You are considering two ways of financing a spring break vacation. You could put it on your credit​ card, at 14% ​APR, compounded​ monthly, or borrow the money from your​ parents, who want an interest payment of 8% every six months. Which is the lower​ rate?

To compare the two financing options, we need to find the effective annual rate (EAR) for both the credit card and the loan from your parents. This will allow us to determine which option has a lower rate.

### 1. **Credit Card (14% APR, compounded monthly):**
The formula for the **Effective Annual Rate (EAR)** when the interest is compounded monthly is:

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

Where:
- \(r = 0.14\) (the annual interest rate of 14%)
- \(n = 12\) (monthly compounding)

\[
\text{EAR}_{credit\ card} = \left( 1 + \frac{0.14}{12} \right)^{12} - 1
\]

### 2. **Loan from Parents (8% every 6 months):**
This is a simple interest case, where the interest is applied every six months. So we calculate the **EAR** for semi-annual compounding:

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

Where:
- \(r = 0.08\) (8% for every six months)
- \(n = 2\) (semi-annual compounding)

\[
\text{EAR}_{parents} = \left( 1 + \frac{0.08}{1} \right)^2 - 1
\]

Now, let's compute the two values.The **Effective Annual Rate (EAR)** for both financing options is as follows:

- **Credit card** (14% APR, compounded monthly): **14.93%**
- **Loan from parents** (8% every six months): **16.64%**

The lower rate is from the **credit card** at 14.93%, which is better than the 16.64% offered by your parents.

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

1. How do different compounding frequencies affect the EAR?
2. What is the formula to calculate EAR for daily compounding interest?
3. Can you explain why the parents' loan has a higher EAR despite a lower nominal rate?
4. How does the APR differ from the EAR?
5. What impact does the loan duration have on interest calculations?

**Tip:** Always compare the **effective annual rate (EAR)** rather than the nominal rate to understand the true cost of borrowing.

Find out how to compare the Effective Annual Rates (EAR) of a 14% APR credit card, compounded monthly, and a semi-annual loan from parents at 8%. Learn how compounding frequency affects the true cost of borrowing. This step-by-step solution is ideal for students in Grades 10-12.

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