Use $$ \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-n t}\right]} $$ to determine the regular payment amount, rounded to the nearest dollar. Consider the following pair of mortgage loan options for a $$ \$ 155,000 $$ mortgage. Which mortgage loan has the larger total cost (closing costs + the amount paid for points + total cost of interest)? By how much? Mortgage $$ \mathrm{A}: 20 $$-year fixed at $$ 7.25 \% $$ with closing costs of $$ \$ 1300 $$ and 1 point. Mortgage $$ \mathrm{B}: 20 $$-year fixed at $$ 5.25 \% $$ with closing costs of $$ \$ 1300 $$ and 2 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 coskthan mortgage B by $$ \$ \square $$. B. Mortgage B has a larger total cost than mortgage A by $$ \$ \square $$.

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To determine the regular payment amount for each mortgage loan option, we can use the formula provided:

$$ \text{Regular Payment Amount} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} $$

where:
- $$ P $$ is the principal amount of the mortgage,
- $$ r $$ is the annual interest rate,
- $$ n $$ is the number of payments per year,
- $$ t $$ is the number of years.

Given:
- Mortgage A: 20-year fixed at 7.25% with closing costs of $1300 and 1 point.
- Mortgage B: 20-year fixed at 5.25% with closing costs of $1300 and 2 points.

Let's calculate the regular payment amount for each mortgage loan option and then determine which one has the larger total cost.
Simplify the expression by following steps:
- step0: Solution:
  $$\frac{P\times \frac{r}{n}}{\left(1-\left(1+\frac{r}{n}\right)^{-nt}\right)}$$
- step1: Remove the parentheses:
  $$\frac{P\times \frac{r}{n}}{1-\left(1+\frac{r}{n}\right)^{-nt}}$$
- step2: Add the terms:
  $$\frac{P\times \frac{r}{n}}{1-\left(\frac{n+r}{n}\right)^{-nt}}$$
- step3: Multiply the terms:
  $$\frac{\frac{Pr}{n}}{1-\left(\frac{n+r}{n}\right)^{-nt}}$$
- step4: Rewrite the expression:
  $$\frac{\frac{Pr}{n}}{\frac{\left(\frac{n+r}{n}\right)^{nt}-1}{\left(\frac{n+r}{n}\right)^{nt}}}$$
- step5: Multiply by the reciprocal:
  $$\frac{Pr}{n}\times \frac{\left(\frac{n+r}{n}\right)^{nt}}{\left(\frac{n+r}{n}\right)^{nt}-1}$$
- step6: Multiply the terms:
  $$\frac{Pr\left(\frac{n+r}{n}\right)^{nt}}{n\left(\left(\frac{n+r}{n}\right)^{nt}-1\right)}$$
- step7: Multiply the terms:
  $$\frac{Pr\left(\frac{n+r}{n}\right)^{nt}}{n\left(\frac{n+r}{n}\right)^{nt}-n}$$
The regular payment amount for a mortgage loan can be calculated using the formula:

$$ \text{Regular Payment Amount} = \frac{P\left(\frac{r}{n}\right)}{\left[1-\left(1+\frac{r}{n}\right)^{-nt}\right]} $$

Now, let's calculate the regular payment amount for each mortgage loan option.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{155000\left(\frac{0.0725}{12}\right)}{\left(1-\left(1+\left(\frac{0.0725}{12}\right)\right)-20\times 12\right)}$$
- step1: Remove the parentheses:
  $$\frac{155000\left(\frac{0.0725}{12}\right)}{1-\left(1+\left(\frac{0.0725}{12}\right)\right)-20\times 12}$$
- step2: Divide the terms:
  $$\frac{155000\times \frac{29}{4800}}{1-\left(1+\left(\frac{0.0725}{12}\right)\right)-20\times 12}$$
- step3: Divide the terms:
  $$\frac{155000\times \frac{29}{4800}}{1-\left(1+\frac{29}{4800}\right)-20\times 12}$$
- step4: Add the numbers:
  $$\frac{155000\times \frac{29}{4800}}{1-\frac{4829}{4800}-20\times 12}$$
- step5: Multiply the numbers:
  $$\frac{155000\times \frac{29}{4800}}{1-\frac{4829}{4800}-240}$$
- step6: Multiply the numbers:
  $$\frac{\frac{22475}{24}}{1-\frac{4829}{4800}-240}$$
- step7: Subtract the numbers:
  $$\frac{\frac{22475}{24}}{-\frac{1152029}{4800}}$$
- step8: Multiply by the reciprocal:
  $$\frac{22475}{24}\left(-\frac{4800}{1152029}\right)$$
- step9: Multiply the numbers:
  $$-\frac{22475}{24}\times \frac{4800}{1152029}$$
- step10: Reduce the numbers:
  $$-22475\times \frac{200}{1152029}$$
- step11: Multiply:
  $$-\frac{22475\times 200}{1152029}$$
- step12: Multiply:
  $$-\frac{4495000}{1152029}$$
The regular payment amount for Mortgage A is approximately $-3.901811.

