approximately 2.71828)
r = the interest rate
t = the time (in years)

In this case, the final amount will be twice the initial investment, so A = 2P. The interest rate, r, is given as 6%, which can be expressed as 0.06.

Substituting these values into the formula:

2P = P * e^(0.06t)

Simplifying the equation by canceling out the initial investment (P) from both sides:

2 = e^(0.06t)

Now, take the natural logarithm (ln) of both sides to isolate t:

ln(2) = ln(e^(0.06t))

Applying the property of logarithms that ln(e^x) = x:

ln(2) = 0.06t

Finally, divide both sides of the equation by 0.06 to solve for t:

t = ln(2) / 0.06
t ≈ 11.55 years

So, it will take approximately 11.55 years for the investment to double with continuous compounding at a 6% interest rate.