Use $$ \frac{P\left(\frac{1}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $$ \$ 145,000 $$ 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: 15-year fixed at $$ 12.25 \% $$ with closing costs of $$ \$ 2600 $$ and 1 point. Mortgage B: 15-year fixed at $$ 11.25 \% $$ with closing costs of $$ \$ 2600 $$ and 5 points. 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 $$ \$ \square $$. B. Mortgage B has a larger total cost than mortgage A by $$ \$ $$

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To determine which mortgage loan has the larger total cost, we need to calculate the total cost for both Mortgage A and Mortgage B. The total cost includes closing costs, the amount paid for points, and the total cost of interest.

### Step 1: Calculate the regular payment amount using the formula

The formula given is:

$$
\frac{P\left(\frac{1}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]}
$$

Where:
- $$ P = 145000 $$ (the principal)
- $$ r $$ is the annual interest rate (as a decimal)
- $$ n $$ is the number of payments per year (12 for monthly payments)
- $$ t $$ is the number of years (15)

### Mortgage A:
- Interest rate $$ r = 12.25\% = 0.1225 $$
- Points = 1 point = 1% of the loan amount = $$ 0.01 \times 145000 $$

### Mortgage B:
- Interest rate $$ r = 11.25\% = 0.1125 $$
- Points = 5 points = 5% of the loan amount = $$ 0.05 \times 145000 $$

### Step 2: Calculate the monthly payment for both mortgages

Let's calculate the monthly payment for both mortgages using the formula.

#### Mortgage A:
- $$ n = 12 $$
- $$ t = 15 $$

#### Mortgage B:
- $$ n = 12 $$
- $$ t = 15 $$

Now, I will calculate the monthly payments for both mortgages.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{145000\times \frac{1}{12}}{\left(1-\left(1+\frac{0.1125}{12}\right)^{-12\times 15}\right)}$$
- step1: Remove the parentheses:
  $$\frac{145000\times \frac{1}{12}}{1-\left(1+\frac{0.1125}{12}\right)^{-12\times 15}}$$
- step2: Divide the terms:
  $$\frac{145000\times \frac{1}{12}}{1-\left(1+\frac{3}{320}\right)^{-12\times 15}}$$
- step3: Add the numbers:
  $$\frac{145000\times \frac{1}{12}}{1-\left(\frac{323}{320}\right)^{-12\times 15}}$$
- step4: Multiply the numbers:
  $$\frac{145000\times \frac{1}{12}}{1-\left(\frac{323}{320}\right)^{-180}}$$
- step5: Multiply the numbers:
  $$\frac{\frac{36250}{3}}{1-\left(\frac{323}{320}\right)^{-180}}$$
- step6: Subtract the numbers:
  $$\frac{\frac{36250}{3}}{\frac{323^{180}-320^{180}}{323^{180}}}$$
- step7: Multiply by the reciprocal:
  $$\frac{36250}{3}\times \frac{323^{180}}{323^{180}-320^{180}}$$
- step8: Multiply the fractions:
  $$\frac{36250\times 323^{180}}{3\left(323^{180}-320^{180}\right)}$$
- step9: Multiply:
  $$\frac{36250\times 323^{180}}{3\times 323^{180}-3\times 320^{180}}$$
Calculate or simplify the expression $$ \frac{145000\left(\frac{1}{12}\right)}{\left[1-\left(1+\frac{0.1225}{12}\right)^{-12\times15}\right]}} $$.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{145000\times \frac{1}{12}}{\left(1-\left(1+\frac{0.1225}{12}\right)^{-12\times 15}\right)}$$
- step1: Remove the parentheses:
  $$\frac{145000\times \frac{1}{12}}{1-\left(1+\frac{0.1225}{12}\right)^{-12\times 15}}$$
- step2: Divide the terms:
  $$\frac{145000\times \frac{1}{12}}{1-\left(1+\frac{49}{4800}\right)^{-12\times 15}}$$
- step3: Add the numbers:
  $$\frac{145000\times \frac{1}{12}}{1-\left(\frac{4849}{4800}\right)^{-12\times 15}}$$
- step4: Multiply the numbers:
  $$\frac{145000\times \frac{1}{12}}{1-\left(\frac{4849}{4800}\right)^{-180}}$$
- step5: Multiply the numbers:
  $$\frac{\frac{36250}{3}}{1-\left(\frac{4849}{4800}\right)^{-180}}$$
- step6: Subtract the numbers:
  $$\frac{\frac{36250}{3}}{\frac{4849^{180}-4800^{180}}{4849^{180}}}$$
- step7: Multiply by the reciprocal:
  $$\frac{36250}{3}\times \frac{4849^{180}}{4849^{180}-4800^{180}}$$
- step8: Multiply the fractions:
  $$\frac{36250\times 4849^{180}}{3\left(4849^{180}-4800^{180}\right)}$$
- step9: Multiply:
  $$\frac{36250\times 4849^{180}}{3\times 4849^{180}-3\times 4800^{180}}$$
The monthly payment calculations for both mortgages are as follows:

