Use 

PMT equals StartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis Over left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt right bracket EndFractionPMT=Prn1−1+rn−nt 

to determine the regular payment​ amount, rounded to the nearest dollar. The price of a small cabin is 

​$85 comma 00085,000. 

The bank requires a​ 5% down payment.  The buyer is offered two mortgage​ options: 20-year fixed at 

88​% 

or​ 30-year fixed at 

88​%. 

 Calculate the amount of interest paid for each option. How much does the buyer save in interest with the​ 20-year option?

Question content area bottom

Part 1

Find the monthly payment for the​ 20-year option.

​$675675

​(Round to the nearest dollar as​ needed.)

Part 2

Find the monthly payment for the​ 30-year option.

​$enter your response here

​(Round to the nearest dollar as​ needed.)

### Step 1: Determine the loan amount

The cabin costs $85,000, and the buyer makes a 5% down payment. Let's first calculate the loan amount.

\[
\text{Down payment} = 5\% \times 85,000 = 0.05 \times 85,000 = 4,250
\]

Now, subtract the down payment from the total price to find the loan amount:

\[
\text{Loan amount} = 85,000 - 4,250 = 80,750
\]

### Step 2: Use the PMT formula

The formula for the regular payment (PMT) is:

\[
PMT = \frac{P \left(\frac{r}{n}\right)}{1 - \left(1 + \frac{r}{n}\right)^{-nt}}
\]

Where:
- $P$ is the loan amount ($80,750),
- $r$ is the annual interest rate (8% or 0.08),
- $n$ is the number of payments per year (12 for monthly payments),
- $t$ is the number of years (either 20 or 30 years depending on the option).

#### Part 1: Monthly Payment for the 20-Year Option
For the 20-year mortgage:
- $P = 80,750$,
- $r = 0.08$,
- $n = 12$,
- $t = 20$.

Now, let's calculate the monthly payment using these values:

\[
PMT = \frac{80,750 \left(\frac{0.08}{12}\right)}{1 - \left(1 + \frac{0.08}{12}\right)^{-12 \times 20}}
\]

We'll compute this step-by-step.

#### Part 2: Monthly Payment for the 30-Year Option
For the 30-year mortgage:
- $P = 80,750$,
- $r = 0.08$,
- $n = 12$,
- $t = 30$.

We will apply the same formula but with $t = 30$:

\[
PMT = \frac{80,750 \left(\frac{0.08}{12}\right)}{1 - \left(1 + \frac{0.08}{12}\right)^{-12 \times 30}}
\]

### Step 3: Calculate the Total Interest Paid for Each Option

The total amount paid is the monthly payment multiplied by the total number of payments:

\[
\text{Total amount paid} = PMT \times n \times t
\]

The total interest paid is the difference between the total amount paid and the loan amount.

\[
\text{Interest paid} = \text{Total amount paid} - P
\]

Finally, we can compare the interest paid for the 20-year and 30-year options to see how much the buyer saves by choosing the 20-year option.

I'll compute these values for you now.### Results:

1. **Monthly payment for the 20-year option**: $675
2. **Monthly payment for the 30-year option**: $593
3. **Total interest paid for the 20-year option**: $81,250
4. **Total interest paid for the 30-year option**: $132,730
5. **Interest savings with the 20-year option**: $51,480

The buyer saves $51,480 in interest by choosing the 20-year mortgage instead of the 30-year option.

Do you need further details or have any questions?

Here are some follow-up questions to explore:

1. How would a change in interest rate affect the monthly payments?
2. What is the effect of increasing the down payment on both options?
3. How much would the buyer pay in total (loan + interest) for each option?
4. What would be the impact if the payments were made biweekly instead of monthly?
5. How does the length of the loan affect the total interest paid?

**Tip**: Shorter loan terms typically result in higher monthly payments but significantly reduce the total interest paid.

Use PMT = P(r/n) / [1 - (1 + r/n)^(-nt)] to determine the regular payment amount, rounded to the nearest dollar. The price of a small cabin is $85,000. The bank requires a 5% down payment. The buyer is offered two mortgage options: 20-year fixed at 8% or 30-year fixed at 8%. Calculate the amount of interest paid for each option. How much does the buyer save in interest with the 20-year option? Part 1: Find the monthly payment for the 20-year option. Part 2: Find the monthly payment for the 30-year option.

Learn how to calculate monthly payments for 20-year and 30-year mortgage options with an 8% interest rate on an $85,000 cabin, using the loan amortization formula. This guide covers key concepts such as interest calculation, loan terms, and how much can be saved by choosing a shorter loan term. Suitable for high school students and anyone interested in finance.

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