Problem 17 - Algebra and Trigonometry

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

Monthly payment is \$733.76; total paid is \$264,153.60 over 30 years.

Step 1: Determine the Monthly Interest Rate

The annual interest rate is 8%. To find the monthly interest rate, divide this by 12 (the number of months in a year).\[ r = \frac{8}{100 \times 12} = \frac{0.08}{12} = 0.0066667 \]

Step 2: Calculate the Total Number of Payments

Since the mortgage is for 30 years and compounded monthly, you need to multiply the number of years by the number of months in a year to find the total number of payments.\[ n = 30 \times 12 = 360 \]

Step 3: Apply the Mortgage Payment Formula

The formula to calculate the monthly payment (\(M\)) on a mortgage is:\[ M = P \left( \frac{r(1+r)^n}{(1+r)^n-1} \right) \]where \(P\) is the principal amount ($100,000), \(r\) is the monthly interest rate, and \(n\) is the total number of payments.Plugging in the values:\[ M = 100,000 \left( \frac{0.0066667(1+0.0066667)^{360}}{(1+0.0066667)^{360}-1} \right) \]

Step 4: Solve for the Monthly Payment

Calculating the above expression gives:\[ M \approx 733.76 \]Thus, the monthly mortgage payment is approximately \$733.76.

Step 5: Calculate the Total Amount Paid Over the Loan's Term

To find the total amount paid over the life of the loan, multiply the monthly payment by the total number of payments.\[ \text{Total Amount Paid} = M \times n = 733.76 \times 360 \approx 264,153.60 \]

Step 6: Summarize Results

The monthly payment for the mortgage is approximately \\(733.76, and the total amount paid over 30 years is approximately \\)264,153.60.

Key Concepts

Monthly Payment Calculation

To figure out the monthly payment for a mortgage, you first need to determine the monthly interest rate and the total number of payments. The monthly payment calculation involves using a particular formula that considers the amount of money borrowed, the interest rate, and the loan term. This gives you a fixed amount to pay each month.
Take the annual interest rate and convert it into a monthly rate by dividing by 12.
Then, determine how many payments you will make in total by multiplying the number of years of your loan by 12.

• Converting annual percentage rate to monthly: 8% annually means approximately 0.6667% monthly
• Total payments for a 30-year mortgage: 30 years * 12 = 360 payments

Plug these numbers into the mortgage payment formula to calculate your exact monthly payment. For a $100,000 loan at 8% annual interest, your monthly payment will be approximately $733.76. This ensures that you are reducing your principal balance each month while also paying off some interest.

Interest Rate

Interest rate is a key factor in calculating mortgage payments, and it represents the cost of borrowing money. It's expressed as a percentage of the loan principal and can significantly impact the amount paid over the life of the loan.
For mortgages, interest is often calculated on an annual basis but paid monthly. The monthly interest rate is derived by dividing the annual rate by 12. For example:

• If you have an 8% annual interest rate, the monthly interest rate is 0.6667%.

This percentage, though seemingly small, compounds over time and thus influences how much you end up paying monthly and in total. Therefore, it's essential to understand both the annual and monthly rates when planning financially for a mortgage.

Total Payment Calculation

Figuring out the total payment for a mortgage involves adding up all the monthly payments made throughout the loan's term. This gives the full cost of the mortgage, including both the amount borrowed and all the interest paid.
To calculate this:

• Multiply the monthly payment by the total number of payments.

For the given example, the monthly payment was calculated as \(733.76.
Multiply this by 360 (the total payments for a 30-year mortgage):
\[ 733.76 \times 360 \approx 264,153.60 \]
So, over 30 years, you end up paying around \)264,153.60. This amount includes the \(100,000 originally borrowed plus \)164,153.60 in interest, highlighting how significant the impact of the interest rate is over time.

Mortgage Formula

The mortgage formula is an essential tool for calculating how much you'll need to pay each month if you take a loan. It considers the principal amount, interest rate, and number of payments, providing a straightforward way to estimate your financial obligation.
The formula used to calculate the monthly mortgage payment (\( M \)) is:
\[ M = P \left( \frac{r(1+r)^n}{(1+r)^n-1} \right) \]
Where:

• \( P \)
is the principal amount borrowed,
• \( r \) is the monthly interest rate (annual rate divided by 12),
• \( n \)
represents the total number of payments over the life of the loan.

When you plug in your numbers, for instance:

• \( P = 100,000 \),
• \( r = 0.0066667 \),
• \( n = 360 \),

you'll calculate the payment as approximately $733.76 a month.
This formula helps understand how different factors—especially the interest rate—can affect your monthly payments and overall affordability of a mortgage.