You borrow $5,830 to buy a car. The terms of the loan call for monthly payments for 6 years a rate ofinterest of 7 percent. What is the amount of each payment?

Question

BlackTom AI · Accepted Answer

Monthly loan payments for a fully amortizing loan use the annuity formula:
PMT = P * i / (1 − (1 + i)^−n),
where P is the loan principal, i is the periodic interest rate, and n is the total number of payments.

1) Relevant data and concepts
- Principal P = $5,830.
- Annual interest rate = 7% → monthly rate i = 0.07/12 = 0.005833333333333333 (about 0.583333% per month).
- Term = 6 years → n = 6 × 12 = 72 monthly payments.
- Use the standard amortizing payment (level-payment annuity) formula above.

2) Step-by-step calculation
Compute (1 + i)^n:
(1 + i) = 1.0058333333333333.
Raise to the 72nd power: (1.0058333333333333)^72 ≈ 1.52065 (using e^{72·ln(1.0058333)}).

Compute the discount factor piece:
(1 + i)^−n = 1 / 1.52065 ≈ 0.657735,
so 1 − (1 + i)^−n ≈ 1 − 0.657735 = 0.342265.

Compute the numerator P·i exactly:
P · i = 5,830 × (7/1200) = 5,830 × 0.005833333333333333 = 34.00833333333333.

Now plug into the formula:
PMT = 34.00833333333333 / 0.342265 ≈ 99.40 (rounded to the nearest cent).

3) Intermediate arithmetic shown
- i = 0.07/12 = 0.005833333333333333
- (1 + i)^72 ≈ 1.52065 → (1 + i)^−72 ≈ 0.657735
- Denominator 1 − (1 + i)^−72 ≈ 0.342265
- PMT ≈ 34.00833333333333 / 0.342265 ≈ 99.40

4) Why this answer is correct
The formula yields the monthly payment that exactly amortizes the loan over 72 months at the stated monthly interest rate. The computed payment of about $99.40 satisfies that condition (interest portion plus principal repayment each month such that the balance reaches zero after 72 payments).

5) Why the other choices are incorrect
- $64.61 and $65.39 are far too small; they wouldn’t even cover the monthly interest plus principal required to amortize $5,830 over 6 years.
- $93.21 and $91.49 are closer but still below the mathematically required payment; they would leave a remaining balance after 72 payments. Only ≈$99.40 meets the amortization equation.

Conclusion: The monthly payment is $99.40, which matches the provided correct answer.

You borrow $5,830 to buy a car. The terms of the loan call for monthly payments for 6 years a rate ofinterest of 7 percent. What is the amount of each payment?单项选择题

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