Now, let's calculate the regular payment amount for Mortgage B.
Calculate the value by following steps:
- step0: Calculate:
  $$\frac{155000\left(\frac{0.0525}{12}\right)}{\left(1-\left(1+\left(\frac{0.0525}{12}\right)\right)-20\times 12\right)}$$
- step1: Remove the parentheses:
  $$\frac{155000\left(\frac{0.0525}{12}\right)}{1-\left(1+\left(\frac{0.0525}{12}\right)\right)-20\times 12}$$
- step2: Divide the terms:
  $$\frac{155000\times \frac{7}{1600}}{1-\left(1+\left(\frac{0.0525}{12}\right)\right)-20\times 12}$$
- step3: Divide the terms:
  $$\frac{155000\times \frac{7}{1600}}{1-\left(1+\frac{7}{1600}\right)-20\times 12}$$
- step4: Add the numbers:
  $$\frac{155000\times \frac{7}{1600}}{1-\frac{1607}{1600}-20\times 12}$$
- step5: Multiply the numbers:
  $$\frac{155000\times \frac{7}{1600}}{1-\frac{1607}{1600}-240}$$
- step6: Multiply the numbers:
  $$\frac{\frac{5425}{8}}{1-\frac{1607}{1600}-240}$$
- step7: Subtract the numbers:
  $$\frac{\frac{5425}{8}}{-\frac{384007}{1600}}$$
- step8: Multiply by the reciprocal:
  $$\frac{5425}{8}\left(-\frac{1600}{384007}\right)$$
- step9: Multiply the numbers:
  $$-\frac{5425}{8}\times \frac{1600}{384007}$$
- step10: Reduce the numbers:
  $$-5425\times \frac{200}{384007}$$
- step11: Multiply:
  $$-\frac{5425\times 200}{384007}$$
- step12: Multiply:
  $$-\frac{1085000}{384007}$$
The regular payment amount for Mortgage B is approximately $-2.825469.

To determine which mortgage loan has the larger total cost, we need to consider the closing costs, points, and total cost of interest for each option. Let's calculate the total cost for each mortgage loan option.
Calculate the value by following steps:
- step0: Calculate:
  $$1300+155000\times 0.0725\times 20+155000\times 0.0725\times 20\times 0.01$$
- step1: Multiply the terms:
  $$1300+224750+155000\times 0.0725\times 20\times 0.01$$
- step2: Multiply the terms:
  $$1300+224750+2247.5$$
- step3: Add the numbers:
  $$228297.5$$
The total cost for Mortgage A is $228,297.50.

Now, let's calculate the total cost for Mortgage B.
Calculate the value by following steps:
- step0: Calculate:
  $$1300+155000\times 0.0525\times 20+155000\times 0.0525\times 20\times 0.02$$
- step1: Multiply the terms:
  $$1300+162750+155000\times 0.0525\times 20\times 0.02$$
- step2: Multiply the terms:
  $$1300+162750+3255$$
- step3: Add the numbers:
  $$167305$$
The total cost for Mortgage B is $167,305.

Comparing the total costs, Mortgage A has a larger total cost than Mortgage B by $228,297.50 - $167,305 = $60,992.50.

Therefore, the correct answer is:
A. Mortgage A has a larger total cost than Mortgage B by $60,992.50.
Mortgage A has a larger total cost than Mortgage B by $60,992.

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