How long will it take money to double if it is invested at​ (A) 13​% compounded​ continuously? (B) 15​% compounded​ continuously? (2024)

There are several ways to do this problem. The easiest way to solve it is to take the ln(2) and divide it by the interest rate. For this problem, assuming that your rates are annual, this formula gives us ln(2)/.13 = 5.33 years and ln(2)/.15 = 4.62 years.

Why does this work? We know that the continuous compounding formula is A = Pe^(rt). Therefore, we can solve this equation to indicate doubling your investment like so: 2 = 1e^(rt). For the first question, we can then plug in our rate of 13%. This would give us 2 = e^(.13t). In order to solve for t, we would take the ln of both sides of the equation (ln 2 = ln e^(.13t)), which would then simplify to ln 2 = .13t. Therefore, we are back to our original equation, which is ln 2 / r = t.