Questions: 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 180,000 loan. Option 1: a 30-year loan at an APR of 10%. Option 2: a 15-year loan at an APR of 9.5%. Find the monthly payment for each option. The monthly payment for option 1 is 1579.63. The monthly payment for option 2 is 1879.60. (Do not round until the final answer. Then round to the nearest cent as needed.) Find the total amount paid for each option. The total payment for option 1 is . The total payment for option 2 is . (Use the answers from the previous step to find this answer. Round to the nearest cent as needed.)

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 180,000 loan. Option 1: a 30-year loan at an APR of 10%. Option 2: a 15-year loan at an APR of 9.5%. Find the monthly payment for each option. The monthly payment for option 1 is 1579.63. The monthly payment for option 2 is 1879.60. (Do not round until the final answer. Then round to the nearest cent as needed.) Find the total amount paid for each option. The total payment for option 1 is . The total payment for option 2 is . (Use the answers from the previous step to find this answer. Round to the nearest cent as needed.)
Transcript text: 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 \$180,000 loan. Option 1: a 30-year loan at an APR of 10\%. Option 2: a 15-year loan at an APR of $9.5 \%$. Find the monthly payment for each option. The monthly payment for option 1 is \$ $\square$ 1579.63 . The monthly payment for option 2 is $\$ 1879.60$. (Do not round until the final answer. Then round to the nearest cent as needed.) Find the total amount paid for each option. The total payment for option 1 is $\$$ $\square$ . The total payment for option 2 is $\$$ $\square$ (Use the answers from the previous step to find this answer. Round to the nearest cent as needed.)
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Solution

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Calculate the monthly payment for Option 1 (30-year loan at an APR of 10\%).

Determine the monthly interest rate for Option 1.

The monthly interest rate is given by \( \frac{10}{100} \div 12 = 0.00833333333 \).

Calculate the total number of payments for Option 1.

The total number of payments is \( 30 \times 12 = 360 \).

Use the loan payment formula to find the monthly payment for Option 1.

The monthly payment is calculated as follows: \[ M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1} \] where \( P = 180000 \), \( r = 0.00833333333 \), and \( n = 360 \). Thus, \[ M = 180000 \cdot \frac{0.00833333333(1+0.00833333333)^{360}}{(1+0.00833333333)^{360} - 1} \approx 1579.63. \]

The monthly payment for Option 1 is \( \boxed{1579.63} \).

Calculate the monthly payment for Option 2 (15-year loan at an APR of 9.5\%).

Determine the monthly interest rate for Option 2.

The monthly interest rate is given by \( \frac{9.5}{100} \div 12 = 0.00791666667 \).

Calculate the total number of payments for Option 2.

The total number of payments is \( 15 \times 12 = 180 \).

Use the loan payment formula to find the monthly payment for Option 2.

The monthly payment is calculated as follows: \[ M = P \cdot \frac{r(1+r)^n}{(1+r)^n - 1} \] where \( P = 180000 \), \( r = 0.00791666667 \), and \( n = 180 \). Thus, \[ M = 180000 \cdot \frac{0.00791666667(1+0.00791666667)^{180}}{(1+0.00791666667)^{180} - 1} \approx 1879.60. \]

The monthly payment for Option 2 is \( \boxed{1879.60} \).

Calculate the total amount paid for each loan option.

Calculate the total payment for Option 1.

The total payment is given by \( 360 \times 1579.63 \approx 568666.38 \).

Calculate the total payment for Option 2.

The total payment is given by \( 180 \times 1879.60 \approx 338328.80 \).

The total payment for Option 1 is \( \boxed{568666.38} \) and for Option 2 is \( \boxed{338328.80} \).

The monthly payment for Option 1 is \( \boxed{1579.63} \).
The monthly payment for Option 2 is \( \boxed{1879.60} \).
The total payment for Option 1 is \( \boxed{568666.38} \).
The total payment for Option 2 is \( \boxed{338328.80} \).

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