To solve this, we use the compound interest formula:

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

Where:

• $A = 157,629.45$ (the matured amount)
• $P = 99,200.00$ (the initial investment)
• $t = 5$ years
• $n = 2$ (since it's compounded semi-annually)
• $r$ is the annual interest rate we're solving for.

Steps:

1. Plug in the known values into the formula:

$157,629.45 = 99,200 \left(1 + \frac{r}{2}\right)^{2 \times 5}$

2. Solve for $r$:

$\frac{157,629.45}{99,200} = \left(1 + \frac{r}{2}\right)^{10}$

$1.5881 = \left(1 + \frac{r}{2}\right)^{10}$

Take the 10th root of both sides:

$1 + \frac{r}{2} = 1.0474$

$\frac{r}{2} = 0.0474$

$r = 0.0948 \quad \text{or} \quad 9.48\%$

Thus, the periodic interest rate is:

$\frac{9.48\%}{2} = 4.74\%$

The correct answer is 4.74%.

Would you like further details or have any questions?

Here are five related questions for further practice:

1. How would the interest rate change if compounded quarterly instead?
2. What is the future value if the investment was compounded annually?
3. How long would it take for the investment to double with this interest rate?
4. What is the difference between compound interest and simple interest for this scenario?
5. How much would an investment of $50,000 grow to in 7 years under the same conditions?

Tip: For compound interest problems, the more frequent the compounding periods, the higher the effective return on investment.