use pmt = \\frac{p(\\frac{r}{n})}{1-(1 + \\frac{r}{n})^{-nt}} to determine the regular payment amount…

use pmt = \\frac{p(\\frac{r}{n})}{1-(1 + \\frac{r}{n})^{-nt}} to determine the regular payment amount, rounded to the nearest dollar. the price of a small cabin is $35,000. the bank requires a 5% down - payment. the buyer is offered two mortgage options: 20 - year fixed at 7% or 30 - year fixed at 7%. calculate the amount of interest paid for each option. how much does the buyer save in interest with the 20 - year option?\nfind the monthly payment for the 20 - year option.\n$258\n(round to the nearest dollar as needed.)\nfind the monthly payment for the 30 - year option.\n$ \n(round to the nearest dollar as needed.)

use pmt = \\frac{p(\\frac{r}{n})}{1-(1 + \\frac{r}{n})^{-nt}} to determine the regular payment amount, rounded to the nearest dollar. the price of a small cabin is $35,000. the bank requires a 5% down - payment. the buyer is offered two mortgage options: 20 - year fixed at 7% or 30 - year fixed at 7%. calculate the amount of interest paid for each option. how much does the buyer save in interest with the 20 - year option?\nfind the monthly payment for the 20 - year option.\n$258\n(round to the nearest dollar as needed.)\nfind the monthly payment for the 30 - year option.\n$ \n(round to the nearest dollar as needed.)

Answer

Explanation:

Step1: Calculate the loan - amount

The price of the cabin is $P = 35000$. The down - payment is $5%$ of $35000$, so the down - payment amount is $0.05\times35000 = 1750$. The loan amount $L=35000 - 1750=33250$.

Step2: Identify the values for the 30 - year option

The annual interest rate $r = 0.07$, the number of times compounded per year $n = 12$ (monthly payments), and the number of years $t = 30$. So $nt=12\times30 = 360$ and $\frac{r}{n}=\frac{0.07}{12}$.

Step3: Use the payment formula

The payment formula is $PMT=\frac{L(\frac{r}{n})}{1-(1 + \frac{r}{n})^{-nt}}$. Substitute $L = 33250$, $\frac{r}{n}=\frac{0.07}{12}$, and $nt = 360$ into the formula: [ \begin{align*} PMT&=\frac{33250\times\frac{0.07}{12}}{1-(1+\frac{0.07}{12})^{-360}}\ &=\frac{33250\times0.0058333}{1-(1.0058333)^{-360}}\ \end{align*} ] First, calculate $(1.0058333)^{-360}\approx0.126267$. Then $1-(1.0058333)^{-360}=1 - 0.126267 = 0.873733$. And $33250\times0.0058333\approx194.95$. So $PMT=\frac{194.95}{0.873733}\approx223$.

Answer:

$223$