Carlin has a mortgage with a remaining balance of \$120,000 at an annual interest rate of 4ext{%} compounded monthly. Her current monthly payment is \$800, but she considers increasing it to \$1,200 per month. If she starts making the higher payment in January 2024, in which month and year will she pay off her house?

• A. August 2029
• B. December 2035
• C. June 2032
• D. March 2030

Answer: (D)

Verified step by step guidance

1. Step 1: Understand the problem. Carlin has a mortgage with a remaining balance of \$120,000, an annual interest rate of 4% compounded monthly, and she is considering increasing her monthly payment from \$800 to \$1,200 starting January 2024. The goal is to determine when she will pay off her mortgage with the higher payment.
2. Step 2: Calculate the monthly interest rate. Since the annual interest rate is 4%, divide it by 12 to find the monthly interest rate: \( \text{Monthly Interest Rate} = \frac{4}{100 \times 12} \).
3. Step 3: Use the formula for the remaining balance of a loan to calculate the time required to pay off the mortgage. The formula is \( B = P \times \frac{1 - (1 + r)^{-n}}{r} \), where \( B \) is the loan balance, \( P \) is the monthly payment, \( r \) is the monthly interest rate, and \( n \) is the number of months. Rearrange the formula to solve for \( n \), the number of months required to pay off the loan.
4. Step 4: Substitute the values into the formula. Use \( B = 120,000 \), \( P = 1,200 \), and \( r \) (calculated in Step 2). Solve for \( n \), which represents the number of months required to pay off the loan with the higher payment.
5. Step 5: Convert the number of months \( n \) into a specific month and year. Add \( n \) months to January 2024 to determine the payoff date. Compare the calculated payoff date to the options provided (August 2029, December 2035, June 2032, March 2030) to identify the correct answer.