Financial institutions often offer lower auto loan rates for new cars than used ones. A local credit union advertises new car loans at 2.79% APR and used car loans at 3.29% both for up to 72 months. Tyresa wants to buy a car but doesn’t want to spend more than $350 a month for a maximum of four years. What is the maximum loan amount she can take out for a new car and a used car using the advertised rates? Use the formula, A=P[(1+rn)nt−1]rn(1+rn)nt where P is the monthly payment, r is the annual interest rate, n is the number times interest is compounded in one year, and t is the number of years. To receive full credit, be sure to show all steps and calculations used, included where you rounded.

Question

"use this to answer the question:
Using the Loan Budget Calculator:New car: M = , N = , R = and T = Maximum new car loan amount = $Used car: M = , N = , R = and T = Maximum used car loan amount = $

Answer 1

To determine the maximum loan amount that Tyresa can take out for both a new and a used car, we will use the formula provided:

\[ A = P \cdot \frac{(1 + r/n)^{nt} - 1}{(r/n)(1 + r/n)^{nt}} \]

Where:

• \(A\) is the total loan amount (principal),
• \(P\) is the monthly payment,
• \(r\) is the annual nominal interest rate (as a decimal),
• \(n\) is the number of times the interest is compounded per year,
• \(t\) is the number of years.

Step 1: Parameters for New Car Loan

• \(M\) (monthly payment) = 350
• \(N\) (number of compounding periods per year) = 12 (monthly payments)
• \(R\) (annual interest rate) = 2.79% = 0.0279 (as a decimal)
• \(T\) (time in years) = 4

Step 2: Calculate New Car Maximum Loan Amount

Substituting these values into the formula:

1. Compute \(r/n\): \[ r/n = \frac{0.0279}{12} \approx 0.002325 \]

2. Compute \(nt\): \[ nt = 12 \cdot 4 = 48 \]

3. Compute \((1 + r/n)^{nt}\): \[ (1 + 0.002325)^{48} \approx 1.119445 \]

4. Now substitute back into the formula to find \(A\): \[ A = 350 \cdot \frac{(1.119445 - 1)}{(0.002325)(1.119445)} \] \[ = 350 \cdot \frac{0.119445}{0.0026044} \approx 350 \cdot 45.828 \approx 16039.80 \]

Thus, the maximum loan amount for a new car is approximately $16,039.80.

Step 3: Parameters for Used Car Loan

• \(M\) (monthly payment) = 350
• \(N\) (number of compounding periods per year) = 12
• \(R\) (annual interest rate) = 3.29% = 0.0329 (as a decimal)
• \(T\) (time in years) = 4

Step 4: Calculate Used Car Maximum Loan Amount

Substituting these values into the formula:

1. Compute \(r/n\): \[ r/n = \frac{0.0329}{12} \approx 0.00274167 \]

2. Compute \(nt\): \[ nt = 12 \cdot 4 = 48 \]

3. Compute \((1 + r/n)^{nt}\): \[ (1 + 0.00274167)^{48} \approx 1.137684 \]

4. Now substitute back into the formula to find \(A\): \[ A = 350 \cdot \frac{(1.137684 - 1)}{(0.00274167)(1.137684)} \] \[ = 350 \cdot \frac{0.137684}{0.003113415} \approx 350 \cdot 44.179 \approx 15462.75 \]

Thus, the maximum loan amount for a used car is approximately $15,462.75.

Summary of Results

Maximum new car loan amount = $16,039.80
Maximum used car loan amount = $15,462.75