Use StartStartFraction Upper P left parenthesis StartFraction r Over n EndFraction right parenthesis OverOver left bracket 1 minus left parenthesis 1 plus StartFraction r Over n EndFraction right parenthesis Superscript negative nt Baseline right bracket EndEndFraction
 to determine the regular payment​ amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a ​$180 comma 000 mortgage. Which mortgage loan has the larger total cost​ (closing costs​ + the amount paid for points​ + total cost of​ interest)? By how​ much?
Mortgage​ A: 20​-year fixed at 7.25​% with closing costs of ​$2900 and 1 point.
Mortgage​ B: 20​-year fixed at 5.25​% with closing costs of ​$2900 and 4 points.
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Part 1
Choose the correct answer​ below, and fill in the answer box to complete your choice.
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)
A.
Mortgage B has a larger total cost than mortgage A by ​$
  
enter your response here.
B.
Mortgage A has a larger total cost than mortgage B by 

To solve this problem and determine the mortgage payments and total costs, we'll follow the given formula for mortgage payments and consider the points and interest rate for each option. Here's how we'll break it down:

### Formula for Regular Payment Amount:
The formula to calculate the regular payment $ P $ is:

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

Where:
- $ A $ is the loan amount (in this case, \$180,000),
- $ r $ is the annual interest rate (as a decimal),
- $ n $ is the number of payments per year (12 for monthly payments),
- $ t $ is the term of the loan in years (20 years for both options),
- $ nt $ is the total number of payments (years multiplied by 12).

We'll compute the total cost for each mortgage option by calculating:
1. Monthly mortgage payments.
2. Total payments made over the life of the mortgage.
3. Additional costs from points and closing costs.

#### **Mortgage A:**
- Interest rate: 7.25% (or 0.0725 as a decimal)
- Points: 1 point (1% of the loan amount, which is 1% of \$180,000 = \$1,800)
- Closing costs: \$2,900
- Loan term: 20 years
- Payments per year: 12

#### **Mortgage B:**
- Interest rate: 5.25% (or 0.0525 as a decimal)
- Points: 4 points (4% of the loan amount, which is 4% of \$180,000 = \$7,200)
- Closing costs: \$2,900
- Loan term: 20 years
- Payments per year: 12

### Step-by-Step Calculations:

1. **Calculate Monthly Payment for Both Mortgages:**
   Use the formula above to calculate the monthly payment for each mortgage.

2. **Calculate Total Payments:**
   Multiply the monthly payment by the total number of payments (20 years × 12 months = 240 payments).

3. **Add Additional Costs:**
   Include the closing costs and points to the total payments for each mortgage.

Let's now perform these calculations step-by-step.

Use the formula P = ( (r / n) * A ) / [1 - (1 + (r / n))^(-nt)] to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $180,000 mortgage. Which mortgage loan has the larger total cost (closing costs + the amount paid for points + total cost of interest)? By how much? Mortgage A: 20-year fixed at 7.25% with closing costs of $2,900 and 1 point. Mortgage B: 20-year fixed at 5.25% with closing costs of $2,900 and 4 points.

Determine which mortgage option for a $180,000 loan has the larger total cost by calculating monthly payments using interest rates of 7.25% and 5.25%. Learn how to account for closing costs, points, and total interest paid over 20 years. This detailed guide is suitable for students and financial learners in Grades 10-12.

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