03- A bank offers an annual rate of 72%. Determine the effective annual interest rate, considering the monthly capitalization of interest. 

To determine the **effective annual interest rate (EAR)** when interest is compounded monthly, we use the following formula:

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

Where:
- \( r \) is the nominal annual interest rate (in decimal form),
- \( n \) is the number of compounding periods per year.

In this case:
- The nominal annual interest rate \( r = 72\% = 0.72 \),
- Interest is compounded monthly, so \( n = 12 \).

Substitute the values into the formula:

\[
\text{EAR} = \left(1 + \frac{0.72}{12}\right)^{12} - 1
\]

Now, let's compute the result.

\[
\text{EAR} = \left(1 + 0.06\right)^{12} - 1 = (1.06)^{12} - 1
\]The effective annual interest rate (EAR) is approximately **101.22%** when interest is compounded monthly on a nominal annual rate of 72%.

Would you like more details or have any questions?

Here are five related questions to explore:
1. How does increasing the number of compounding periods affect the effective interest rate?
2. What is the effective rate if the interest is compounded quarterly instead of monthly?
3. How does the effective annual rate compare to the nominal rate for different interest rates?
4. Can you explain the difference between nominal and effective interest rates?
5. What happens if interest is compounded continuously?

**Tip:** The more frequently interest is compounded, the higher the effective annual interest rate becomes, even if the nominal rate stays the same.

A bank offers an annual rate of 72%. Determine the effective annual interest rate, considering the monthly capitalization of interest.

Calculate the effective annual interest rate for a nominal rate of 72% with monthly compounding. This detailed solution uses the compound interest formula EAR = (1 + r/n)^n - 1 and explains how monthly compounding affects the effective rate. Suitable for high school students in Grades 10-12.

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