You have been told that the maturity value of a Treasury Bill that matures in 3 months is $10,000. The simple interest rate offered by the Treasury Bill is 10% per annum. Calculate the principal of the investment.
你被告知，3个月到期的国库券的到期价值是1万美元。财政部法案提供的简单利率为每年10%。计算投资本金。

Question

BlackTom AI · Accepted Answer

Simple interest formula: M = P(1 + r t), so P = M / (1 + r t).

1) Relevant knowledge points and concepts
- Simple interest: interest computed only on the original principal. For time measured in years, the maturity (future) value M equals principal P multiplied by (1 + r t), where r is annual rate (decimal) and t is time in years.
- Given: Maturity value M = $10,000; annual simple interest rate r = 10% = 0.10; time t = 3 months = 3/12 = 0.25 years. We solve for P.

2) Complete reasoning process step by step
- Start from P = M / (1 + r t).
- Compute r t, then 1 + r t, then divide M by that factor to get P.

3) Calculations with intermediate steps
- r = 0.10
- t = 0.25
- r t = 0.10 × 0.25 = 0.025
- 1 + r t = 1 + 0.025 = 1.025
- P = 10,000 / 1.025
- 10,000 ÷ 1.025 = 9,756.09756097561...
- Round to cents: P ≈ $9,756.10

4) Why the correct answer is correct
- The principal must be smaller than the maturity value because interest accrues. Dividing the maturity by the growth factor (1 + r t) correctly backs out the original amount invested under simple interest. Using the given 10% annual rate for 0.25 years yields a 2.5% total interest, so the principal is $10,000 / 1.025 = $9,756.10.

5) Why the other options are incorrect
- $9,750.00: slightly lower than the correct value; this would understate the principal (off by about $6.10) and does not match exact division by 1.025.
- $2,500.00: far too small; this would imply a 300% growth to $10,000, which is inconsistent with a 10% annual simple rate over 3 months.
- $9,764.54: slightly higher than the correct result; it likely comes from a miscalculation or different rounding method and does not equal 10,000 / 1.025.

6) Conclusion
The principal of the investment is $9,756.10, which matches the provided correct answer.

类似问题

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