If John takes out a 30-year mortgage of \$200,000 at an annual interest rate of 5\% compounded monthly, how much total interest will he pay over the life of the mortgage?

• A. \$80,000
• B. \$120,000
• C. \$186,512
• D. \$300,000

Answer: (C)

Verified step by step guidance

1. Step 1: Understand the problem. John has taken out a 30-year mortgage of \$200,000 with an annual interest rate of 5% compounded monthly. The goal is to calculate the total interest paid over the life of the mortgage.
2. Step 2: Calculate the monthly interest rate by dividing the annual interest rate by 12. Use the formula: \( r = \frac{5\%}{12} \), where \( r \) is the monthly interest rate.
3. Step 3: Determine the total number of payments over the life of the mortgage. Since the mortgage is for 30 years and payments are made monthly, the total number of payments is \( n = 30 \times 12 \).
4. Step 4: Use the formula for the monthly payment of a fixed-rate mortgage: \( M = P \cdot \frac{r(1 + r)^n}{(1 + r)^n - 1} \), where \( M \) is the monthly payment, \( P \) is the loan principal (\$200,000), \( r \) is the monthly interest rate, and \( n \) is the total number of payments.
5. Step 5: Multiply the monthly payment \( M \) by the total number of payments \( n \) to find the total amount paid over the life of the mortgage. Subtract the original loan principal \( P \) from this total to calculate the total interest paid.