Emilia and Liam are purchasing a home. They wish to save money for 10 years and purchase a house that has a value of \( \$ 150,000 \) with cash. If they deposit money into an account paying \( 4 \% \) interest. compounded monthly, how much do they need to deposit each month in order to make the purchase? Round your answer to the nearest cent, if necessary. Formulas Answer How to enter your answer (opens in new window)

To calculate how much Emilia and Liam need to deposit each month, we can use the future value of an ordinary annuity formula: \[ FV = P \times \frac{(1 + r)^n - 1}{r} \] Where: - \( FV \) is the future value of the annuity (in this case, \$150,000) - \( P \) is the monthly deposit - \( r \) is the monthly interest rate (annual rate divided by 12 months, so \( 0.04/12 \)) - \( n \) is the total number of deposits (10 years times 12 months per year = 120). Rearranging the formula to solve for P gives: \[ P = \frac{FV \cdot r}{(1 + r)^n - 1} \] Substituting the values provides: \[ r = 0.04 / 12 = 0.003333 \] Thus: \[ P = \frac{150,000 \cdot 0.003333}{(1 + 0.003333)^{120} - 1} \approx \frac{500}{0.491825} \approx 1,016.68 \] So, Emilia and Liam need to deposit approximately \$1,016.68 each month. For those who love numbers, this problem shows how saving a little each month can lead to big dreams! If Emilia and Liam stick to this plan, they can turn their vision of homeownership into reality in just a decade. Keep in mind, financial planning isn't just about numbers; it's about transforming aspirations into tangible outcomes! Now that we’ve laid out the math, remember to account for other costs of home buying, like taxes, insurance, and potential repairs. It’s easy to focus solely on the purchase price, but homeownership often comes with many unexpected expenses. Budgeting for these extras can make for a smoother transition into your new abode!