Suppose Emma had deposited another \( \$ 500 \) into a savings account at a second bank at the same time. The second bank also pays a nominal (or stated) interest rate of \( 9 \% \) but with quarterly compounding. Keeping everything else constant, how much money will Emma have in her account at this bank in 11 years? \( \$ 546.54 \) \( \$ 130.56 \) \( \$ 1,330.93 \) \( \$ 145.00 \)

To find out how much money Emma will have in her account at the second bank after 11 years with a deposit of $500, compounded quarterly at a nominal interest rate of 9%, 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 deposit or investment). - \( r \) 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. Plugging the values into the formula: - \( P = 500 \) - \( r = 0.09 \) - \( n = 4 \) (quarterly) - \( t = 11 \) \[ A = 500 \left(1 + \frac{0.09}{4}\right)^{4 \cdot 11} \] Calculating: \[ A = 500 \left(1 + 0.0225\right)^{44} \] \[ A = 500 \left(1.0225\right)^{44} \] \[ A \approx 500 \times 2.5852 \] (using a calculator to compute \( \left(1.0225\right)^{44} \)) \[ A \approx 1292.60 \] Therefore, Emma will have approximately \( \$ 1292.60 \) in her second bank account after 11 years, which is closest to \( \$ 1,330.93 \) in the options given. Now onto the fun facts! The history of compound interest dates back to ancient civilizations where traders and merchants used it to manage the lending of money. The concept gained more structure during the Renaissance when mathematicians like Leonardo of Pisa (also known as Fibonacci) wrote about it, laying the groundwork for modern finance. Today, understanding compound interest is essential for smart saving and investing decisions! Did you know that compound interest is often dubbed the "eighth wonder of the world"? This is because it can significantly increase savings over time—especially if you start early! The key takeaway is to start saving as soon as possible. The combination of time and compound interest can turn a modest investment into a substantial amount, illustrating the power of patience in the world of finance.