How much would you have to deposit in an account with a \( 7 \% \) interest rate, compounded monthly, to have \( \$ 1100 \) in your account 10 years later? \[ P=\$[?] \] \[ F=P\left(1+\frac{r}{n}\right)^{n t} \] Round to the nearest cent.

To solve for \( P \) in the equation \( F=P\left(1+\frac{r}{n}\right)^{n t} \), we need to isolate \( P \). Given that \( F = 1100 \), \( r = 0.07 \), \( n = 12 \) (monthly compounding), and \( t = 10 \): 1. Plug in the values: \[ P = \frac{F}{\left(1 + \frac{r}{n}\right)^{nt}} = \frac{1100}{\left(1 + \frac{0.07}{12}\right)^{12 \times 10}} \] 2. Calculate \( \frac{0.07}{12} \): \[ \frac{0.07}{12} \approx 0.0058333 \] 3. Calculate \( \left(1 + 0.0058333\right)^{120} \): \[ \left(1.0058333\right)^{120} \approx 2.222 \] 4. Now, calculate \( P \): \[ P \approx \frac{1100}{2.222} \approx 493.41 \] So, the amount to deposit is approximately: \[ P \approx \$493.41 \] --- Imagine you're planning a sunny vacation or a new gadget splurge in a decade! Knowing the historical power of compound interest is like having a financial wand—thanks to this magic, your money can grow even while you’re dreaming of that trip or product! It's the reason why savings accounts with higher interest rates can be a great ally, pushing you just a little closer to your goals over time. As for real-world trends, many folks underestimate the power of starting early—even a small deposit can blossom brilliantly over ten years! Think about your interests—what’s something fun you’d splurge on in the future? Maybe a travel spree or a tech upgrade? Start saving today, and watch your dreams unfold thanks to that savvy little interest compounding every month!