\( 1< \) The price of a home is \( \$ 230,000 \). The bank requires a \( 15 \% \) down payment. The buyer is offered two mortgage options: 15 -year fixed at \( 6.5 \% \) or 30 -year fixed at \( 6.5 \% \). Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 15 -year option? Use the following formula to determine the regular payment amount. PMT \( =\frac{\mathrm{P}\left(\frac{\mathrm{r}}{\mathrm{n}}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} \) Find the monthly payment for the 15 -year option. \( \$ 1,703 \) (Round to the nearest dollar as needed.) Find the monthly payment for the 30-year option. \( \$ \square \) (Round to the nearest dollar as needed.)
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To calculate the monthly payment for the 30-year option, we first need to determine the principal amount after the down payment. The down payment for a \( \$ 230,000 \) home at \( 15\% \) is \( \$ 34,500 \), making the principal \( \$ 195,500 \). Now using the provided formula where \( P \) is the principal (\$ 195,500), \( r \) is the annual interest rate \( 0.065 \), \( n \) is the number of payments per year (12), and \( t \) is the term in years (30): \[ PMT = \frac{195500 \left(\frac{0.065}{12}\right)}{\left[1-\left(1+\frac{0.065}{12}\right)^{-12 \cdot 30}\right]} \] Calculating this, we find the monthly payment for the 30-year option to be approximately \( \$ 1,244 \). Now, to find the amount saved in interest between the two options, you would need to calculate the total payments over the life of each loan and subtract the two. In the 15-year option, the total paid would be \( 1,703 \times 180 \) (15 years), while for the 30-year option, it's \( 1,244 \times 360 \). Subtract the total from the 15-year option from the total for the 30-year option to find the savings in interest. Over the loan period, you save a significant amount in interest with the shorter term. To summarize: - Monthly payment for the 30-year option: \( \$ 1,244 \) - Savings in interest with the 15-year option: Simply calculate both totals and subtract!
