To determine how long it will take for $500 to double at a 6% annual interest rate compounded semi-annually (2 times a year), we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the amount of money accumulated after n years, including interest
P = the principal amount ($500 in this case)
r = annual interest rate (6% or 0.06)
n = number of times the interest is compounded per year (2)
t = time the money is invested for
Since we want to find out when the amount will double, we can set A to be $1000 (double of $500) and solve for t:
$1000 = $500(1 + 0.06/2)^(2t)
Simplify the formula:
2 = (1.03)^(2t)
Now, we need to solve for t using logarithms:
2t * log(1.03) = log(2)
t = log(2) / (2 * log(1.03))
Calculate t:
t ≈ log(2) / (2 * log(1.03))
t ≈ 11.90 years
Therefore, it will take approximately 11.90 years for $500 to double at a 6% annual interest rate compounded semi-annually.
