Des Grimm had $3000 invested at 8% p.a. compound interest for 3 years. He then invested for a further 2 years at 10%.   How much interest has he gained?

To determine the total interest Des Grimm has gained from his investments, we need to calculate the amount of money he has after each period of investment and then find the total interest.

1. **First 3 years at 8% p.a. compound interest:**

   The formula for compound interest is:
   \[
   A = P \left(1 + \frac{r}{100}\right)^n
   \]
   where:
   - $A$ is the amount of money accumulated after $n$ years, including interest.
   - $P$ is the principal amount (initial investment).
   - $r$ is the annual interest rate (in percentage).
   - $n$ is the number of years the money is invested or borrowed for.

   Given:
   - $P = 3000$
   - $r = 8\%$
   - $n = 3$

   Plugging in these values:
   \[
   A_1 = 3000 \left(1 + \frac{8}{100}\right)^3
   \]
   \[
   A_1 = 3000 \left(1 + 0.08\right)^3
   \]
   \[
   A_1 = 3000 \left(1.08\right)^3
   \]
   \[
   A_1 = 3000 \times 1.259712
   \]
   \[
   A_1 \approx 3779.14
   \]

2. **Next 2 years at 10% p.a. compound interest:**

   Now, using the amount $A_1$ as the principal for the next period:
   \[
   P = 3779.14
   \]
   - $r = 10\%$
   - $n = 2$

   Using the same formula:
   \[
   A_2 = 3779.14 \left(1 + \frac{10}{100}\right)^2
   \]
   \[
   A_2 = 3779.14 \left(1 + 0.10\right)^2
   \]
   \[
   A_2 = 3779.14 \left(1.10\right)^2
   \]
   \[
   A_2 = 3779.14 \times 1.21
   \]
   \[
   A_2 \approx 4573.76
   \]

3. **Total interest gained:**

   The total interest gained is the final amount minus the initial principal:
   \[
   \text{Total Interest} = A_2 - P_{\text{initial}}
   \]
   \[
   P_{\text{initial}} = 3000
   \]
   \[
   \text{Total Interest} = 4573.76 - 3000
   \]
   \[
   \text{Total Interest} \approx 1573.76
   \]

Therefore, Des Grimm has gained approximately $1573.76 in interest over the 5-year period.

Would you like further details or have any questions?

Here are 8 related questions for further exploration:

1. How would the total interest change if the interest rates were reversed?
2. What would be the final amount if the initial investment was $5000 instead of $3000?
3. How much interest would be gained if the entire 5 years were at 8% interest?
4. What is the difference in total amount if the second period was invested at simple interest instead of compound interest?
5. How does the interest calculation change if the interest is compounded monthly instead of annually?
6. What would be the total interest if the second investment period was 5 years instead of 2?
7. How can we calculate the effective annual rate for the combined investment periods?
8. What is the impact of a 2% increase in the annual interest rate during the first 3 years?

**Tip:** Always double-check your calculations, especially when dealing with compound interest over multiple periods, to ensure accuracy.

Learn how to calculate compound interest with Des Grimm's investments. Understand the application of compound interest formulas over different periods and rates. Suitable for advanced high school students.

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