Problem 51 - 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. Car loan An automobile dealer offers customers no-downpayment 3 -year loans at an interest rate of \(10 \%\). If a customer can afford to pay \(\$ 500\) per month, find the price of the most expensive car that can be purchased.

Step-by-Step Solution

Answer

The maximum car price is about $15,516.78.

Step 1: Identifying Variables and Given Values

We need to determine the maximum loan amount (or car price 'L') the customer can afford. We know the monthly payment, \(M = 500\), the interest rate, \(r = 0.10\) (since 10% is expressed as a decimal), and the loan term \(t = 3\) years.

Step 2: Calculate k

The formula for \(k\) is \(k = [1 + (r / 12)]^{12t}\). Substituting the values: \(r = 0.10\), and \(t = 3\), the expression becomes:\[k = \left[1 + \frac{0.10}{12}\right]^{12 \times 3} = [1 + 0.0083]^{36} \approx 1.34885.\]

Step 3: Substitute Values into the Monthly Payment Formula

We rearrange the formula to solve for \(L\):\[L = \frac{M \times 12 \times (k-1)}{r \times k}.\]Substituting \(M = 500\), \(r = 0.10\), and the calculated \(k \approx 1.34885\):\[L = \frac{500 \times 12 \times (1.34885 - 1)}{0.10 \times 1.34885}.\]

Step 4: Perform the Calculation for L

Calculate the numerator:\[500 \times 12 \times (1.34885 - 1) = 500 \times 12 \times 0.34885 \approx 2093.1.\]Calculate the denominator:\[0.10 \times 1.34885 \approx 0.134885.\]Finally, divide the results to find \(L\):\[L \approx \frac{2093.1}{0.134885} \approx 15516.78.\]

Step 5: Conclusion

The maximum price of the car that the customer can afford is approximately $15,516.78.

Key Concepts

Interest Rate

The interest rate is a key component in loan payment calculations. It represents the cost of borrowing money, expressed as a percentage of the loan amount. For example, a 10% interest rate means the borrower must pay back an additional 10% of the original loan value over the loan term. When expressed as a decimal for calculation purposes, 10% becomes 0.10.
Understanding the interest rate allows you to determine the total cost of a loan over time. In calculations, this is vital for determining the monthly payment amount or the total repayment.

• Higher interest rates increase monthly payments and total loan costs.
• Lower interest rates make loans more affordable by reducing these payments.

Loan contracts specify whether the interest rate is fixed or variable. A fixed rate remains the same throughout the loan term, providing payment stability. A variable rate may change, usually in response to economic conditions, which can make budgeting more challenging.

Monthly Payment

The monthly payment is the amount the borrower is required to pay each month to repay the loan over the agreed-upon time frame. This amount includes both the principal and interest.
In our car loan example, the customer can make a monthly payment of $500. Knowing the maximum you can afford to pay monthly helps you determine the loan's total size and whether the loan fits your budget.

• Monthly payments are calculated using the loan amount, interest rate, and loan term.
• A higher monthly payment can reduce the total interest paid over the life of the loan.

It's important to ensure monthly payments fit comfortably within your budget to avoid financial strain. Missing payments can lead to penalties and affect your credit score, increasing future borrowing costs.

Loan Term

The loan term is the period over which the borrower agrees to repay the loan. Loan terms are typically expressed in years.
In our exercise, the loan term is 3 years, meaning the total debt is to be paid off in 36 monthly installments. The choice of a loan term can significantly influence the monthly payment and total interest paid.

• Longer loan terms generally result in smaller monthly payments but can increase cumulative interest costs.
• Shorter loan terms typically mean higher monthly payments but less total interest over the loan's lifespan.

Choosing the right loan term should balance your ability to meet higher payments with the desire to reduce overall costs.

Mathematical Formula

Calculating loan payments involves specific mathematical formulas. For instance, the formula to determine the monthly payment in our example incorporates variables like the loan amount, interest rate, and term.
The formula used is \(M=\frac{L r k}{12(k-1)}\) where \(k=[1+(r / 12)]^{12 t}\). This captures the compounding nature of interest, making it suitable for calculating monthly payments.

• Understanding these formulas helps decipher how changes in parameters affect payments.
• Practice with examples enhances familiarity with these calculations.

It's important to input the correct values for each variable, such as converting percentages to decimals and ensuring all units align. Mastery of this formula empowers you to make informed borrowing decisions and optimize loan conditions.