John is going to make a house purchase and borrow $300,000 and has two options. John can have a mortgage for 30 years and interest rate of 7.5% or a mortgage for 15 years at an interest rate of 7.25%. What is the interest rate written as a decimal for the 30 loan?

Question

What was your equation to find the monthly payment for the 30 year loan.
Show how you find the total paid for the 30 year loan.
What was your equation to find the monthly payment for the 15 year loan?
How do you find the total paid for the 15 year loan?
How much money is saved if the minimum payment was paid for 15 years instead of the minimum payment for 30 years

Answer 1

Let's go through each part of your question step by step.

1. Interest Rate as a Decimal for the 30-Year Loan

The interest rate given for the 30-year loan is 7.5%. To express this as a decimal, divide by 100: \[ 7.5% = \frac{7.5}{100} = 0.075 \]

2. Equation to Find the Monthly Payment for the 30-Year Loan

The formula to calculate the monthly payment (M) for a fixed-rate mortgage is: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] Where:

• \( P \) = loan amount (principal)
• \( r \) = monthly interest rate (annual interest rate / 12)
• \( n \) = number of payments (loan term in months)

For the 30-year mortgage:

• \( P = 300,000 \)
• Annual interest rate = 7.5%, thus monthly interest rate \( r = \frac{0.075}{12} = 0.00625 \)
• \( n = 30 \times 12 = 360 \)

Substituting in the values: \[ M = 300,000 \frac{0.00625(1 + 0.00625)^{360}}{(1 + 0.00625)^{360} - 1} \]

3. Total Paid for the 30-Year Loan

To find the total amount paid over the life of the loan, multiply the monthly payment by the total number of payments: \[ \text{Total Paid} = M \times n = M \times 360 \]

Calculating \( M \): Plugging the value into a calculator can be required. Here's the calculation: \[ M \approx 300,000 \frac{0.00625(1 + 0.00625)^{360}}{(1 + 0.00625)^{360} - 1} \approx 2090.37 \]

Thus: \[ \text{Total Paid} = 2090.37 \times 360 \approx 752,133.2 \]

4. Equation to Find the Monthly Payment for the 15-Year Loan

Using the same payment formula: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \]

For the 15-year mortgage:

• \( P = 300,000 \)
• Annual interest rate = 7.25%, thus monthly interest rate \( r = \frac{0.0725}{12} \approx 0.00604167 \)
• \( n = 15 \times 12 = 180 \)

Substituting these values: \[ M = 300,000 \frac{0.00604167(1 + 0.00604167)^{180}}{(1 + 0.00604167)^{180} - 1} \]

5. Total Paid for the 15-Year Loan

Similar to before: \[ \text{Total Paid} = M \times n = M \times 180 \]

Calculating \( M \): \[ M \approx 300,000 \frac{0.00604167(1 + 0.00604167)^{180}}{(1 + 0.00604167)^{180} - 1} \approx 2,574.18 \]

Thus: \[ \text{Total Paid} = 2,574.18 \times 180 \approx 463,353.40 \]

6. Money Saved by Choosing the 15-Year Loan

To find the amount saved by choosing the 15-year plan over the 30-year plan: \[ \text{Savings} = \text{Total Paid for 30-Year Loan} - \text{Total Paid for 15-Year Loan} \] \[ \text{Savings} = 752,133.20 - 463,353.40 \approx 288,779.80 \]

Summary of Results

• Monthly payment for 30-year loan: Approx $2,090.37
• Total paid for 30-year loan: Approx $752,133.20
• Monthly payment for 15-year loan: Approx $2,574.18
• Total paid for 15-year loan: Approx $463,353.40
• Savings by choosing 15-year loan: Approx $288,779.80

Make sure to double-check calculations and possibly use a financial calculator or spreadsheet for precise results.