To solve this problem, we need to determine three different rates of return: the simple rate of return, the effective (compound) rate of return, and the continuous rate of return.

1. Simple Rate of Return

The formula for simple interest is:

$A = P(1 + rt)$

Where:

• $A$ is the final amount ($11,500)
• $P$ is the initial principal ($5,000)
• $r$ is the simple rate of return per annum (what we need to find)
• $t$ is the time in years (6 years and 9 months = 6.75 years)

Rearranging the formula to solve for $r$:

$r = \frac{A - P}{Pt}$

Substitute the values:

$r = \frac{11,500 - 5,000}{5,000 \times 6.75}$

Now, let's calculate the simple rate of return.

2. Effective (Compound) Rate of Return

The formula for compound interest is:

$A = P \left(1 + r_{\text{eff}}\right)^t$

Where:

• $r_{\text{eff}}$ is the effective rate of return per annum (what we need to find)

Rearranging the formula to solve for $r_{\text{eff}}$:

$r_{\text{eff}} = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1$

Substitute the values:

$r_{\text{eff}} = \left(\frac{11,500}{5,000}\right)^{\frac{1}{6.75}} - 1$

Let's calculate the effective rate of return.

3. Continuous Rate of Return

The formula for continuous compounding is:

$A = Pe^{rt}$

Where:

• $r$ is the continuous rate of return per annum (what we need to find)
• $e$ is the base of the natural logarithm (approximately 2.71828)

Rearranging the formula to solve for $r$:

$r = \frac{\ln\left(\frac{A}{P}\right)}{t}$

Substitute the values:

$r = \frac{\ln\left(\frac{11,500}{5,000}\right)}{6.75}$

Now, let's calculate the continuous rate of return.

I'll compute these values now.The rates of return are as follows:

1. Simple Rate of Return (per annum): 19.3% (or 0.193)
2. Effective (Compound) Rate of Return (per annum): 13.1% (or 0.131)
3. Continuous Rate of Return (per annum): 12.3% (or 0.123)

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

Here are 5 related questions:

1. How does the simple rate of return compare to the compound rate of return in long-term investments?
2. What are the differences between simple interest and compound interest?
3. How does the continuous compounding formula differ in its application compared to annual compounding?
4. If Bertha reinvests her returns yearly, how would her future value change under compound interest?
5. How would the rates change if the investment duration was longer or shorter?

Tip: For long-term investments, compound interest generally yields higher returns than simple interest due to the compounding effect.