If $25,000.00 is invested in an account for 30 years. Find the value of the investment at the end of 30 years if the interest is:

(a) 6% simple interest:

(b) 6% compounded monthly:

To solve these problems, we'll use the formulas for simple interest and compound interest.

### (a) 6% Simple Interest

The formula for simple interest is:

\[
A = P(1 + rt)
\]

where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount (the initial investment).
- $ r $ is the annual interest rate (in decimal form).
- $ t $ is the time the money is invested for (in years).

Given:
- $ P = 25,000.00 $ dollars
- $ r = 0.06 $ (6%)
- $ t = 30 $ years

Substituting the values:

\[
A = 25,000 \times (1 + 0.06 \times 30)
\]

\[
A = 25,000 \times (1 + 1.8) = 25,000 \times 2.8 = 70,000.00
\]

So, the value of the investment at the end of 30 years with simple interest is **$70,000.00**.

### (b) 6% Compounded Monthly

The formula for compound interest is:

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

where:
- $ A $ is the amount of money accumulated after $ t $ years, including interest.
- $ P $ is the principal amount.
- $ r $ is the annual interest rate (in decimal form).
- $ n $ is the number of times that interest is compounded per year.
- $ t $ is the time the money is invested for (in years).

Given:
- $ P = 25,000.00 $ dollars
- $ r = 0.06 $ (6%)
- $ n = 12 $ (since interest is compounded monthly)
- $ t = 30 $ years

Substituting the values:

\[
A = 25,000 \left(1 + \frac{0.06}{12}\right)^{12 \times 30}
\]

\[
A = 25,000 \left(1 + 0.005\right)^{360} = 25,000 \times (1.005)^{360}
\]

Now, calculate the compound factor:

\[
(1.005)^{360} \approx 6.0226
\]

\[
A \approx 25,000 \times 6.0226 \approx 150,565.00
\]

So, the value of the investment at the end of 30 years with compound interest is approximately **$150,565.00**.

---

Would you like further details on these calculations, or do you have any questions?

Here are 5 related questions:
1. How would the final amount change if the interest rate was 5% instead of 6%?
2. What would the value be if the interest was compounded quarterly instead of monthly?
3. How much interest is earned with simple interest versus compound interest?
4. What if the investment period was only 20 years? How would that affect the outcomes?
5. Can you explain the difference between nominal and effective interest rates?

**Tip:** When comparing different investment options, compound interest generally yields higher returns than simple interest, especially over longer periods.

If $25,000.00 is invested in an account for 30 years. Find the value of the investment at the end of 30 years if the interest is:
(a) 6% simple interest:
(b) 6% compounded monthly:

Calculate the value of a $25,000 investment over 30 years with both 6% simple and 6% monthly compounded interest. Learn how simple interest yields $70,000 while compound interest grows the investment to approximately $150,565. Perfect for high school students exploring finance math concepts.

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