Question12 Solve for r: You pay $2,000 today and receive $2,508.80 in 2 years. The implied annualrate r is:Select one alternative: 6% 12% 25% 10% ResetMaximum marks: 3 Flag question undefined单项选择题
A
6%
B
12%
C
25%
D
10%

Question

Question12 Solve for r: You pay $2,000 today and receive $2,508.80 in 2 years. The implied annualrate r is:Select one alternative: 6% 12% 25% 10% ResetMaximum marks: 3 Flag question undefined

BlackTom AI · Accepted Answer

The future value formula FV = PV × (1 + r)^n applies, so use FV = 2,508.80, PV = 2,000, n = 2 to solve for r.

1. Relevant knowledge points and concepts
- Compound interest with annual compounding: FV = PV(1 + r)^n.
- Solving for r involves isolating (1 + r) and taking the nth root (or using logarithms).
- The implied annual rate r is the rate that, compounded annually for n years, grows PV to FV.

2. Step-by-step reasoning
Start with the equation:
2,508.80 = 2,000 × (1 + r)^2

Divide both sides by 2,000 to isolate the compound factor:
(1 + r)^2 = 2,508.80 / 2,000

3. Intermediate calculations
Compute the ratio:
2,508.80 / 2,000 = 1.2544

Take the square root (since n = 2) to find 1 + r:
1 + r = sqrt(1.2544)

Compute the square root:
sqrt(1.2544) = 1.12 (because 1.12 × 1.12 = 1.2544)

Thus:
r = 1.12 − 1 = 0.12 = 12%

(Alternate method: r = (FV/PV)^(1/n) − 1 = 1.2544^(1/2) − 1 = 1.12 − 1 = 0.12.)

4. Why the correct answer is correct
The calculation shows that an annual rate of 12%, compounded annually for 2 years, exactly grows $2,000 to $2,508.80 (2000 × 1.12^2 = 2000 × 1.2544 = 2,508.80). Therefore 12% is the implied annual rate.

5. Why the other options are incorrect
- 6%: FV = 2000 × 1.06^2 = 2000 × 1.1236 = $2,247.20 (too low).
- 10%: FV = 2000 × 1.10^2 = 2000 × 1.21 = $2,420.00 (too low).
- 25%: FV = 2000 × 1.25^2 = 2000 × 1.5625 = $3,125.00 (too high).

6. Conclusion
The implied annual rate is 12%, which matches the provided correct answer.

Question12 Solve for r: You pay $2,000 today and receive $2,508.80 in 2 years. The implied annualrate r is:Select one alternative: 6% 12% 25% 10% ResetMaximum marks: 3 Flag question undefined单项选择题

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