the price of a small cabin is $45,000. the bank requires a 5% down - payment. the buyer can save in interest…

the price of a small cabin is $45,000. the bank requires a 5% down - payment. the buyer can save in interest by choosing a 20 - year mortgage option instead of a 30 - year mortgage option. calculate the amount of interest paid for each option. the buyer saves in interest with the 20 - year mortgage. how much? use pmt = p\\(\\frac{\\frac{r}{n}}{1-(1 + \\frac{r}{n})^{-nt}}\\) to determine the regular payment amount, rounded to the nearest dollar. the bank offers two mortgage options: 20 - year fixed at 7% or 30 - year fixed at 7%. find the monthly payment for the 20 - year option. round to the nearest dollar as needed. find the monthly payment for the 30 - year option. round to the nearest dollar as needed.

the price of a small cabin is $45,000. the bank requires a 5% down - payment. the buyer can save in interest by choosing a 20 - year mortgage option instead of a 30 - year mortgage option. calculate the amount of interest paid for each option. the buyer saves in interest with the 20 - year mortgage. how much? use pmt = p\\(\\frac{\\frac{r}{n}}{1-(1 + \\frac{r}{n})^{-nt}}\\) to determine the regular payment amount, rounded to the nearest dollar. the bank offers two mortgage options: 20 - year fixed at 7% or 30 - year fixed at 7%. find the monthly payment for the 20 - year option. round to the nearest dollar as needed. find the monthly payment for the 30 - year option. round to the nearest dollar as needed.

Answer

Explanation:

Step1: Identify the mortgage - payment formula variables

The mortgage - payment formula is $PMT = P\frac{\frac{r}{n}}{1-(1 + \frac{r}{n})^{-nt}}$, where $P$ is the principal amount, $r$ is the annual interest rate (in decimal), $n$ is the number of payments per year, and $t$ is the number of years. The principal amount $P=45000$, the annual interest rate $r = 0.07$, and $n = 12$ (monthly payments).

Step2: Calculate for the 20 - year option

For $t = 20$ years, we first calculate the exponent $-nt=-12\times20=-240$ and $\frac{r}{n}=\frac{0.07}{12}$. $PMT_{20}=45000\times\frac{\frac{0.07}{12}}{1-(1+\frac{0.07}{12})^{-240}}$ Let $x=\frac{0.07}{12}\approx0.005833$ and $y = 1 + x=1.005833$. $PMT_{20}=45000\times\frac{0.005833}{1 - y^{-240}}$ $y^{-240}=\frac{1}{y^{240}}\approx\frac{1}{4.0387}=0.2476$ $1 - y^{-240}=1 - 0.2476 = 0.7524$ $PMT_{20}=45000\times\frac{0.005833}{0.7524}=45000\times0.007753\approx348.89\approx349$

Step3: Calculate for the 30 - year option

For $t = 30$ years, the exponent $-nt=-12\times30=-360$ $PMT_{30}=45000\times\frac{\frac{0.07}{12}}{1-(1+\frac{0.07}{12})^{-360}}$ Let $x=\frac{0.07}{12}\approx0.005833$ and $y = 1 + x = 1.005833$ $y^{-360}=\frac{1}{y^{360}}\approx\frac{1}{8.1177}=0.1232$ $1 - y^{-360}=1 - 0.1232=0.8768$ $PMT_{30}=45000\times\frac{0.005833}{0.8768}=45000\times0.006652\approx299.34\approx299$

Answer:

For the 20 - year option: 349 For the 30 - year option: 299