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 $45,000. the bank requires a 5% down payment. the buyer is offered two mortgage options: 20 - year fixed at 7.5% or 30 - year fixed at 7.5%. calculate the amount of interest paid for each option. how much does the buyer save in interest with the 20 - year option? find the monthly payment for the 20 - year option. $344 (round to the nearest dollar as needed.) find the monthly payment for the 30 - year option. (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 $45,000. the bank requires a 5% down payment. the buyer is offered two mortgage options: 20 - year fixed at 7.5% or 30 - year fixed at 7.5%. calculate the amount of interest paid for each option. how much does the buyer save in interest with the 20 - year option? find the monthly payment for the 20 - year option. $344 (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: Calculate the loan - amount

The price of the cabin is $45000. The down - payment is 5% of $45000, so the down - payment $D = 0.05\times45000=2250$. The loan amount $P=45000 - 2250 = 42750$.

Step2: Identify the values for the 30 - year option

For a 30 - year mortgage, $n = 12$ (monthly payments) and $t = 30$ years, so the number of payments $mt=12\times30 = 360$. The annual interest rate $r = 0.075$, so the monthly interest rate $i=\frac{r}{n}=\frac{0.075}{12}=0.00625$.

Step3: Calculate the monthly payment for the 30 - year option using the formula $PMT=\frac{P(\frac{r}{n})}{1-(1 + \frac{r}{n})^{-nt}}$

Substitute $P = 42750$, $i = 0.00625$, and $nt = 360$ into the formula: [ \begin{align*} PMT&=\frac{42750\times0.00625}{1-(1 + 0.00625)^{-360}}\ &=\frac{267.1875}{1 - 0.10757}\ &=\frac{267.1875}{0.89243}\ &\approx299 \end{align*} ]

Answer:

$299$