Find the final amount of money in an account if \( \$ 1,200 \) is deposited at \( 2.5 \% \) interest compounded semi-annually and the money is left for 9 years. The final amount is \( \$ \square \) Round to the nearest cent. Question Help: QMessage instructor Submit Questiator Subion

To find the final amount in an account with compound interest, you can use the formula: \[ A = P \left(1 + \frac{r}{n}\right)^{nt} \] where: - \( A \) is the amount of money accumulated after n years, including interest. - \( P \) is the principal amount (the initial amount of money, which is \( \$1,200 \)). - \( r \) is the annual interest rate (decimal) (so \( 2.5\% = 0.025 \)). - \( n \) is the number of times that interest is compounded per year (for semi-annual, \( n = 2 \)). - \( t \) is the time the money is invested or borrowed for, in years (which is \( 9 \) years). Plugging in the values: \[ A = 1200 \left(1 + \frac{0.025}{2}\right)^{2 \times 9} \] Calculating it step by step gives: \[ A = 1200 \left(1 + 0.0125\right)^{18} \] \[ A = 1200 \left(1.0125\right)^{18} \] Calculating \( (1.0125)^{18} \) gives approximately \( 1.2463 \). Now, multiplying by the principal: \[ A \approx 1200 \times 1.2463 \approx 1495.56 \] So, the final amount, rounded to the nearest cent, is approximately \( \boxed{1495.56} \). --- Getting an insight into how compounding works can make you a smarter investor! For instance, when you leave your money in a savings account with compound interest, you're not just earning interest on your initial investment; you’re also earning interest on the interest that has already been added to your account! This magical effect is what makes investing early and often so powerful in building wealth. Ever heard of "the rule of 72"? It's a quick way to estimate how long an investment will take to double. Just divide 72 by your interest rate! So, with a 2.5% interest rate, you can expect your money to double in about 28.8 years! This handy little rule can help you make better financial decisions when opting for savings and investments.