Final answer:
Using the compound interest formula, it would take approximately 23 years for a $6,000 investment to grow to $24,000 with a 6% annual interest rate compounded monthly.
Step-by-step explanation:
To determine how long it will take to grow an initial investment of $6,000 to $24,000 at a 6% interest rate, compounded monthly, we can use the formula for compound interest:
A = P (1 + \frac{r}{n})^{nt}
Where:
- A is the amount of money accumulated after n years, including interest.
- P is the principal amount (the initial amount of money).
- r is the annual interest rate (decimal).
- n is the number of times that interest is compounded per year.
- t is the time the money is invested for, in years.
First, we need to solve for t when A is $24,000, P is $6,000, r is 0.06 (since 6% is 0.06 as a decimal), and n is 12 (since the interest is compounded monthly).
Putting these values into the formula:
$24,000 = $6,000 (1 + \frac{0.06}{12})^{12t}
Now we solve for t. We first divide both sides by $6,000:
4 = (1 + \frac{0.06}{12})^{12t}
We then take the natural logarithm of both sides:
ln(4) = ln((1 + \frac{0.06}{12})^{12t})
ln(4) = 12t * ln(1 + \frac{0.06}{12})
Then, we divide both sides by (12 * ln(1 + 0.06/12)) to solve for t:
t = \frac{ln(4)}{12 * ln(1 + \frac{0.06}{12})}
Using a calculator, we find that t is approximately 23.45 years, which can be rounded to the nearest whole number as 23 years.
It will take approximately 23 years for the $6,000 investment to grow to $24,000 with a 6% interest rate, compounded monthly.