Problem 12 - Precalculus: Mathematical for Calculus

Question

Mortgage \(\quad\) 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

Monthly payments: $643.27 for 30 years; $811.65 for 15 years.

Step 1: Understanding the Problem

We need to determine the monthly payment for a mortgage with different time periods. The principal amount is $80,000, the interest rate is 9% per annum, and we are given a choice of repaying it over 30 years or 15 years.

Step 2: Identify the Formula

The formula for the monthly mortgage payment is given by: \( M = P \frac{r(1+r)^n}{(1+r)^n-1} \) where \( M \) is the monthly payment, \( P \) is the principal loan amount, \( r \) is the monthly interest rate, and \( n \) is the number of payments.

Step 3: Calculate the Monthly Interest Rate

Convert the annual interest rate of 9% to a monthly rate by dividing it by 12: \( r = \frac{9\%}{12} = 0.0075 \).

Step 4: Calculate Number of Payments (30-year mortgage)

For a 30-year mortgage, calculate the total number of payments: \( n = 30 \times 12 = 360 \) payments.

Step 5: Calculate Monthly Payment (30-year mortgage)

Substitute \( P = 80000 \), \( r = 0.0075 \), and \( n = 360 \) into the formula. \[ M = 80000 \frac{0.0075(1+0.0075)^{360}}{(1+0.0075)^{360}-1} \]. Compute to find \( M \approx 643.27 \).

Step 6: Calculate Number of Payments (15-year mortgage)

For a 15-year mortgage, calculate the total number of payments: \( n = 15 \times 12 = 180 \) payments.

Step 7: Calculate Monthly Payment (15-year mortgage)

Substitute \( P = 80000 \), \( r = 0.0075 \), and \( n = 180 \) into the formula. \[ M = 80000 \frac{0.0075(1+0.0075)^{180}}{(1+0.0075)^{180}-1} \]. Compute to find \( M \approx 811.65 \).

Key Concepts

Monthly Payment Formula

Understanding how to calculate the monthly payment on a mortgage is an essential skill when dealing with loans or mortgages. The monthly payment formula is your guide to figuring out how much you need to pay each month to fully pay off a mortgage within a specific time. The formula is:

• \( M = P \frac{r(1+r)^n}{(1+r)^n-1} \)

Here is what each symbol in the formula means:

• \( M \) is the monthly payment you will make.
• \( P \) stands for the principal amount, which is the total loan amount.
• \( r \) is the monthly interest rate, which you get by dividing the annual interest rate by 12.
• \( n \) represents the number of total payments, typically the total amount of months you'll be paying.

In a nutshell, this formula helps you see how your principal, interest rate, and loan term interact with each other to determine what you'll owe each month. It incorporates the idea of compound interest growing each month. Keep this formula handy whenever you need to calculate or understand monthly payments for different mortgage options.

Interest Rate Conversion

When dealing with mortgages, it's crucial to convert the annual interest rate into a monthly rate. This conversion gives a clearer picture of how much interest you'll actually be paying each month. To convert an annual interest rate to a monthly one, you should divide the annual rate by 12. For instance, if you have an annual rate of 9%, the calculation would be:

• Annual interest: 9%
• Monthly interest rate: \( r = \frac{9\%}{12} = 0.0075 \)

It's worth noting that this conversion is valid as mortgage payments are typically made monthly. This smaller division also helps to compound the effect of interest, making it easier to fit into the monthly payment calculator.
Remember, understanding this conversion is fundamental as it directly influences the interest you will pay over time and affects your monthly payment amount.

Number of Payments

The number of payments in a mortgage loan refers to how many times you will be making a payment over the life of the loan. This count is important because it affects both the size of your monthly payment and the total interest paid over the loan’s life. Mortgages can be structured in numerous ways, but some common examples questioned in exercises are 15-year and 30-year mortgages. Here's how you would calculate the number of payments:

• For a 15-year mortgage: Multiply 15 years by 12 months per year to get \( n = 15 \times 12 = 180 \) payments.
• For a 30-year mortgage: Multiply 30 years by 12 months per year to get \( n = 30 \times 12 = 360 \) payments.

This factor plays a vital role in the monthly payment formula mentioned above. Essentially, the longer the term of the mortgage, the more payments you make, often resulting in smaller monthly payments but more interest paid over time. Conversely, a shorter term results in fewer payments, typically larger monthly amounts, but less overall interest. Determining your ideal number of payments can help tailor a mortgage plan that best fits your financial situation and long-term goals.