You bought a new car today which cost you $20,000. You financed the entire cost with a 5-year loan at 4.00%. If you make payments at the end of each month starting a month from now, how much is your monthly payment?

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BlackTom AI · Accepted Answer

Restating the scenario: You bought a car for 20,000 and financed it with a 5-year loan (60 months) at an annual rate of 4.00%, with payments due at the end of each month starting one month from now. The question asks for the monthly payment amount.

First, identify the formula for calculating the fixed monthly payment on an amortizing loan: PMT = P · r / (1 − (1 + r)^(−n)), where
- P is the loan principal (20,000),
- r is the monthly interest rate, and
- n is the number of payments (60).

Compute the monthly interest rate: r = annual rate / 12 = 0.04 / 12 = 0.0033333333 (approximately 0.3333% per month).

Next, plug the values into the formula:
- Numerator: P · r = 20,000 × 0.0033333333 ≈ 66.6666667.
- Denominator: 1 − (1 + r)^(−n) = 1 − (1.0033333333)^(−60).

Estimate or compute (1.0033333333)^(−60). This is equivalent to 1 / (1.0033333333)^(60). The factor (1.0033333333)^(60) is roughly around 1.221, so its reciprocal is about 0.818. Therefore, 1 − (1.0033333333)^(−60) is approximately 1 − 0.818 ≈ 0.182.

Now divide the numerator by the denominator to obtain the monthly payment: 66.6666667 ÷ 0.182 ≈ 366–368 range depending on the precise power calculation. With more precise computation, the monthly payment evaluates to about $368.33.

Why this result makes sense: The payment amount balances paying interest each month and reducing the principal over 60 months, ensuring the loan is paid off exactly at the end of the term. The amount is sensitive to the interest rate and the number of payments, but at 4% APR over 60 months, a payment in the high $360s is typical for a $20,000 loan.

Why the other potential numbers (if present) would be incorrect: If a number were notably lower, it would imply either a much shorter loan term, a lower interest rate, or a miscalculation of the amortization factor. If a number were much higher, it would imply a longer term or higher rate. The precise PMT formula accounts for both interest accrual and principal repayment over the full term, so it yields the exact amount shown here. In this case, the provided option $368.33 aligns with the standard amortization calculation for P = 20,000, r = 0.04/12, n = 60.

MSB-250-300-002 Topic 5 Quiz

You bought a new car today which cost you $20,000. You financed the entire cost with a 5-year loan at 4.00%. If you make payments at the end of each month starting a month from now, how much is your monthly payment?

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