Problem 16 - Precalculus Mathematics for Calculus

Question

What is the monthly payment on a 30 -year mortgage of \(\$ 80,000\) at \(9 \%\) interest? What is the monthly payment on this same mortgage if it is to be repaid over a 15 -year period?

Step-by-Step Solution

Answer

30-year mortgage payment: $643.28; 15-year mortgage payment: $811.12.

Step 1: Understand the Formula for Monthly Payment

The formula to calculate the monthly payment on a loan is given by \[ M = P \cdot \frac{r(1+r)^n}{(1+r)^n-1} \]where, - \( M \) is the monthly payment,- \( P \) is the loan principal,- \( r \) is the monthly interest rate, and- \( n \) is the number of payments.

Step 2: Calculate Monthly Interest Rate

The annual interest rate given is 9%. To get the monthly interest rate \( r \), divide by 12 months: \[ r = \frac{9\%}{12} = 0.0075 \]

Step 3: Calculate Number of Payments for 30-year Mortgage

For a 30-year mortgage, multiply the number of years by 12 (the number of months in a year) to find the number of payments \( n \):\[ n = 30 \times 12 = 360 \]

Step 4: Calculate Monthly Payment for 30-year Mortgage

Substitute \( P = 80000 \), \( r = 0.0075 \), and \( n = 360 \) into the formula:\[ M = 80000 \cdot \frac{0.0075(1+0.0075)^{360}}{(1+0.0075)^{360}-1} \]After calculating, you find:\[ M \approx 643.28 \]

Step 5: Calculate Number of Payments for 15-year Mortgage

For a 15-year mortgage, multiply the number of years by 12 to find the number of payments \( n \):\[ n = 15 \times 12 = 180 \]

Step 6: Calculate Monthly Payment for 15-year Mortgage

Substitute \( P = 80000 \), \( r = 0.0075 \), and \( n = 180 \) into the formula:\[ M = 80000 \cdot \frac{0.0075(1+0.0075)^{180}}{(1+0.0075)^{180}-1} \]After calculating, you find:\[ M \approx 811.12 \]

Key Concepts

Monthly Payment Formula

Calculating the monthly payment for a mortgage involves a specific formula that considers several key financial elements:

• Loan Principal
• Monthly Interest Rate
• Loan Term (number of payments)

To understand the formula, it's important to break down each element. The formula itself is given by:\[M = P \cdot \frac{r(1+r)^n}{(1+r)^n-1}\]Here:

• \( M \) is the monthly payment, the consistent amount you must pay each month.
• \( P \) is the loan principal, the initial amount borrowed which is \( \$80,000 \) in this scenario.
• \( r \) is the monthly interest rate, not the annual rate, which needs to be calculated separately.
• \( n \) is the total number of payments, derived from the loan term in years multiplied by 12.

Now, by using this formula, you can plug in the values for these variables to determine the monthly payment on any mortgage. It helps in comparing different loan terms and interest rates to find the best financial option for your situation.

Interest Rate Calculation

Understanding and calculating the interest rate is crucial for evaluating a mortgage. The interest rate represents the cost of borrowing funds. Here’s how you calculate it on a monthly basis:

• The given annual interest rate is \( 9\% \).
• To convert an annual rate to a monthly rate, divide by 12 (months):

\[ r = \frac{9\%}{12} = 0.0075\] This conversion is necessary because mortgage payments are usually made monthly, thus requiring the interest to be expressed in the same time frame. Monthly interest calculation helps in determining how much of each payment is going toward interest versus the principal. Consistently applying this rate across the loan term is critical in ensuring accurate calculations of total interest paid over the life of the loan.

Loan Repayment Period

The loan repayment period, or the loan term, significantly impacts the monthly payment amount. Simply put, it is the duration over which you agree to repay the mortgage, often broken down into established time frames such as 15, 20, or 30 years. This period establishes the total number of monthly payments:For a 30-year mortgage:

• The loan term is 30 years. Multiply by the number of months in a year:
• \( n = 30 \times 12 = 360 \)

For a 15-year mortgage:

• The loan term is 15 years. This results in:
• \( n = 15 \times 12 = 180 \)

A longer term generally means lower monthly payments, but more interest paid over the loan's life. Conversely, a shorter term means higher monthly payments, but less interest. Understanding this trade-off can help you make informed decisions about your borrowing capacity and long-term financial health.