How long does it take your money to double if it is left in an account for 20 years and earns 8.5% interest compounded continuously?
2 = e^(.085 t)
ln(2) = .085 t
To determine how long it takes for your money to double with continuous compounding, you can use the formula to calculate exponential growth:
A = P * e^(rt)
Where:
A is the future value (amount of money after a certain period)
P is the principal amount (initial investment)
e is the mathematical constant approximately equal to 2.71828
r is the interest rate per year (in decimal form)
t is the time period in years
In this case, you want to find the time it takes for your money to double, so A = 2P (twice the initial amount). The interest rate (r) is 8.5% or 0.085, and the time period (t) is 20 years.
Plugging in these values, we get:
2P = P * e^(0.085 * 20)
Now we can solve for t by simplifying the equation:
2 = e^(1.7)
To find t, we need to take the natural logarithm (ln) of both sides of the equation:
ln(2) = ln(e^(1.7))
Using the property of logarithms, ln(a^b) = b * ln(a), the equation simplifies to:
ln(2) = 1.7
Now we can solve for t by dividing both sides of the equation by 1.7:
t = ln(2) / 1.7
Using a calculator or a computer program, we can determine that ln(2) is approximately 0.6931. Dividing this value by 1.7, we find:
t ≈ 0.4071
Therefore, it takes approximately 0.4071 years (or approximately 4.85 months) for your money to double in an account with continuous compounding at an interest rate of 8.5% per year over a 20-year period.