You are considering the purchase of new living room furniture that costs $1,540. The store will allow you to make weekly payments of $32.82 for one year to pay off the loan. What is the EAR of this arrangment?

Multiple Choice

20.55%

19.52%

22.76%

21.65%

23.90%

To calculate the **Effective Annual Rate (EAR)** for the loan, we can use the following formula:

\[
EAR = \left(1 + \frac{i}{m}\right)^m - 1
\]

Where:
- $i$ is the nominal interest rate for the year.
- $m$ is the number of compounding periods in a year.

To determine the EAR, we need to first calculate the nominal interest rate (APR). Here's how we do it:

### Step 1: Calculate the total amount paid.
You make weekly payments of $32.82 for one year, so the total number of payments in a year is 52 weeks.

\[
\text{Total paid} = 32.82 \times 52 = 1,706.64
\]

### Step 2: Find the interest paid.
Subtract the original cost of the furniture ($1,540) from the total amount paid ($1,706.64):

\[
\text{Interest paid} = 1,706.64 - 1,540 = 166.64
\]

### Step 3: Calculate the interest rate over the year.
The interest paid over the year is $166.64, so the nominal interest rate can be approximated as:

\[
\text{Nominal interest rate (APR)} = \frac{166.64}{1,540} = 0.1082 \text{ or } 10.82\%
\]

### Step 4: Use the APR to find the EAR.
Assuming weekly compounding (52 compounding periods), we can now use the formula for the EAR:

\[
EAR = \left(1 + \frac{0.1082}{52}\right)^{52} - 1
\]

Let's calculate this.It seems like I can’t do more advanced data analysis right now. Please try again later.

Let me walk you through the steps so you can calculate it manually:

1. Use the formula for **Effective Annual Rate (EAR)**:
   \[
   EAR = \left(1 + \frac{0.1082}{52}\right)^{52} - 1
   \]
2. Perform the calculations step by step:
   - Divide the APR by 52 (weekly compounding).
   - Add 1 to that result.
   - Raise the result to the 52nd power.
   - Subtract 1 from the final result and convert to a percentage.

If you have a calculator or a spreadsheet, you can apply these steps easily.

Let me know if you need further clarification or help with any specific part of the calculation!

---

Here are 5 related questions for further practice:

1. What would the EAR be if the loan was compounded monthly instead of weekly?
2. How does the nominal interest rate (APR) compare to the EAR in this case?
3. How much interest would you pay if the payment schedule was bi-weekly instead?
4. What would the total interest paid be if the loan was for 2 years at the same payment rate?
5. How would increasing the weekly payment affect the EAR?

**Tip**: EAR tends to be higher than the nominal interest rate when compounding occurs more frequently.

You are considering the purchase of new living room furniture that costs $1,540. The store will allow you to make weekly payments of $32.82 for one year to pay off the loan. What is the EAR of this arrangement?

Learn how to calculate the Effective Annual Rate (EAR) for a loan with weekly payments of $32.82 on a $1,540 furniture purchase. This detailed guide covers key financial concepts, including nominal interest rates, compounding periods, and the formula for EAR. Suitable for advanced high school or college-level students.

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