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 find the APR (Annual Percentage Rate) for this installment loan, we can use the following steps:

### 1. Breakdown of Information:
- **Cost of hot tub:** $5,070
- **Down payment:** $1,300
- **Amount financed:** $ 5,070 - 1,300 = 3,770 $ dollars
- **Monthly payment:** $175.77
- **Number of payments (term):** 24 months

### 2. Formula for APR (approximation):
We will use an approximate formula for APR for installment loans:

\[
\text{APR} \approx \frac{2 \times \text{Number of Payments} \times \text{Total Finance Charge}}{\text{Loan Amount} \times (\text{Total Number of Payments} + 1)} \times 100
\]

First, we need to calculate the **total finance charge**.

### 3. Total Finance Charge:
\[
\text{Total Payments} = 175.77 \times 24 = 4,218.48
\]
\[
\text{Finance Charge} = \text{Total Payments} - \text{Amount Financed} = 4,218.48 - 3,770 = 448.48
\]

### 4. Applying the Formula:
\[
\text{APR} \approx \frac{2 \times 24 \times 448.48}{3,770 \times (24 + 1)} \times 100
\]

Let's calculate this step by step.

\[
\text{APR} \approx \frac{2 \times 24 \times 448.48}{3,770 \times 25} \times 100
\]
\[
\text{APR} \approx \frac{21,527.04}{94,250} \times 100
\]
\[
\text{APR} \approx 0.2284 \times 100 = 22.84\%
\]

### Final Answer:
The APR is approximately **22.84%**.

Would you like further details or have any questions?

Here are some related questions:
1. What is the definition of APR?
2. How can the finance charge be reduced?
3. How does a down payment affect the APR?
4. What is the difference between APR and interest rate?
5. How can changing the loan term affect the APR?

**Tip:** Always double-check monthly payments, the finance charge, and terms when calculating APR to ensure accuracy.

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 Annual Percentage Rate (APR) for an installment loan with this step-by-step example. Miguel purchased a hot tub costing $5,070 with a down payment of $1,300 and financed the remaining $3,770 over 24 months with monthly payments of $175.77. This problem covers key financial mathematics concepts and the formula used to find the APR.

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