Question

When Gustavo and Serrana bought their home, they had a 5.9% loan with monthly payments of $870.60 for 30 years. After making 78 monthly payments, they plan to refinance for an amount that includes an additional $35,000 to remodel their kitchen. They can refinance at 4.8% compounded monthly for 25 years with refinancing costs of $625 included with the amount refinanced.

(a) Find the amount refinanced. (Round your answer to the nearest cent.)
(b) Find their new monthly payment. (Round your answer to the nearest cent.) $
(c) How long will it take to pay off this new loan if they pay $1200 each month? (Round your answer up to the next whole number.)

Answer 1

Gustavo and Serrana will need to refinance an amount of $199,497.85. Their new monthly payment will be $1,227.45. It will take them approximately 187 months to pay off this new loan.

Step-by-step explanation:

(a) Amount refinanced:

To find the amount refinanced, we need to calculate the remaining balance on the original loan after making 78 monthly payments. Using an annuity formula:

Calculate the monthly interest rate:

• i = 5.9% / 12 = 0.00492
• Calculate the remaining balance:
• B = $870.60 * [(1 - (1 + 0.00492)³⁰ˣ¹² - 78}) / 0.00492] = $163,872.85

Calculate the amount refinanced:

Refinanced Amount = $163,872.85 + $35,000 + $625 = $199,497.85

(b) New Monthly Payment:

To calculate the new monthly payment, we use the annuity formula with the refinanced amount and the new interest rate:

• Calculate the new monthly interest rate:

• i = 4.8% / 12 = 0.004

• Calculate the new monthly payment:

• New Monthly Payment = $199,497.85 * [0.004 * (1 + 0.004)²⁵ˣ¹²] / [(1 + 0.004)²⁵ˣ¹²} - 1] = $1,227.45

(c) Time to Pay Off New Loan:

To calculate the time it will take to pay off the new loan with a monthly payment of $1,200, we rearrange the annuity formula to solve for N:

• Calculate the monthly interest rate:

• i = 4.8% / 12 = 0.004

Calculate the time to pay off the new loan:

• N = log[(1.005 * $199,497.85 - $1,200) / (0.004 * $199,497.85)] / log(1 + 0.004) = 186 months
• Round up to the nearest whole number:
• Time to Pay Off New Loan = 187 months

Answer 2

b

Step-by-step explanation:

The formula used to calculate the fixed monthly payment (P) required to fully amortize a loan of L dollars over a term of n months at a monthly interest rate of r is P = L [r(1 +r)n]/[(1 + r)n- 1]