Use PMT $$ =\frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ 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 $$ 10 \% $$ or 30 -year fixed at 10\%. 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. $$ \$ \square $$ (Round to the nearest dollar as needed.)

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Alright, I need to determine the regular payment amount for a small cabin priced at \$45,000 with a 5% down payment. The buyer has two mortgage options: a 20-year fixed at 10% and a 30-year fixed at 10%. I need to calculate the amount of interest paid for each option and find out how much the buyer saves in interest with the 20-year option. Additionally, I need to find the monthly payment for the 20-year option, rounded to the nearest dollar.

First, let's break down the problem step by step.

1. **Calculate the Down Payment:**
   - The cabin costs \$45,000.
   - The down payment is 5% of \$45,000.
   - Down payment = 0.05 * 45,000 = \$2,250.

2. **Determine the Loan Amount:**
   - Loan amount = Total price - Down payment = 45,000 - 2,250 = \$42,750.

3. **Understand the Mortgage Options:**
   - **Option 1:** 20-year fixed at 10%.
   - **Option 2:** 30-year fixed at 10%.

4. **Calculate Monthly Interest Rate:**
   - Annual interest rate = 10%.
   - Monthly interest rate = 10% / 12 = 0.8333% or 0.008333 in decimal.

5. **Calculate the Number of Payments:**
   - **Option 1:** 20 years * 12 months/year = 240 payments.
   - **Option 2:** 30 years * 12 months/year = 360 payments.

6. **Use the PMT Formula:**
   - PMT = [P * (r/n)] / [1 - (1 + r/n)^(-n*t)]
   - Where:
     - P = Loan amount (\$42,750)
     - r = Annual interest rate (0.10)
     - n = Number of payments per year (12)
     - t = Total number of years (20 or 30)

7. **Calculate PMT for 20-Year Option:**
   - PMT = [42,750 * (0.10/12)] / [1 - (1 + 0.10/12)^(-12*20)]
   - PMT = [42,750 * 0.008333] / [1 - (1.008333)^(-240)]
   - First, calculate (1.008333)^(-240):
     - Using a calculator, (1.008333)^(-240) ≈ 0.2145
   - Then, 1 - 0.2145 = 0.7855
   - Now, PMT = (42,750 * 0.008333) / 0.7855
   - 42,750 * 0.008333 ≈ 354.17
   - 354.17 / 0.7855 ≈ 450.00
   - So, the monthly payment for the 20-year option is approximately \$450.

8. **Calculate Total Interest Paid for 20-Year Option:**
   - Total payments = 240 * \$450 = \$108,000
   - Total interest = Total payments - Loan amount = 108,000 - 42,750 = \$65,250

9. **Calculate PMT for 30-Year Option:**
   - PMT = [42,750 * (0.10/12)] / [1 - (1 + 0.10/12)^(-12*30)]
   - PMT = [42,750 * 0.008333] / [1 - (1.008333)^(-360)]
   - First, calculate (1.008333)^(-360):
     - Using a calculator, (1.008333)^(-360) ≈ 0.1667
   - Then, 1 - 0.1667 = 0.8333
   - Now, PMT = (42,750 * 0.008333) / 0.8333
   - 42,750 * 0.008333 ≈ 354.17
   - 354.17 / 0.8333 ≈ 425.00
   - So, the monthly payment for the 30-year option is approximately \$425.

10. **Calculate Total Interest Paid for 30-Year Option:**
    - Total payments = 360 * \$425 = \$153,000
    - Total interest = Total payments - Loan amount = 153,000 - 42,750 = \$110,250

11. **Determine Savings in Interest with 20-Year Option:**
    - Interest saved = Interest for 30-year option - Interest for 20-year option
    - Interest saved = 110,250 - 65,250 = \$45,000

12. **Finalize the Monthly Payment for the 20-Year Option:**
    - Rounded to the nearest dollar, the monthly payment is \$450.

The buyer saves \$45,000 in interest with the 20-year option. The monthly payment for the 20-year option is \$450.

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