To determine the Annual Percentage Rate (APR) that Audrey has paid for her car loan, we can follow these steps:
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Calculate the loan amount after the down payment: \[ \text{Loan Amount} = \text{Car Price} - \text{Down Payment} \] \[ \text{Loan Amount} = 32,998.00 - 4,200.00 = 28,798.00 \]
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Determine the total amount paid over the course of the loan: \[ \text{Total Payments} = \text{Monthly Payment} \times \text{Number of Payments} \] Since she plans to make monthly payments for 5 years (60 months): \[ \text{Total Payments} = 525 \times 60 = 31,500.00 \]
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Find the total interest paid: \[ \text{Total Interest} = \text{Total Payments} - \text{Loan Amount} \] \[ \text{Total Interest} = 31,500.00 - 28,798.00 = 2,702.00 \]
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Estimate the APR using the formula for the monthly payment of an installment loan: The formula for the monthly payment can be rearranged to achieve the APR. The formula for monthly payments \(M\) is given as: \[ M = P \frac{r(1 + r)^n}{(1 + r)^n - 1} \] where:
- \(P\) = principal (loan amount)
- \(r\) = monthly interest rate (APR/12)
- \(n\) = number of total payments (months)
In this case:
- \(M = 525\)
- \(P = 28,798\)
- \(n = 60\)
We can solve for \(r\). It is easier to use spreadsheet software or a financial calculator for this.
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Finding \(r\): You can use methods like the trial-and-error approach, numerical methods, or optimization functions in a calculator to estimate \(r\). For simplicity, let's use online loan calculators or financial calculators to derive \(r\). After applying it or adjusting the input to approximate, you will find that:
- A monthly payment of $525 on a loan amount of approximately $28,798 over 60 months corresponds to an APR of about 3.7%.
Given the options:
A) 0.37%
B) 37%
C) 3%
D) 3.7%
The correct answer is: D) 3.7%.
1 answer