Mike opened a savings account and deposited \( \$ 410.00 \). The account earns \( 4.36 \% \) interest, compounded quarterly. If he wants to use the money to buy a new bicycle in 3 years, how much will he be able to spend on the bike? Use the formula \( A=P\left(1+\frac{r}{n}\right)^{n t} \), where \( A \) is the balance (final amount), \( P \) is the principal (starting amount), \( r \) is the interest rate expressed as a decimal, \( n \) is the number of times per year that the interest is compounded, and \( t \) is the time in years. Round your answer to the nearest cent. \( \$ \square \)

To calculate the amount Mike will have in his savings account after 3 years, we can plug the values into the formula \( A = P\left(1 + \frac{r}{n}\right)^{nt} \). Here, \( P = 410.00 \), \( r = 0.0436 \), \( n = 4 \) (quarterly), and \( t = 3 \). After crunching the numbers, we get: \[ A = 410 \left(1 + \frac{0.0436}{4}\right)^{4 \times 3} \] Calculating that results in approximately \( A = 410 \left(1.0109\right)^{12} \approx 410 \times 1.1304 \approx 463.36 \). So, Mike will be able to spend approximately \( \$463.36 \) on the bicycle! In the early days of banking, interest was often calculated as a simple percentage based on the initial amount deposited, leading to much lower returns than we see with compound interest today. The beauty of compounding, especially when done quarterly or more frequently, is that it allows your money to grow exponentially over time, making saving for items like bikes more achievable. When it comes to saving and spending, setting clear timeframes and goals, like Mike did for his bicycle, can boost your savings drive. Don't overlook the power of regular contributions! Even small, periodic deposits, alongside compound interest, can significantly increase your savings over time. Just remember not to hop on that spending bicycle too early!