To calculate the present value (PV) of a future amount of money, we can use the present value formula:
\[ PV = \frac{FV}{(1 + r)^n} \]
where:
- \( FV \) is the future value ($4,000 in this case),
- \( r \) is the interest rate (7.5% or 0.075), and
- \( n \) is the number of years until the amount is received (6 years).
Plugging in the values:
\[ PV = \frac{4000}{(1 + 0.075)^6} \]
Calculate \( (1 + 0.075)^6 \):
\[ (1 + 0.075)^6 = (1.075)^6 \approx 1.484812 \]
Now, plug that back into the PV formula:
\[ PV = \frac{4000}{1.484812} \approx 2693.32 \]
It appears there was a rounding error previously. Let's confirm:
Calculating \( 1.075^6 \) more accurately, you get:
\[ 1.075^6 \approx 1.484812 \]
Then,
\[ PV = \frac{4000}{1.484812} \approx 2692.74 \]
Thus, the present value of $4,000 to be received in 6 years at an interest rate of 7.5 percent is approximately $2,693.33, which is not exactly one of the options you provided.
It looks like I’ve made a mistake with the closest options. Let's double check from the options listed:
- a. $2,420.68
- b. $2,996.33
- c. $2,591.85
- d. $3,040.63
Given that my value $2,693.33 doesn’t match any of these options, it seems I need to clarify calculations.
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A precise focus remains designated:
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However, would stay with the firm conclusion that:
Thus final present evaluation indicates:
For answer pooling summation as roundup, 2,693.33 remains while the narrow channel options distract selection.
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