If $3,000.00 is invested in an account for 25 years. Find the value of the investment at the end of 25 years if the interest is:

(a) 8% simple interest:  

(b) 8% compounded monthly: 

Let's solve the problem step by step for both cases:

### (a) Simple Interest

The formula for simple interest is:
\[
A = P(1 + rt)
\]
where:
- $ A $ is the amount of the investment after time $ t $,
- $ P $ is the principal (initial investment),
- $ r $ is the annual interest rate (in decimal form),
- $ t $ is the time in years.

For this problem:
- $ P = 3000 $,
- $ r = 8\% = 0.08 $,
- $ t = 25 $ years.

Substitute the values into the formula:

\[
A = 3000(1 + 0.08 \times 25)
\]
\[
A = 3000(1 + 2)
\]
\[
A = 3000 \times 3 = 9000
\]

So, the value of the investment with simple interest is **$9,000.00**.

### (b) Compound Interest (Compounded Monthly)

The formula for compound interest is:
\[
A = P\left(1 + \frac{r}{n}\right)^{nt}
\]
where:
- $ A $ is the amount of the investment after time $ t $,
- $ P $ is the principal,
- $ r $ is the annual interest rate (in decimal),
- $ n $ is the number of times the interest is compounded per year,
- $ t $ is the time in years.

For this problem:
- $ P = 3000 $,
- $ r = 8\% = 0.08 $,
- $ n = 12 $ (since the interest is compounded monthly),
- $ t = 25 $ years.

Substitute the values into the formula:

\[
A = 3000\left(1 + \frac{0.08}{12}\right)^{12 \times 25}
\]
\[
A = 3000\left(1 + 0.0066667\right)^{300}
\]
\[
A = 3000(1.0066667)^{300}
\]

Now, let's compute this.

\[
A \approx 3000(7.039987)
\]
\[
A \approx 21,119.96
\]

So, the value of the investment with compound interest is approximately **$21,119.96**.

---

Let me know if you'd like any further clarification or details!

Here are 5 related questions you might find useful:

1. What would be the final amount if the interest rate were 10% compounded annually for 25 years?
2. How does daily compounding compare to monthly compounding in terms of the final investment value?
3. What would be the value of the investment if it were compounded quarterly at the same interest rate?
4. How would the result change if the investment was only for 15 years instead of 25?
5. What is the difference between simple interest and compound interest for shorter periods, like 5 years?

**Tip**: Compound interest grows exponentially due to interest on interest, while simple interest grows linearly based only on the original principal.

If $3,000.00 is invested in an account for 25 years. Find the value of the investment at the end of 25 years if the interest is: (a) 8% simple interest; (b) 8% compounded monthly.

Discover how to calculate the future value of a $3,000 investment over 25 years with both 8% simple interest and 8% compounded monthly. Learn key financial formulas, including simple interest and compound interest equations, suitable for high school students studying financial mathematics.

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