To solve this problem, we need to find three different rates of return based on Bertha's investment: the simple rate of return per annum, the compound (effective) rate of return per annum, and the continuous per annum rate of return. Let's break down the problem step by step:

Given Data:

• Initial investment $P = 5,000$
• Final value $A = 11,500$
• Time period $t = 6 \text{ years and } 9 \text{ months}$

Convert 9 months into years: $9 \text{ months} = \frac{9}{12} = 0.75 \text{ years}$ So, the total time $t = 6.75 \text{ years}$.

1. Simple Rate of Return per Annum

The formula for simple interest is: $A = P(1 + rt)$ Where:

• $A$ is the final amount,
• $P$ is the principal amount,
• $r$ is the annual interest rate,
• $t$ is the time in years.

Rearrange the formula to solve for $r$: $r = \frac{A - P}{P \times t}$ Substitute the values: $r = \frac{11,500 - 5,000}{5,000 \times 6.75}$

Let's calculate $r$.

2. Effective (Compound) Rate of Return Per Annum

The formula for compound interest is: $A = P \left(1 + \frac{r}{n}\right)^{nt}$ For annual compounding, $n = 1$, so the formula simplifies to: $A = P(1 + r)^t$ Rearrange the formula to solve for $r$: $r = \left(\frac{A}{P}\right)^{\frac{1}{t}} - 1$ Substitute the values: $r = \left(\frac{11,500}{5,000}\right)^{\frac{1}{6.75}} - 1$

Let's calculate $r$.

3. Continuous Rate of Return Per Annum

The formula for continuous compounding is: $A = P e^{rt}$ Rearrange the formula to solve for $r$: $r = \frac{\ln\left(\frac{A}{P}\right)}{t}$ Substitute the values: $r = \frac{\ln\left(\frac{11,500}{5,000}\right)}{6.75}$

Let's calculate $r$.

I'll calculate all three rates now.The calculated 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 how these were calculated, or do you have any questions?

Here are five related questions to deepen your understanding:

1. How does the simple interest formula differ from compound interest?
2. What is the significance of the continuous rate of return compared to the compound rate?
3. How would the rates of return change if the time period was longer or shorter?
4. Can you compare the simple and compound rates over a different compounding frequency (e.g., semi-annually)?
5. What assumptions do these calculations make about the consistency of the returns?

Tip: When comparing investments, always consider how interest is compounded, as it significantly affects the overall return.