To solve this problem, we'll use the formula for compound interest:

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

Where:

• $A$ is the amount of money accumulated after interest (in this case, $76,000).
• $P$ is the principal amount (in this case, $46,000).
• $r$ is the annual interest rate (which we are solving for).
• $n$ is the number of times interest is compounded per year (in this case, monthly, so $n = 12$).
• $t$ is the number of years the money is invested (in this case, 8 years).

Step 1: Plug in the known values into the compound interest formula.

$76,000 = 46,000 \left( 1 + \frac{r}{12} \right)^{12 \times 8}$

Step 2: Simplify the equation.

First, divide both sides of the equation by 46,000:

$\frac{76,000}{46,000} = \left( 1 + \frac{r}{12} \right)^{96}$

$1.65217391 = \left( 1 + \frac{r}{12} \right)^{96}$

Step 3: Take the 96th root of both sides.

To isolate $1 + \frac{r}{12}$, take the 96th root (since the interest is compounded monthly for 8 years, so $n \times t = 96$):

$\left( 1.65217391 \right)^{\frac{1}{96}} = 1 + \frac{r}{12}$

Using a calculator:

$1.65217391^{\frac{1}{96}} \approx 1.006428$

Now, subtract 1 from both sides:

$1.006428 - 1 = \frac{r}{12}$

$0.006428 = \frac{r}{12}$

Step 4: Solve for $r$.

Multiply both sides by 12:

$r = 0.006428 \times 12 = 0.077136$

Step 5: Convert $r$ to a percentage.

To get the interest rate as a percentage, multiply by 100:

$r = 0.077136 \times 100 \approx 7.71\%$

So, the interest rate required is approximately 7.71%.

---

Do you need further clarification or would you like to explore another example?

Here are some additional questions you can explore:

1. How would the final amount change if the interest were compounded quarterly instead of monthly?
2. If Abdoulaye invested $46,000 at an interest rate of 6% compounded monthly for 8 years, how much would he have after the 8 years?
3. What happens if the investment period were to be doubled to 16 years? How would the interest rate change?
4. How does increasing the number of compounding periods (e.g., daily) affect the required interest rate?
5. What other methods are there to calculate compound interest in real-life scenarios?

Tip: Remember that higher compounding frequencies (like monthly or daily) result in a slightly higher final amount, so it’s important to consider compounding frequency when planning investments!