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 ​$185 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: 30​-year fixed at 6.25​% with closing costs of ​$2400 and 1 point.
Mortgage​ B: 30​-year fixed at 5.25​% with closing costs of ​$2400 and 3 points.
Question content area bottom
Part 1
Choose the correct answer​ below, and fill in the answer box to complete your choice.
​(Do not round until the final answer. Then round to the nearest dollar as​ needed.)
A.
Mortgage A has a larger total cost than mortgage B by ​$
  
enter your response here.
B.
Mortgage B has a larger total cost than mortgage A by ​$
  
enter your response here.

To determine which mortgage loan has the larger total cost, we need to calculate the regular payment amount for each option using the formula provided and then compute the total cost of each mortgage, including the interest, closing costs, and points.

### Step 1: Understanding the Formula
The formula provided is for calculating the monthly mortgage payment:

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

Where:
- $M$ = Monthly mortgage payment
- $P$ = Principal loan amount
- $r$ = Annual interest rate (as a decimal)
- $n$ = Number of payments per year (typically 12 for monthly payments)
- $t$ = Number of years

### Step 2: Calculate the Monthly Payment for Each Mortgage

**Mortgage A:**
- Principal, $P = 185,000$
- Annual interest rate, $r = 6.25\%$ or $0.0625$
- Number of payments per year, $n = 12$
- Term, $t = 30$ years
- Points: 1 point = 1% of the principal = $0.01 \times 185,000 = 1,850$
- Closing costs = $2400

The monthly interest rate = $\frac{r}{n} = \frac{0.0625}{12}$.

**Mortgage B:**
- Principal, $P = 185,000$
- Annual interest rate, $r = 5.25\%$ or $0.0525$
- Number of payments per year, $n = 12$
- Term, $t = 30$ years
- Points: 3 points = 3% of the principal = $0.03 \times 185,000 = 5,550$
- Closing costs = $2400

The monthly interest rate = $\frac{r}{n} = \frac{0.0525}{12}$.

Let's calculate the monthly payment $M$ for both mortgages.

### Step 3: Total Cost Calculation

1. **Total Payment Over 30 Years**:
   \[
   \text{Total Payment} = M \times 12 \times 30
   \]

2. **Total Cost (Including Points and Closing Costs)**:
   \[
   \text{Total Cost} = \text{Total Payment} + \text{Points} + \text{Closing Costs}
   \]

Now, let's perform these calculations.### Results:

- **Total cost of Mortgage A:** $414,318 (rounded to the nearest dollar)
- **Total cost of Mortgage B:** $375,718 (rounded to the nearest dollar)
- **Difference in total cost:** $38,600 (rounded to the nearest dollar)

**Conclusion:**  
Mortgage A has a larger total cost than Mortgage B by **$38,600**.

Would you like more details or have any questions?

### Additional Related Questions:
1. What impact does increasing the number of points have on the total cost?
2. How would the monthly payment change if the interest rate of Mortgage B was slightly higher?
3. What would be the effect of a shorter loan term (e.g., 15 years) on the total interest paid?
4. How do closing costs compare to the total interest paid over the loan term?
5. What is the breakeven point for paying points on a mortgage?

### Tip:
Paying points on a mortgage can lower your interest rate, but it's important to calculate the breakeven point to determine if it's worth the upfront cost.

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 ​$185,000 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: 30​-year fixed at 6.25​% with closing costs of ​$2400 and 1 point. Mortgage​ B: 30​-year fixed at 5.25​% with closing costs of ​$2400 and 3 points.

Learn how to calculate the regular payment amount for a $185,000 mortgage and compare two 30-year fixed mortgage options with interest rates of 6.25% and 5.25%. This detailed guide covers how to compute the total cost of each mortgage, including closing costs, points, and interest, with step-by-step instructions.

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