An amount of money is invested at 12%/a interest. Write an equation and solve it to determine the length of time it will take for the original amouunt to double if the interest is compounded annually.
The rule of 72 gives the rough period as 72/12=6 years.
The accurate calculation is as follows:
P=principal
R=interest rate= 1.12 for 12%
n=number of years required.
So
PRn=2P
Rn=2
Take log on both sides,
log(Rn=log(2)
n*log(R) = log(2)
n=log(2)/log(R)
=log(2)/log(1.12)
=0.3010/0.0492
=6.116 years.
To determine the length of time it will take for the original amount to double at a 12% annual interest rate when compounded annually, we can use the formula for compound interest:
A = P(1 + r/n)^(nt)
Where:
A = the future value of the investment (in this case, double the original amount)
P = the initial investment (original amount)
r = the annual interest rate (12% as a decimal, which is 0.12)
n = the number of times the interest is compounded in a year (since it is compounded annually, n = 1)
t = the time (in years)
We can rewrite the formula for doubling the original amount as follows:
2P = P(1 + 0.12/1)^(1*t)
Simplifying the equation:
2 = (1.12)^t
Now, we can solve for t by taking the logarithm of both sides of the equation. Since the base of the logarithm isn't specified, we can use the natural logarithm (ln) or the common logarithm (log base 10).
Using the natural logarithm (ln):
ln(2) = ln(1.12)^t
t * ln(1.12) = ln(2)
t = ln(2) / ln(1.12)
Using a calculator, evaluate ln(2) / ln(1.12) to find the value of t. The result will give you the number of years it takes for the original amount to double at a 12% annual interest rate compounded annually.