To solve these questions, let's break down each part.

Part (a) - Finding the Annual Simple Interest Rate

Given:

• The future value of the T-bill is $10,000.
• The present value (or the price you pay now) is $9,700.
• The time until maturity is 6 months, which is $\frac{1}{2}$ year.

To calculate the annual simple interest rate, we use the formula:

$\text{Interest Rate (r)} = \frac{\text{Interest}}{\text{Principal} \times \text{Time}}$

First, calculate the interest earned:

$\text{Interest} = \text{Future Value} - \text{Principal} = 10,000 - 9,700 = 300$

Now, apply the formula for the interest rate. The time is 0.5 years:

$r = \frac{300}{9,700 \times 0.5}$

Let's compute this.The annual simple interest rate for the T-bill is approximately 6.2%.

Part (b) - Finding the Actual Interest Rate (Including the Fee)

If the bank charges a fee of $40 to buy the T-bill, the total cost becomes:

$\text{Total Cost} = 9,700 + 40 = 9,740$

Now, we need to calculate the actual interest rate earned based on this new principal of $9,740. We still use the same interest of $300 and time of 0.5 years. The formula remains:

$r = \frac{300}{9,740 \times 0.5}$

Let's compute the new interest rate.The actual annual simple interest rate, considering the $40 fee, is approximately 6.2%.

Summary:

• (a) The annual simple interest rate without the fee is 6.2%.
• (b) The actual annual interest rate after accounting for the fee is 6.2%.

Would you like more details or have any questions?

Here are 5 related questions:

1. How does compound interest differ from simple interest in similar scenarios?
2. What would happen to the interest rate if the maturity period was 1 year?
3. How does the purchase of a T-bill compare to other short-term investments?
4. How would a larger fee affect the actual interest rate?
5. What are other ways to calculate the yield on T-bills?

Tip: Always consider transaction fees when evaluating the true return on investment.