Questions: Suppose Santiago wants to buy a car. The dealer offers a financing package consisting of a 6% APR compounded monthly for a term of 7 years. Suppose Santiago wants his monthly payments to be at most 330. What is the maximum amount that he should finance? How much interest will he pay on the loan? a) The maximum amount Santiago should finance is . (Do not round until the final answer. Then round to the nearest dollar as needed.) b) The total paid for the car under these terms would be 27,720, including in total interest, (Use the answer from part a to answer part b. Write whole dollar answers.)

Suppose Santiago wants to buy a car. The dealer offers a financing package consisting of a 6% APR compounded monthly for a term of 7 years. Suppose Santiago wants his monthly payments to be at most 330. What is the maximum amount that he should finance? How much interest will he pay on the loan? a) The maximum amount Santiago should finance is . (Do not round until the final answer. Then round to the nearest dollar as needed.) b) The total paid for the car under these terms would be 27,720, including in total interest, (Use the answer from part a to answer part b. Write whole dollar answers.)
Transcript text: Suppose Santiago wants to buy a car. The dealer offers a financing package consisting of a $6 \%$ APR compounded monthly for a term of 7 years. Suppose Santiago wants his monthly payments to be at most $\$ 330$. What is the maximum amount that he should finance? How much interest will he pay on the loan? a) The maximum amount Santiago should finance is $\$$ $\square$ . (Do not round until the final answer. Then round to the nearest dollar as needed.) b) The total paid for the car under these terms would be $\$ 27,720$, including $\$$ $\square$ in total interest, (Use the answer from part a to answer part b. Write whole doliar answers.)
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Solution

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Solution Steps

To solve this problem, we need to use the formula for the present value of an annuity to determine the maximum amount Santiago should finance. The formula is:

\[ PV = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \]

where:

  • \( PV \) is the present value (the amount to be financed)
  • \( PMT \) is the monthly payment
  • \( r \) is the monthly interest rate (APR divided by 12)
  • \( n \) is the total number of payments (loan term in years multiplied by 12)

Once we have the present value, we can calculate the total interest paid by subtracting the principal from the total amount paid over the term of the loan.

Solution Approach
  1. Calculate the monthly interest rate by dividing the annual rate by 12.
  2. Calculate the total number of payments by multiplying the number of years by 12.
  3. Use the present value formula to find the maximum amount Santiago should finance.
  4. Calculate the total amount paid over the term of the loan.
  5. Subtract the principal from the total amount paid to find the total interest paid.
Step 1: Calculate Monthly Interest Rate

The annual percentage rate (APR) is given as \( 6\% \). To find the monthly interest rate, we divide the APR by \( 12 \):

\[ r = \frac{0.06}{12} = 0.005 \]

Step 2: Calculate Total Number of Payments

The loan term is \( 7 \) years. The total number of monthly payments is calculated as:

\[ n = 7 \times 12 = 84 \]

Step 3: Calculate Present Value

Using the present value formula for an annuity, we find the maximum amount Santiago should finance:

\[ PV = \frac{PMT \times (1 - (1 + r)^{-n})}{r} \]

Substituting the values:

\[ PV = \frac{330 \times (1 - (1 + 0.005)^{-84})}{0.005} \approx 22590 \]

Step 4: Calculate Total Amount Paid

The total amount paid over the term of the loan is:

\[ \text{Total Amount Paid} = PMT \times n = 330 \times 84 = 27720 \]

Step 5: Calculate Total Interest Paid

The total interest paid on the loan is the difference between the total amount paid and the present value:

\[ \text{Total Interest Paid} = \text{Total Amount Paid} - PV = 27720 - 22590 = 5130 \]

Final Answer

a) The maximum amount Santiago should finance is \(\boxed{22590}\).

b) The total interest paid on the loan is \(\boxed{5130}\).

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