### Mortgage A:
- Monthly Payment $$ \approx 14397.01 $$

### Mortgage B:
- Monthly Payment $$ \approx 14852.44 $$

### Step 3: Calculate the total payments over the loan term

The total payment over the loan term can be calculated as:

$$
\text{Total Payment} = \text{Monthly Payment} \times n \times t
$$

Where:
- $$ n = 12 $$ (number of payments per year)
- $$ t = 15 $$ (number of years)

Now, let's calculate the total payments for both mortgages.
Calculate the value by following steps:
- step0: Calculate:
  $$14852.44\times 12\times 15$$
- step1: Multiply the terms:
  $$178229.28\times 15$$
- step2: Multiply the numbers:
  $$2673439.2$$
Calculate or simplify the expression $$ 14397.01 \times 12 \times 15 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$14397.01\times 12\times 15$$
- step1: Multiply the terms:
  $$172764.12\times 15$$
- step2: Multiply the numbers:
  $$2591461.8$$
The total payments over the loan term for both mortgages are as follows:

### Mortgage A:
- Total Payment $$ \approx 2591461.80 $$

### Mortgage B:
- Total Payment $$ \approx 2673439.20 $$

### Step 4: Calculate the total cost for each mortgage

The total cost includes:
1. Total payments over the loan term
2. Closing costs
3. Amount paid for points

#### Mortgage A:
- Closing Costs: $$ 2600 $$
- Points: $$ 0.01 \times 145000 = 1450 $$
- Total Cost for Mortgage A:

$$
\text{Total Cost A} = \text{Total Payment A} + \text{Closing Costs} + \text{Points}
$$

#### Mortgage B:
- Closing Costs: $$ 2600 $$
- Points: $$ 0.05 \times 145000 = 7250 $$
- Total Cost for Mortgage B:

$$
\text{Total Cost B} = \text{Total Payment B} + \text{Closing Costs} + \text{Points}
$$

Now, let's calculate the total costs for both mortgages.
Calculate the value by following steps:
- step0: Calculate:
  $$2591461.8+2600+1450$$
- step1: Add the numbers:
  $$2595511.8$$
Calculate or simplify the expression $$ 2673439.20 + 2600 + 7250 $$.
Calculate the value by following steps:
- step0: Calculate:
  $$2673439.2+2600+7250$$
- step1: Add the numbers:
  $$2683289.2$$
The total costs for both mortgages are as follows:

### Mortgage A:
- Total Cost $$ \approx 2595511.80 $$

### Mortgage B:
- Total Cost $$ \approx 2683289.20 $$

### Step 5: Determine which mortgage has the larger total cost and by how much

To find out which mortgage has the larger total cost, we will subtract the total costs:

$$
\text{Difference} = \text{Total Cost B} - \text{Total Cost A}
$$

Now, let's calculate the difference.
Calculate the value by following steps:
- step0: Calculate:
  $$2683289.2-2595511.8$$
- step1: Subtract the numbers:
  $$87777.4$$
The difference in total costs between Mortgage B and Mortgage A is:

$$
\text{Difference} \approx 87777.40
$$

### Conclusion:
Mortgage B has a larger total cost than Mortgage A by approximately $$ \$ 87777 $$.

Thus, the correct answer is:
**B. Mortgage B has a larger total cost than Mortgage A by $$ \$ 87777 $$.**
Mortgage B has a larger total cost than Mortgage A by \$87,777.

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