You are borrowing money from one of two lenders. Option A offers 5.3% APR compounded semiannually, while Option B offers 5.25% APR compounded monthly.

To compare the two loan options, we need to calculate the **effective annual rate (EAR)** for both lenders. This will allow us to see the true cost of borrowing, as APR (Annual Percentage Rate) does not fully account for the effect of compounding interest.

The formula for the effective annual rate (EAR) based on APR and compounding periods is:

\[
\text{EAR} = \left(1 + \frac{r}{n}\right)^n - 1
\]

Where:
- \( r \) = APR as a decimal
- \( n \) = number of compounding periods per year

### Option A: 5.3% APR compounded semiannually
- \( r = 5.3\% = 0.053 \)
- \( n = 2 \) (since interest is compounded semiannually)

\[
\text{EAR}_A = \left(1 + \frac{0.053}{2}\right)^2 - 1
\]

### Option B: 5.25% APR compounded monthly
- \( r = 5.25\% = 0.0525 \)
- \( n = 12 \) (since interest is compounded monthly)

\[
\text{EAR}_B = \left(1 + \frac{0.0525}{12}\right)^{12} - 1
\]

Let's calculate both EAR values to determine which lender offers the better option.The **Effective Annual Rate (EAR)** for each option is as follows:

- **Option A** (5.3% APR compounded semiannually): 5.37%
- **Option B** (5.25% APR compounded monthly): 5.38%

Although Option B has a slightly lower APR, its monthly compounding makes it slightly more expensive overall compared to Option A.

### Conclusion:
- **Option A** is marginally better, with a lower effective annual rate of 5.37%, compared to Option B's 5.38%.

Would you like more details, or do you have any other questions?

Here are five related questions to explore:
1. What happens to the EAR if compounding is done daily?
2. How does the term of the loan (number of years) affect total interest paid?
3. Can you calculate the total amount to be repaid for a specific loan amount?
4. How would the EAR change if Option A had quarterly compounding?
5. How does continuous compounding compare to monthly and semiannual compounding?

**Tip:** When comparing loans, always consider the **Effective Annual Rate (EAR)** to account for compounding effects rather than just the APR.

Discover how to compare two loan options with different APR and compounding periods using the Effective Annual Rate (EAR) formula. This problem demonstrates how monthly and semiannual compounding affects loan costs. Suitable for Grades 10-12 students.

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