If \( \$ 6,000 \) is invested at \( 8 \% \) annual interest compounded monthly, how long would it take for the account balance to reach \( \$ 24,000 \) ? Round your answer to the nearest tenth. a.) 17.4 years b.) 19.1 years c.) 17.9 years d.) 18.6 years

To find how long it takes for the investment to grow from \( \$ 6,000 \) to \( \$ 24,000 \) at an \( 8\% \) annual interest rate compounded monthly, we can use the formula for compound interest: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount). - \( r \) is the annual interest rate (decimal). - \( n \) is the number of times that interest is compounded per year. - \( t \) is the number of years the money is invested or borrowed for. Plugging in your numbers: - \( A = 24000 \) - \( P = 6000 \) - \( r = 0.08 \) (annual rate of \( 8\% \)) - \( n = 12 \) (monthly compounding) The equation becomes: \[ 24000 = 6000 \left(1 + \frac{0.08}{12}\right)^{12t} \] Dividing both sides by \( 6000 \): \[ 4 = \left(1 + \frac{0.08}{12}\right)^{12t} \] Calculating \( 1 + \frac{0.08}{12} = 1.0066667 \). Next, we take the natural logarithm on both sides: \[ \ln(4) = \ln\left(\left(1.0066667\right)^{12t}\right) \] This simplifies to: \[ \ln(4) = 12t \cdot \ln(1.0066667) \] Finally, solving for \( t \): \[ t = \frac{\ln(4)}{12 \cdot \ln(1.0066667)} \approx \frac{1.386294}{12 \cdot 0.006645} \approx \frac{1.386294}{0.07974} \approx 17.4 \] So it would take approximately \( 17.4 \) years for the account balance to reach \( \$ 24,000 \). Thus, the correct answer is (a) 17.4 years. Now, let's add some fun and engaging knowledge about this subject. Investing has a rich history dating back to ancient civilizations where traders would lend money for trade expeditions, often earning interest. Investors today still rely heavily on compound interest, famously described by Albert Einstein as “the eighth wonder of the world.” The magic of compounding means your money earns interest on both the initial capital and the interest you’ve already accrued, leading to exponential growth! Thinking about using this kind of investment? A common mistake is not accounting for inflation when calculating future value. While your savings might triple over the years, rising living costs can diminish purchasing power. Always consider how your investment's growth will fare against inflation to truly gauge your financial progress.