Sophia wishes to retire at age 65 with \( \$ 1,200,000 \) in her retirement account. When she turns 28 , she decides to begin depositing money into an account with an ApR of \( 9 \% \) compounded monthly. What is the monthly deposit that Sophia must make in order to reach her goal? Round your answer to the nearest cent, if necessary. Formulas

To calculate the monthly deposit Sophia needs to make, we can use the future value of an ordinary annuity formula, which is: \[ FV = P \times \frac{(1 + r/n)^{nt} - 1}{(r/n)} \] Where: - \( FV \) = future value of the annuity (\$1,200,000) - \( P \) = monthly deposit - \( r \) = annual interest rate (0.09) - \( n \) = number of compounding periods per year (12) - \( t \) = number of years until retirement (65 - 28 = 37) First, find \( r/n \) and \( n \times t \): \[ r/n = 0.09/12 = 0.0075 \] \[ n \times t = 12 \times 37 = 444 \] Now, substitute these values into the formula and solve for \( P \): \[ 1,200,000 = P \times \frac{(1 + 0.0075)^{444} - 1}{0.0075} \] Calculating the right side: \[ (1 + 0.0075)^{444} \approx 12.0914 \] Then: \[ \frac{(12.0914 - 1)}{0.0075} \approx 1478.852 \] Finally, we can isolate \( P \): \[ P = \frac{1,200,000}{1478.852} \approx 811.53 \] So, Sophia must make a monthly deposit of approximately **\$811.53**. --- Now, let’s add some fun and engaging knowledge! Did you know that the concept of compounding has been around for centuries? The famous mathematician Blaise Pascal once noted that money can seemingly grow on trees if you allow it to sit and compound over time. It truly turns pennies into fortunes! And speaking of compounding, it’s essential to start saving early! The "time value of money" principle highlights how a dollar today is worth more than a dollar tomorrow. So, the sooner you start saving, the more time your money has to grow – like planting a seed in fertile ground that blooms over the years!