Michelle bought a new car for $31,000. She paid a 10% down payment and financed the remaining balance for 36 months with an APR of 3.5%. Assuming she makes monthly payments, determine the total interest Michelle pays over the life of the loan. Round your answer to the nearest cent, if necessary.

To find the total interest Michelle pays over the life of the loan, we can break down the problem into several steps.

### Step 1: Calculate the loan amount
Michelle paid a 10% down payment on the car, so the remaining balance is the amount she financed.

The down payment is:
\[
\text{Down payment} = 10\% \times 31,000 = 0.10 \times 31,000 = 3,100
\]

The remaining balance (the loan amount) is:
\[
\text{Loan amount} = 31,000 - 3,100 = 27,900
\]

### Step 2: Use the formula for monthly payments
We will use the formula for calculating the monthly payment for an amortized loan:

\[
M = P \times \frac{r(1+r)^n}{(1+r)^n - 1}
\]
Where:
- $M$ is the monthly payment.
- $P$ is the loan amount ($27,900).
- $r$ is the monthly interest rate, which is the annual rate divided by 12.
- $n$ is the total number of payments (36 months in this case).

First, let's calculate the monthly interest rate:
\[
r = \frac{3.5\%}{12} = \frac{0.035}{12} = 0.00291667
\]

The number of payments $n$ is:
\[
n = 36 \text{ months}
\]

### Step 3: Calculate the monthly payment
Now, we substitute the values into the formula:

\[
M = 27,900 \times \frac{0.00291667(1+0.00291667)^{36}}{(1+0.00291667)^{36} - 1}
\]

Let me calculate the monthly payment.The monthly payment Michelle makes is approximately $817.53.

### Step 4: Calculate the total amount paid over the loan
The total amount paid over the 36 months is:

\[
\text{Total payments} = M \times n = 817.53 \times 36
\]

Let me calculate the total amount paid.The total amount Michelle will pay over 36 months is approximately \$29,431.01.

### Step 5: Calculate the total interest paid
The total interest paid is the difference between the total amount paid and the loan amount:

\[
\text{Total interest} = \text{Total payments} - \text{Loan amount} = 29,431.01 - 27,900
\]

Let me calculate the total interest.The total interest Michelle pays over the life of the loan is approximately \$1,531.01.

### Summary:
- **Loan amount**: \$27,900
- **Monthly payment**: \$817.53
- **Total interest paid**: \$1,531.01

Would you like more details or further clarification? Here are some related questions:

1. How would the interest change if the loan term was 48 months instead of 36?
2. What would happen if the interest rate were 4% instead of 3.5%?
3. How much would Michelle save by paying an extra \$50 per month?
4. What would the total cost be if Michelle decided to lease the car instead?
5. How does the down payment percentage affect the total interest?

**Tip**: The longer your loan term, the more interest you'll end up paying, even with the same monthly payment.

Learn how to calculate the total interest Michelle pays on her $31,000 car loan after making a 10% down payment and financing the remaining $27,900 for 36 months at a 3.5% APR. This problem involves key concepts like loan amortization, monthly payments, and interest calculation. Suitable for Grades 10-12.

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