Todd can afford to pay $440 per month for the next 5 years in order to purchase a new car. The interest rate is 7.8 percent compounded monthly. What is the most he can afford to pay for a new car today?

Question

BlackTom AI · Accepted Answer

Present value of an ordinary annuity (monthly payments at end of each month) is used: PV = PMT * [1 - (1 + r)^(-n)] / r.

1) Relevant knowledge points and concepts
- PMT (payment) = $440 per month.
- Annual nominal interest rate = 7.8% compounded monthly → monthly rate r = 0.078/12 = 0.0065 (0.65% per month).
- Number of payments n = 5 years × 12 months = 60.
- Use the present-value-of-an-annuity formula because Todd makes a uniform series of monthly payments and wants the equivalent lump-sum today.

2) Complete reasoning process step by step
- Compute the monthly interest rate and number of periods.
- Compute the discount factor (1 + r)^(-n).
- Compute the annuity present-value factor [1 - (1 + r)^(-n)]/r.
- Multiply that factor by the monthly payment 440 to get the maximum present value (most he can afford today).

3) Calculations with intermediate steps
- r = 0.078/12 = 0.0065
- n = 5 × 12 = 60
- (1 + r)^(-n) = (1.0065)^(-60) ≈ 0.67791581
- Numerator: 1 - (1.0065)^(-60) ≈ 1 - 0.67791581 = 0.32208419
- Annuity factor: 0.32208419 / 0.0065 ≈ 49.55156818
- PV = 440 × 49.55156818 ≈ 21,802.89

4) Why the correct answer is correct
The formula directly converts the known monthly payment stream into its present value using the exact monthly interest rate and number of months. The computed annuity factor (≈49.551568) accurately discounts each of the 60 monthly payments at the monthly rate 0.65%, and multiplying by $440 gives $21,802.89 — the largest lump-sum purchase price affordable given those payments and interest terms.

5) Why the other options are incorrect
- $20,712.74 and $21,504.61 are slightly lower; they result from either incorrect rounding, wrong discount factor, or using a different rate/term (e.g., mistaking annual for monthly compounding).
- $32,161.83 is much larger and would correspond to a drastically lower discount rate or treating the payments as lasting far longer (or computing a future value rather than present value). None of these match the correct monthly-compounded PV calculation.

6) Conclusion and confirmation
Using the present-value-of-an-annuity formula with r = 0.0065 and n = 60 gives PV ≈ $21,802.89, which matches the provided correct answer.

Todd can afford to pay $440 per month for the next 5 years in order to purchase a new car. The interest rate is 7.8 percent compounded monthly. What is the most he can afford to pay for a new car today?单项选择题

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