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 

​$40 comma 00040,000. 

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

7.57.5​% 

or​ 30-year fixed at 

7.57.5​%. 

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

Let's solve the problem step by step using the given formula to calculate the monthly payment (PMT) for both the 20-year and 30-year mortgage options. Then, we'll calculate the total interest paid for each option and determine the savings with the 20-year option.

### Step 1: Calculate the Loan Amount
First, calculate the down payment and the loan amount.

- **Price of the cabin**: \$40,000
- **Down payment percentage**: 5%
- **Interest rate (r)**: 7.5% per year or 0.075
- **Number of payments per year (n)**: 12 (monthly payments)

#### Down Payment:
\[
\text{Down Payment} = 40,000 \times \frac{5}{100} = 40,000 \times 0.05 = \$2,000
\]

#### Loan Amount:
\[
\text{Loan Amount} = \text{Price of the Cabin} - \text{Down Payment} = 40,000 - 2,000 = \$38,000
\]

### Step 2: Calculate the Monthly Payment (PMT) for Both Options

The formula for the monthly payment (PMT) is:

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

Where:
- $ P $ is the loan amount
- $ r $ is the annual interest rate (decimal)
- $ n $ is the number of payments per year
- $ t $ is the number of years for the loan

#### Option 1: 20-Year Mortgage
- **Loan term (t)**: 20 years

Substituting the values:

\[
\text{PMT}_{20} = \frac{38,000 \times \frac{0.075}{12}}{1 - \left(1 + \frac{0.075}{12}\right)^{-12 \times 20}}
\]

Let's calculate this value.

#### Option 2: 30-Year Mortgage
- **Loan term (t)**: 30 years

Substituting the values:

\[
\text{PMT}_{30} = \frac{38,000 \times \frac{0.075}{12}}{1 - \left(1 + \frac{0.075}{12}\right)^{-12 \times 30}}
\]

Let's calculate this value as well.

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

The total interest paid can be calculated as:

\[
\text{Total Interest} = (\text{PMT} \times \text{Total Number of Payments}) - \text{Loan Amount}
\]

For the 20-year option:

\[
\text{Total Interest}_{20} = (\text{PMT}_{20} \times 12 \times 20) - 38,000
\]

For the 30-year option:

\[
\text{Total Interest}_{30} = (\text{PMT}_{30} \times 12 \times 30) - 38,000
\]

Finally, calculate the interest savings:

\[
\text{Interest Savings} = \text{Total Interest}_{30} - \text{Total Interest}_{20}
\]

Now, I'll perform these calculations.### Results:

1. **Monthly Payment for 20-Year Mortgage**: \$306 (rounded to the nearest dollar)
2. **Monthly Payment for 30-Year Mortgage**: \$266 (rounded to the nearest dollar)
3. **Total Interest Paid for 20-Year Mortgage**: \$35,470
4. **Total Interest Paid for 30-Year Mortgage**: \$57,653
5. **Interest Savings with 20-Year Option**: \$22,182

### Summary:
By choosing the 20-year mortgage, the buyer would save **\$22,182** in interest compared to the 30-year mortgage.

Do you have any questions about this, or would you like to explore further details?

Here are five follow-up questions related to this problem:

1. How would the monthly payments change if the interest rate was 6% instead of 7.5%?
2. What would be the total interest paid if the down payment was 10% instead of 5%?
3. How much would the buyer save if the 20-year mortgage had a 7% interest rate?
4. What is the effect of increasing the monthly payments on the total interest paid?
5. How would the loan amount change if the price of the cabin was \$50,000?

**Tip:** Opting for a shorter loan term can significantly reduce the total interest paid, even if the monthly payments are higher.

Learn how to calculate mortgage payments and compare interest savings between 20-year and 30-year fixed mortgage options. Use the PMT formula to determine monthly payments. Calculate total interest paid for each option and discover potential savings.

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