Problem 13 - Precalculus: Mathematical for Calculus

Question

Mortgage What is the monthly payment on a 30 -year mortgage of \(\$ 100,000\) at \(8 \%\) interest per year, compounded monthly? What is the total amount paid on this loan over the 30 -year period?

Step-by-Step Solution

Answer

The monthly payment is \( \$733.76 \) and the total amount paid is \( \$264,153.60 \).

Step 1: Understand the Formula

The monthly payment for a loan can be computed using the formula: \[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]where \( M \) is the monthly payment, \( P \) is the principal loan amount, \( r \) is the monthly interest rate, and \( n \) is the total number of payments. In this exercise, \( P = 100,000 \), \( r = \frac{0.08}{12} = 0.0066667 \), and \( n = 30 \times 12 = 360 \).

Step 2: Substitute Values and Calculate

Plug the values into the formula: \[ M = 100,000 \frac{0.0066667(1+0.0066667)^{360}}{(1+0.0066667)^{360} - 1} \]Calculate the expression to find the monthly payment.

Step 3: Solve for Monthly Payment

When you compute the expression, you find:\[ M \approx 733.76 \]Therefore, the monthly payment is approximately \( \$733.76 \).

Step 4: Calculate Total Payment

The total amount paid over the loan period is the monthly payment multiplied by the total number of payments:\[ \text{Total Payment} = 733.76 \times 360 \]Calculate this to find the total payment.

Step 5: Solve for Total Amount Paid

After computing, the total payment is:\[ 733.76 \times 360 = 264,153.60 \]Thus, the total amount paid on this loan over the 30-year period is approximately \( \$264,153.60 \).

Key Concepts

Monthly Payment Formula

Mortgage loans often require understanding a crucial equation called the **monthly payment formula**. This formula helps determine how much you will need to pay each month to completely pay off a loan over a specified period of time. The formula is given by:\[ M = P \frac{r(1+r)^n}{(1+r)^n - 1} \]where:

• \( M \) is the monthly payment,
• \( P \) is the principal or initial amount of the loan,
• \( r \) is the monthly interest rate, which is the annual rate divided by 12 months,
• \( n \) is the total number of payments or months over which the loan is amortized.

Understanding this formula is essential because it considers the interest that accumulates on the mortgage over the life of the loan. A precise calculation requires plugging in the figures for \( P \), \( r \), and \( n \) into the formula, then solving for \( M \). Remember, the principal \( P \) is the amount borrowed, and in the example provided, it is $100,000. The monthly interest rate \( r \) is calculated from the annual rate, so an 8% annual interest rate results in a monthly rate of \( \frac{0.08}{12} \approx 0.0066667 \). Lastly, \( n \) for a 30-year mortgage is \( 30 \times 12 = 360 \) months.

Compounded Interest

The concept of **compounded interest** is fundamental when working with loans such as mortgages. Unlike simple interest where interest is only calculated on the initial principal, compounded interest is calculated on the initial principal and also on the accumulated interest from previous periods. This means interest is essentially { re-invested}, compounding multiple times during the loan's lifetime. In a mortgage scenario, compounding typically occurs monthly. Each month, interest is added to the unpaid balance. This new balance becomes the basis on which the next month's interest is calculated. Over time, this can significantly increase the total amount paid because each month's interest is charged on a larger "compounded" figure rather than just the loan's principal. Breaking it down:

• Monthly compounding adds interest to the loan balance every month.
• The mortgage payment covers both the interest and a fraction of the principal.
• Over time, more of the payment is applied to the principal as less is needed to cover interest.

Understanding compounded interest is key to comprehending why a long-term loan often costs significantly more in total payments than just its principal amount.

Total Loan Payment

Calculating the **total loan payment** helps you to understand the overall cost of taking a loan. The total payment on a mortgage is the sum of all monthly payments made over the life of the loan.The formula for finding the total payment is:\[ \text{Total Payment} = M \times n \]where:

• \( M \) is the monthly payment amount.
• \( n \) is the total number of payments over the mortgage period.

In the given example, the monthly payment \( M \) was calculated to be about \(733.76. With a 30-year mortgage, which entails 360 monthly payments, the total payment is:\[ 733.76 \times 360 = 264,153.60 \]Thus, over 30 years, you would pay approximately \)264,153.60 for a $100,000 loan. What's interesting here is recognizing the impact of interest: your total payment covers not just the principal, but also a significant amount of interest accrued over 30 years. This illustrates how essential it is to understand the terms of a mortgage agreement, and how interest rates and time affect the amount you end up repaying on your loan.