Questions: For the car loan described, give the following information. A newspaper advertisement offers a 4,000 used car for nothing down and 36 easy monthly payments of 147.62. (a) amount to be paid (b) amount of interest (c) interest rate (Round your answer to two decimal places.) % (d) APR (rounded to the nearest tenth of a percent) %

For the car loan described, give the following information. A newspaper advertisement offers a 4,000 used car for nothing down and 36 easy monthly payments of 147.62. (a) amount to be paid (b) amount of interest (c) interest rate (Round your answer to two decimal places.) % (d) APR (rounded to the nearest tenth of a percent) %
Transcript text: For the car loan described, give the following information. A newspaper advertisement offers a $\$ 4,000$ used car for nothing down and 36 easy monthly payments of $\$ 147.62$. (a) amount to be paid \$ $\qquad$ (b) amount of interest \$ $\qquad$ (c) interest rate (Round your answer to two decimal places.) $\qquad$ \% (d) APR (rounded to the nearest tenth of a percent) $\qquad$ \%
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Solution

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Solution Steps

To solve the given problem, we need to follow these steps:

(a) Calculate the total amount to be paid by multiplying the monthly payment by the number of payments. (b) Calculate the amount of interest by subtracting the principal (initial loan amount) from the total amount paid. (c) Calculate the interest rate using the formula for the monthly interest rate and then annualize it. (d) Calculate the APR, which is the annual percentage rate, using the formula for APR.

Step 1: Total Amount to be Paid

To find the total amount to be paid over the duration of the loan, we calculate: \[ \text{Total Amount Paid} = \text{Monthly Payment} \times \text{Number of Payments} = 147.62 \times 36 = 5314.32 \]

Step 2: Amount of Interest

The amount of interest paid over the life of the loan is given by: \[ \text{Amount of Interest} = \text{Total Amount Paid} - \text{Principal} = 5314.32 - 4000 = 1314.32 \]

Step 3: Interest Rate

The monthly interest rate can be calculated using the formula: \[ r = \left( \frac{A}{P} \right)^{\frac{1}{n}} - 1 \] where \( A \) is the total amount paid, \( P \) is the principal, and \( n \) is the number of payments. Substituting the values: \[ r = \left( \frac{5314.32}{4000} \right)^{\frac{1}{36}} - 1 \approx 0.0079231876 \] To find the annual interest rate: \[ \text{Annual Interest Rate} = r \times 12 \times 100 \approx 9.51\% \]

Step 4: APR

The APR is typically approximated by the annual interest rate: \[ \text{APR} \approx 9.5\% \]

Final Answer

\[ \text{(a) Total amount to be paid: } \boxed{5314.32} \] \[ \text{(b) Amount of interest: } \boxed{1314.32} \] \[ \text{(c) Interest rate: } \boxed{9.51\%} \] \[ \text{(d) APR: } \boxed{9.5\%} \]

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