Miguel purchased a hot tub costing $5,070 by taking out an installment loan. He made a down payment of $1,300 and financed the balance for 24 months. If the payments are $175.77 each month, use the APR formula to find the APR. Round to the nearest hundredth of a percent.

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To calculate the Annual Percentage Rate (APR) using the information provided, we can apply the following steps. The formula for APR in installment loans is not direct, but we can approximate it through the following process:

### Information given:
- **Total Cost of Hot Tub**: $5,070
- **Down Payment**: $1,300
- **Loan Amount**: $5,070 - $1,300 = $3,770
- **Monthly Payment**: $175.77
- **Number of Payments**: 24 months

### Step 1: Calculate Total Paid in Installments
The total amount paid over the 24 months will be:
\[
\text{Total Paid} = 24 \times 175.77 = 4,218.48
\]

### Step 2: Calculate Total Interest Paid
The total interest paid over the life of the loan is:
\[
\text{Total Interest} = \text{Total Paid} - \text{Loan Amount}
\]
\[
\text{Total Interest} = 4,218.48 - 3,770 = 448.48
\]

### Step 3: Estimate the APR
The approximate formula for APR (Annual Percentage Rate) for installment loans is:
\[
\text{APR} \approx \frac{2 \times \text{Number of Payments per Year} \times \text{Total Interest}}{\text{Loan Amount} \times (\text{Number of Payments} + 1)}
\]
Where:
- Number of payments per year = 12
- Total interest = 448.48
- Loan amount = 3,770
- Number of payments = 24

Now, let's plug in the numbers:
\[
\text{APR} \approx \frac{2 \times 12 \times 448.48}{3,770 \times (24 + 1)}
\]

I'll calculate this.The estimated APR (Annual Percentage Rate) is approximately **11.42%**, rounded to the nearest hundredth of a percent.

Would you like more details or have any questions?

Here are 5 related questions to explore:
1. How is APR different from the nominal interest rate?
2. What happens if the payment term is extended to 36 months?
3. How would a larger down payment affect the APR?
4. What other fees might affect the total cost of the loan?
5. How can you calculate the total cost of a loan with varying interest rates?

**Tip**: Always compare the APR of different loans, as it reflects the true cost of borrowing, including interest and fees.

Miguel purchased a hot tub costing $5,070 by taking out an installment loan. He made a down payment of $1,300 and financed the balance for 24 months. If the payments are $175.77 each month, use the APR formula to find the APR. Round to the nearest hundredth of a percent.

Learn how to calculate the APR for an installment loan using a real-life example. Miguel financed a $5,070 hot tub with a $1,300 down payment and 24 monthly payments of $175.77. Understand the steps to estimate the APR, including calculating the total interest paid. Suitable for high school and college students.

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