You are borrowing money to buy a car. If you can make payments of $285 per month starting one month from now at an interest rate of 7.5% per annum, you will you be able to borrow $14,200 (to the nearest hundred dollars) for the car today if you finance the amount over five years.[Fill in the blank]判断题
A
True
B
False

Question

You are borrowing money to buy a car. If you can make payments of $285 per month starting one month from now at an interest rate of 7.5% per annum, you will you be able to borrow $14,200 (to the nearest hundred dollars) for the car today if you finance the amount over five years.[Fill in the blank]判断题

A

True

B

False

BlackTom AI · Accepted Answer

Present value of an ordinary annuity formula: PV = PMT × [1 − (1 + i)^−n] / i, because payments of $285 start one month from now (an ordinary monthly annuity).

1) Relevant knowledge points and concepts
- Convert the annual nominal rate to a monthly rate: i = 7.5%/12.
- Number of monthly payments over five years: n = 5 × 12.
- Because payments begin one month from now, use the ordinary annuity PV formula above.
- Round the final PV to the nearest hundred dollars for the question’s comparison.

2) Complete reasoning process and 3) intermediate calculations
- Monthly interest rate: i = 0.075 / 12 = 0.00625.
- Number of payments: n = 60.
- Compute the discount factor: (1 + i)^−n = (1.00625)^−60.
  - ln(1.00625) ≈ 0.00623055, so −60·ln(1.00625) ≈ −0.373833.
  - Thus (1.00625)^−60 ≈ e^(−0.373833) ≈ 0.6882.
- Annuity factor: [1 − (1.00625)^−60] / 0.00625 = (1 − 0.6882) / 0.00625 = 0.3118 / 0.00625 ≈ 49.888.
- Present value: PV = 285 × 49.888 ≈ 14,218.1.

4) Why the correct answer is correct
- The present value of the 60 monthly payments of $285 at 7.5% p.a. (compounded monthly) is approximately $14,218. Rounding to the nearest hundred dollars gives $14,200, which matches the stated borrowing amount.

5) Why the other option is incorrect
- “False” would imply the PV is not about $14,200. Since the calculated PV rounds to $14,200, the “False” choice contradicts the arithmetic and the correct annuity valuation.

6) Conclusion
- The statement is True; the present value of the payments is about $14,218, which to the nearest hundred dollars is $14,200, confirming the provided answer True.

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