Question 1151563
f = p * (1 + r) ^ n
f is the future value
p is the present value
r is the interest rate per time period
n is the number of time periods.


with annual compounding and f = 2 * p and r = .04, the formula becomes:
2 = (1 + .04) ^ n
take the natural log of both sides of this equation to get:
ln(2) = ln(1.04^n)
since ln(1.04^n) is equal to n * ln(1.04), this becomes:
ln(2) = n * ln(1.04)
divide both sides of this equation by ln(1.04) and solve for n to get:
n = ln(2) / ln(1.04) = 17.67298769
confirm by replacing n with that in the original equation to get:
f = (1 + .04) ^ 17.67298769
solve for f to get:
f = 2
this confirms the solution is correct.
the solution, assuming annual compounding, is equal to 17.67298769 years.


if you assume monthly compounding, the formula becomes:
2 = (1 + .04/12) ^ n
n becomes the number of months.
take the natural log of both sides of this equation to get:
ln(2) = ln((1+ .04/12) ^ n)
this becomes ln(2) = n * ln(1 + .04/12)
divide both sides of the equation by ln(1 + .04/12) to get:
ln(2) / ln(1 + .04/12) = n
solve for n to get n = 208.2905355
that's the number of months.
divide that by 12 to get number of years = 17.35754463


with monthly compoounding, the number of years to double the money is slightly less because monthly compounding gives a higher effective interest rate per year than annual compounding.


with annual compounding, the effective annual interest rate is .04.
with monthly compounding, the effective annual interest rate is (1 + .04/12) ^ 12 - 1 = .040741543.


that's the rate,
the percent is 100 times that.
.04 = 4%
.040741543 = 4.0741543%