To calculate the present value (PV) of an amount to be received in the future, we use the formula for present value with compound interest:

$PV = \frac{FV}{(1 + \frac{r}{n})^{nt}}$

Where:

• $PV$ = Present Value (what we are calculating)
• $FV$ = Future Value ($4000)
• $r$ = Annual interest rate (8% or 0.08)
• $n$ = Number of times the interest is compounded per year (quarterly compounding means $n = 4$)
• $t$ = Time in years (9 years)

Step-by-Step Calculation:

1. Future Value (FV): $4000
2. Interest rate (r): 8% or 0.08
3. Compounding frequency (n): Quarterly, so $n = 4$
4. Time (t): 9 years

Using the formula:

$PV = \frac{4000}{(1 + \frac{0.08}{4})^{4 \times 9}} = \frac{4000}{(1 + 0.02)^{36}} = \frac{4000}{1.02^{36}}$

Now let's calculate $1.02^{36}$ and then the present value.

$PV = \frac{4000}{1.9826} \approx 2017.25$

None of the given choices match exactly, but let me double-check the options and clarify further.

The present value (PV) of $4000 to be received in 9 years, with an 8% annual interest rate compounded quarterly, is approximately $1960.89.

Thus, the closest answer is:

c. $1961

Would you like further details or explanations on any part of this process?

Here are 5 related questions to expand this topic:

1. How does the compounding frequency affect the present value?
2. What would the present value be if the interest was compounded annually instead of quarterly?
3. How do you calculate future value when given present value and interest rate?
4. What is the effect of increasing the interest rate on the present value?
5. How does the time period (years) influence the present value?

Tip: The more frequently interest is compounded, the lower the present value for a given future amount.