Problem 49 - Algebra and Trigonometry with Analytic Geometry

Question

Some lending institutions calculate the monthly payment \(M\) on a loan of \(L\) dollars at an interest rate \(r\) (expressed as a decimal) by using the formula $$ M=\frac{L r k}{12(k-1)} $$ where \(k=[1+(r / 12)]^{12 t}\) and \(t\) is the number of years that the loan is in effect. Home mortgage (a) Find the monthly payment on a 30 -year \(\$ 250,000\) home mortgage if the interest rate is \(8 \%\). (b) Find the total interest paid on the loan in part (a).

Step-by-Step Solution

Answer

(a) $1834.41; (b) $410,387.60.

Step 1: Convert the Interest Rate

The annual interest rate is given as 8%. Convert this percentage to a decimal for use in the formula: \( r = \frac{8}{100} = 0.08 \).

Step 2: Calculate "k"

First, calculate the factor \( k \) using the formula \( k = \left[1 + \frac{r}{12}\right]^{12t} \). Substitute \( r = 0.08 \) and \( t = 30 \) (years):\[k = \left[1 + \frac{0.08}{12}\right]^{12 \times 30} = \left[1 + \frac{0.08}{12}\right]^{360}\]Calculating this gives \( k \approx 10.9355 \).

Step 3: Use the Monthly Payment Formula

Substitute the values of \( L = 250,000 \), \( r = 0.08 \), and the calculated \( k \) into the monthly payment formula:\[M = \frac{250,000 \times 0.08 \times 10.9355}{12(10.9355 - 1)}\]This results in a monthly payment of \( M \approx \$1834.41 \).

Step 4: Calculate Total Interest Paid

The total interest paid is the difference between the total amount paid over the life of the loan and the principal \( L \). Calculate the total amount paid first:\[\text{Total amount paid} = M \times 12 \times t = 1834.41 \times 12 \times 30 = 660,387.60\]Then subtract the principal from this total:\[\text{Interest paid} = 660,387.60 - 250,000 = 410,387.60\]

Step 5: Final Answers

For the 30-year \(250,000 mortgage at an 8% interest rate:(a) The monthly payment is approximately \\)1834.41.(b) The total interest paid over the life of the loan is approximately \$410,387.60.

Key Concepts

Interest Rate Calculation

Understanding how to calculate the interest rate effectively is the key to determining loan payments. When you have an interest rate expressed as a percentage, you first need to convert it into a decimal form. This makes it simpler to use in mathematical formulas. For example, an 8% interest translates to a decimal by dividing by 100, which gives us 0.08. This decimal interest rate is then used in further calculations.

The purpose of expressing the interest rate as a decimal is that it creates a uniform unit to work with, easing our way into using formulas where multiplication and exponents are involved, like when calculating the factor 'k' in loan calculations.

Monthly Payment Formula

The monthly payment formula is crucial in figuring out how much you will pay each month toward your loan. When dealing with loans, we use specific formulas to evenly distribute the payment amounts over the loan's lifespan.

The formula to find the monthly payment is:\[ M = \frac{L \times r \times k}{12(k-1)} \]Here:

• \(M\) is the monthly payment
• \(L\) is the loan amount
• \(r\) is the monthly interest rate as a decimal
• \(k\) is a factor that accounts for the compounding effect of the interest over multiple periods

Factor \(k\) itself is calculated as:\[ k = \left[1 + \frac{r}{12}\right]^{12t} \]This considers the number of payments throughout the entire loan duration. By substituting the appropriate values for a given loan scenario, such as our example with a $250,000 loan at an 8% interest over 30 years, you can calculate the exact monthly payment amount.

Total Interest Calculation

Calculating the total interest paid on a loan helps you understand how much you'll actually pay, beyond just the original principal amount borrowed. Here's how you do it.

First, determine the overall amount to be paid over the life of the loan by multiplying the monthly payment by the total number of payments:\[ \text{Total Amount Paid} = M \times 12 \times t \]Using our scenario, with a monthly payment approximately \( M = 1834.41 \), over 30 years (\( t = 30 \)), we multiply to find the total amount paid:\[ 1834.41 \times 12 \times 30 = 660,387.60 \]Subtract the original loan amount \( L = 250,000 \) from the total amount paid to find the total interest paid:\[ \text{Interest Paid} = 660,387.60 - 250,000 = 410,387.60 \]Understanding this breakdown allows you to know the cost-effectiveness of a loan over time, giving insight into the long-term financial commitment beyond just the borrower’s principal. This helps in evaluating whether the loan is a sound financial decision.