This question involves calculating the present value (PV) based on the future value (FV), the interest rate (r), and the compounding frequency. The formula for present value varies depending on the type of compounding.

Here are the relevant formulas for different compounding methods:

1. Yearly Compounding (Case A): $PV = \frac{FV}{(1 + r)^t}$ Where:

   • $FV = 8000$
   • $r = 0.05$ (5% interest)
   • $t = 1$ (1 year)

2. Quarterly Compounding (Case B): $PV = \frac{FV}{\left(1 + \frac{r}{n}\right)^{n \cdot t}}$ Where:

   • $n = 4$ (since it’s compounded quarterly)
   • $r = 0.05$
   • $t = 1$

3. Continuous Compounding (Case C): $PV = FV \cdot e^{-r \cdot t}$ Where:

   • $e$ is the base of the natural logarithm (approximately 2.71828)
   • $r = 0.05$
   • $t = 1$

Now, I'll calculate the present value for each case:

Case A: Yearly Compounding

$PV_A = \frac{8000}{(1 + 0.05)^1} = \frac{8000}{1.05} = 7619.05$

Case B: Quarterly Compounding

$PV_B = \frac{8000}{\left(1 + \frac{0.05}{4}\right)^{4 \cdot 1}} = \frac{8000}{\left(1 + 0.0125\right)^4} = \frac{8000}{(1.0125)^4} = \frac{8000}{1.050945} \approx 7612.19$

Case C: Continuous Compounding

$PV_C = 8000 \cdot e^{-0.05 \cdot 1} = 8000 \cdot e^{-0.05} = 8000 \cdot 0.951229 \approx 7609.84$

Conclusion:

• $A \approx 7619.04$
• $B \approx 7612.19$
• $C \approx 7609.84$

Thus, the correct answer is (d).

Do you have any questions or need more details?

Here are some related questions to expand your understanding:

1. How does compounding frequency affect the present value?
2. What happens to the present value if the interest rate increases?
3. How would you calculate the future value if given the present value?
4. What would the present value be for a different time period, say 2 years?
5. How is continuous compounding different from yearly or quarterly compounding?

Tip: The more frequently interest is compounded, the smaller the present value will be for the same future value and interest rate.