Arnett College predicts that in 18 years it will take \( \$ 350,000 \) to attend the college for four years. Emma has a substantial amount of cash and wishes to invest a lump sum of money for her child's college fund. How much should Emma put aside in an account with an APR of \( 12 \% \) compounded monthly in order to have \( \$ 350,000 \) in the account in 18 years? Round your answer to the nearest cent, if necessary. Formulas

To find out how much Emma needs to invest today (the present value), we can use the formula for future value with compound interest, which is: \[ FV = PV \times (1 + \frac{r}{n})^{nt} \] Where: - \( FV \) is the future value (\$350,000 in this case), - \( PV \) is the present value (the amount to invest today), - \( r \) is the annual interest rate (0.12), - \( n \) is the number of times the interest is compounded per year (12 for monthly), - \( t \) is the number of years until the future value is needed (18). Rearranging the formula to solve for \( PV \): \[ PV = \frac{FV}{(1 + \frac{r}{n})^{nt}} \] Plugging in the values: \[ PV = \frac{350,000}{(1 + \frac{0.12}{12})^{12 \times 18}} \] Now calculating: \[ PV = \frac{350,000}{(1 + 0.01)^{216}} \] \[ = \frac{350,000}{(1.01)^{216}} \] Using a calculator, \( (1.01)^{216} \approx 6.680 \): \[ PV = \frac{350,000}{6.680} \approx 52,300.80 \] Emma needs to invest approximately **$52,300.80** now to reach her goal of $350,000 in 18 years. Saving for college can feel daunting, but it’s a journey worth taking! Knowing how to calculate your future savings will help demystify the process and give you peace of mind. Plus, consider that these investments can grow significantly over time, so starting early is the key! Want to dive deeper into finance? Check out books like "The Intelligent Investor" by Benjamin Graham, which offers timeless principles on investing, or "Your Money or Your Life" by Vicki Robin, which transforms your relationship with money and prioritizes financial independence!