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? first, find the monthly payment for the 20 - year option. $amount (round to the nearest dollar as needed.) find the monthly payment for the 30 - year option. $amount (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? first, find the monthly payment for the 20 - year option. $amount (round to the nearest dollar as needed.) find the monthly payment for the 30 - year option. $amount (round to the nearest dollar as needed.)

Answer

Explanation:

Step1: Calculate the loan - amount

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

Step2: Calculate the monthly interest rate and number of payments for 20 - year option

The annual interest rate $r_1 = 7.5%=0.075$. The monthly interest rate $i_1=\frac{0.075}{12}=0.00625$. The number of payments $n_1 = 20\times12 = 240$. Using the formula $PMT=\frac{L\times i_1}{1-(1 + i_1)^{-n_1}}$, we substitute $L = 42750$, $i_1=0.00625$ and $n_1 = 240$: [ \begin{align*} PMT_1&=\frac{42750\times0.00625}{1-(1 + 0.00625)^{-240}}\ &=\frac{267.1875}{1 - 0.224929}\ &=\frac{267.1875}{0.775071}\ &\approx345 \end{align*} ] The total amount paid over 20 years is $345\times240 = 82800$. The total interest paid $I_1=82800 - 42750=40050$.

Step3: Calculate the monthly interest rate and number of payments for 30 - year option

The annual interest rate $r_2 = 7.5% = 0.075$. The monthly interest rate $i_2=\frac{0.075}{12}=0.00625$. The number of payments $n_2=30\times12 = 360$. Using the formula $PMT=\frac{L\times i_2}{1-(1 + i_2)^{-n_2}}$, we substitute $L = 42750$, $i_2 = 0.00625$ and $n_2 = 360$: [ \begin{align*} PMT_2&=\frac{42750\times0.00625}{1-(1 + 0.00625)^{-360}}\ &=\frac{267.1875}{1-0.107467}\ &=\frac{267.1875}{0.892533}\ &\approx300 \end{align*} ] The total amount paid over 30 years is $300\times360 = 108000$. The total interest paid $I_2=108000 - 42750 = 65250$.

Step4: Calculate the interest savings

The interest savings with the 20 - year option is $I_2 - I_1=65250 - 40050 = 25200$.

Answer:

The monthly payment for the 20 - year option is $$345$. The monthly payment for the 30 - year option is $$300$. The interest paid for the 20 - year option is $$40050$. The interest paid for the 30 - year option is $$65250$. The buyer saves $$25200$ in interest with the 20 - year option.