$4000 are invested in a bank account at an interest rate of 5 percent per year.


You have a principal amount of $4000 invested at an annual interest rate of 5%. To analyze this, we can compute the interest earned and the total balance in the account after a certain period of time, based on different interest types (simple or compound).

### 1. **Simple Interest**:
Simple interest is calculated using the formula:
\[
I = P \times r \times t
\]
where:
- $I$ is the interest,
- $P$ is the principal ($4000),
- $r$ is the annual interest rate (5% or 0.05),
- $t$ is the time in years.

For example, after 1 year, the interest earned will be:
\[
I = 4000 \times 0.05 \times 1 = 200
\]
So, the total amount after 1 year will be:
\[
A = P + I = 4000 + 200 = 4200
\]

### 2. **Compound Interest**:
Compound interest is calculated using the formula:
\[
A = P \times \left(1 + \frac{r}{n}\right)^{n \times t}
\]
where:
- $A$ is the amount after $t$ years,
- $n$ is the number of times interest is compounded per year,
- $P$ is the principal ($4000),
- $r$ is the annual interest rate (5% or 0.05),
- $t$ is the time in years.

If the interest is compounded annually ($n = 1$) for 1 year:
\[
A = 4000 \times \left(1 + \frac{0.05}{1}\right)^{1 \times 1} = 4000 \times (1.05) = 4200
\]

If the interest is compounded monthly ($n = 12$) for 1 year:
\[
A = 4000 \times \left(1 + \frac{0.05}{12}\right)^{12 \times 1} = 4000 \times (1.05116) \approx 4204.64
\]

The amount increases slightly more with monthly compounding due to more frequent interest calculation.

Would you like more details on either calculation? Here are five related questions to consider:

1. What happens if the investment period is extended to 5 or 10 years?
2. How does quarterly compounding compare to annual or monthly compounding?
3. What is the difference in interest earned between simple and compound interest over time?
4. How would the total amount change with a higher or lower interest rate?
5. What would the final amount be if you made additional contributions to the investment?

**Tip**: Compounding frequency plays a crucial role in determining the growth of an investment. The more frequent the compounding, the greater the total interest earned.

$4000 are invested in a bank account at an interest rate of 5 percent per year.

Explore how to calculate both simple and compound interest on a $4000 investment at a 5% annual interest rate. This solution explains key interest formulas, showing examples of how much the investment grows over 1 year, with both simple and compound interest, and discusses compounding frequency.

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