Emma is going to invest in an account paying an interest rate of \( 5.5 \% \) compounded annually. How much would Emma need to invest, to the nearest hundred dollars, for the value of the account to reach \( \$ 480 \) in 8 years?

To find out how much Emma needs to invest now, we can use the formula for compound interest: \[ A = P(1 + r)^n \] 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 years the money is invested. In this case, \( A = 480 \), \( r = 0.055 \), and \( n = 8 \). Rearranging the formula to solve for \( P \): \[ P = \frac{A}{(1 + r)^n} = \frac{480}{(1 + 0.055)^8} \] Calculating \( (1 + 0.055)^8 \): Calculating \( (1.055)^8 \) gives approximately \( 1.48024 \). Substituting back into the equation: \[ P = \frac{480}{1.48024} \approx 324.35 \] Rounding to the nearest hundred dollars, Emma would need to invest approximately \( \$ 300 \). If you round more precisely, you can recognize that Emma would need to set aside a little extra cushion, just in case hey? So let’s keep it around $300—because it’s safer with investments!