An investment is doubled over a period of 12 years. Determine the rate of
interest which would make this possible if interest is calculated as:
1.2.1 Simple interest.
1.2.2 Compound interest.

To determine the rate of interest required to double an investment over a period of 12 years, we will consider both simple interest and compound interest scenarios.

### 1.2.1 Simple Interest

The formula for simple interest is:

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

Where:
- \( A \) is the final amount (which is double the initial investment, so \( A = 2P \)),
- \( P \) is the principal amount,
- \( r \) is the rate of interest (which we need to find),
- \( t \) is the time period in years (here, \( t = 12 \)).

Given that \( A = 2P \), we can substitute into the equation:

\[
2P = P(1 + 12r)
\]

Dividing both sides by \( P \):

\[
2 = 1 + 12r
\]

Solving for \( r \):

\[
12r = 2 - 1 = 1
\]

\[
r = \frac{1}{12} \approx 0.0833
\]

So, the rate of interest required for simple interest is approximately \( 8.33\% \) per annum.

### 1.2.2 Compound Interest

The formula for compound interest is:

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

Where:
- \( A \) is the final amount (again, \( A = 2P \)),
- \( P \) is the principal amount,
- \( r \) is the annual rate of interest (which we need to find),
- \( n \) is the number of times interest is compounded per year (assuming it's compounded annually, \( n = 1 \)),
- \( t \) is the time period in years (here, \( t = 12 \)).

Given \( n = 1 \), the formula simplifies to:

\[
2P = P\left(1 + r\right)^{12}
\]

Dividing both sides by \( P \):

\[
2 = \left(1 + r\right)^{12}
\]

Taking the 12th root of both sides:

\[
\sqrt[12]{2} = 1 + r
\]

Subtracting 1 from both sides:

\[
r = \sqrt[12]{2} - 1
\]

Now, calculating \( r \):

\[
r \approx 2^{\frac{1}{12}} - 1 \approx 1.059463 - 1 \approx 0.059463
\]

So, the rate of interest required for compound interest is approximately \( 5.95\% \) per annum.

### Summary

1. **Simple Interest Rate**: \( 8.33\% \) per annum.
2. **Compound Interest Rate**: \( 5.95\% \) per annum.

Would you like more details or have any questions?

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**Related Questions:**

1. What happens to the required rate of interest if the investment period changes?
2. How does the rate of interest compare if compounded monthly instead of annually?
3. How can you calculate the future value of an investment with continuous compounding?
4. What would the simple interest rate be for tripling the investment over 12 years?
5. How does inflation affect the real rate of return on an investment?

**Tip:** When calculating compound interest, always check how often the interest is compounded, as this can significantly impact the final amount.

Explore how to determine the rate of interest needed to double an investment over 12 years. Learn about simple interest and compound interest calculations with step-by-step solutions and formulas. Suitable for high school students and financial enthusiasts.

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