Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs.
You need a ​$140 comma 000 loan.
Option​ 1: a​ 30-year loan at an APR of 8.25​%.
Option​ 2: a​ 15-year loan at an APR of 7.8​%.
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Part 1
Find the monthly payment for each option.
The monthly payment for option 1 is ​$
  
enter your response here.
The monthly payment for option 2 is ​$
  
enter your response here.
​(Do not round until the final answer. Then round to the nearest cent as​ needed.)

To find the monthly payments for both loan options, we can use the formula for monthly payments of a fixed-rate loan:

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

Where:
- $M$ is the monthly payment.
- $P$ is the loan amount.
- $r$ is the monthly interest rate (annual interest rate divided by 12).
- $n$ is the total number of payments (loan term in years multiplied by 12).

### For Option 1 (30-year loan, APR = 8.25%):
- $P = 140,000$
- Annual interest rate = 8.25% = 0.0825
- Monthly interest rate = $\frac{0.0825}{12}$
- Loan term = 30 years = 360 months

\[
r = \frac{0.0825}{12} = 0.006875
\]
\[
n = 30 \times 12 = 360
\]

Now we plug these values into the formula to calculate the monthly payment for Option 1:

\[
M_1 = \frac{140,000 \times 0.006875 \times (1 + 0.006875)^{360}}{(1 + 0.006875)^{360} - 1}
\]

### For Option 2 (15-year loan, APR = 7.8%):
- $P = 140,000$
- Annual interest rate = 7.8% = 0.078
- Monthly interest rate = $\frac{0.078}{12}$
- Loan term = 15 years = 180 months

\[
r = \frac{0.078}{12} = 0.0065
\]
\[
n = 15 \times 12 = 180
\]

Now we plug these values into the formula to calculate the monthly payment for Option 2:

\[
M_2 = \frac{140,000 \times 0.0065 \times (1 + 0.0065)^{180}}{(1 + 0.0065)^{180} - 1}
\]

Let me calculate both monthly payments for you.The monthly payments for each option are as follows:

- **Option 1 (30-year loan at 8.25% APR):** $1051.77
- **Option 2 (15-year loan at 7.8% APR):** $1321.80

Would you like more details or have any questions?

Here are five related questions to expand on this information:
1. What is the total interest paid over the life of the loan for each option?
2. How does the total loan cost compare between the two options?
3. What is the difference in monthly payment between the two loan options?
4. How does the length of the loan affect the total interest paid?
5. How would a slight change in interest rates affect the monthly payment?

**Tip:** A shorter loan term reduces the total interest paid but increases the monthly payment.

Compare the monthly payments and total loan costs for the following pairs of loan options. Assume that both loans are fixed rate and have the same closing costs. You need a $140,000 loan. Option 1: a 30-year loan at an APR of 8.25%. Option 2: a 15-year loan at an APR of 7.8%.

Compare the monthly payments and total loan costs for a $140,000 loan with two options: a 30-year loan at 8.25% APR and a 15-year loan at 7.8% APR. This problem uses loan amortization formulas to calculate monthly payments and highlights the difference in total interest paid. Suitable for Grades 10-12.

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