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The image shows a compound interest problem where you are tasked with finding the value of an investment of $22,500 at an interest rate of 5% per year, compounded using different methods. The compounding periods are:

- (a) Annually
- (b) Semiannually
- (c) Monthly
- (d) Daily

To solve this, we can use the compound interest formula:

\[
A = P \left(1 + \frac{r}{n}\right)^{nt}
\]

Where:
- $A$ is the future value of the investment/loan, including interest.
- $P$ is the principal investment amount ($22,500).
- $r$ is the annual interest rate (5% or 0.05).
- $n$ is the number of times that interest is compounded per year (varies depending on the case).
- $t$ is the time the money is invested for (5 years in this case).

Let's calculate for each compounding method:

### (a) Annually (n = 1)
\[
A = 22,500 \left(1 + \frac{0.05}{1}\right)^{1 \times 5} = 22,500 \times (1.05)^5
\]
\[
A \approx 22,500 \times 1.2762815625 \approx 28,716.33
\]

### (b) Semiannually (n = 2)
\[
A = 22,500 \left(1 + \frac{0.05}{2}\right)^{2 \times 5} = 22,500 \times \left(1.025\right)^{10}
\]
\[
A \approx 22,500 \times 1.280084512 \approx 28,802.00
\]

### (c) Monthly (n = 12)
\[
A = 22,500 \left(1 + \frac{0.05}{12}\right)^{12 \times 5} = 22,500 \times \left(1.00416667\right)^{60}
\]
\[
A \approx 22,500 \times 1.283358678 \approx 28,875.57
\]

### (d) Daily (n = 365)
\[
A = 22,500 \left(1 + \frac{0.05}{365}\right)^{365 \times 5} = 22,500 \times \left(1.000136986\right)^{1825}
\]
\[
A \approx 22,500 \times 1.28400337 \approx 28,890.08
\]

Here are the answers rounded to the nearest cent:
- (a) Annual: **$28,716.33**
- (b) Semiannual: **$28,802.00**
- (c) Monthly: **$28,875.57**
- (d) Daily: **$28,890.08**

Would you like more details or further clarifications?

Here are 5 related questions:
1. How would the future value change if the interest rate was 6% instead of 5%?
2. What is the difference in future value if the investment period is 10 years instead of 5?
3. How is compound interest different from simple interest in terms of final value?
4. What effect does increasing the compounding frequency (e.g., from annually to daily) have on the final amount?
5. How can the compound interest formula be used for continuous compounding?

**Tip:** The more frequently interest is compounded, the larger the final amount will be, due to more frequent application of interest on interest.

If $22,500 is invested at an interest rate of 5 percent per year, find the value of the investment at the end of 5 years for the following compounding methods, to the nearest cent: (a) Annual, (b) Semiannual, (c) Monthly, (d) Daily.

Calculate the future value of a $22,500 investment with a 5% interest rate over 5 years using different compounding methods: annual, semiannual, monthly, and daily. Explore key concepts such as compound interest and exponential growth with detailed step-by-step calculations.

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