Mathematics Advanced • Year 12 • Module 7 • Lesson 17
Car Loans, Personal Loans and Credit Cards
Apply consumer-loan mathematics to dealer-finance, credit-card and BNPL decisions.
Problem 1, Cash vs "0% finance" (the "Think First" scenario)
A car dealer offers two ways to buy the same vehicle:
Option A: Cash price $32,000, paid today.
Option B: $0 down, "0% interest", $583/month for 60 months (total payments $34,980).
Set up: What are we solving for?
(i) Find the total cost of Option B and the dollar premium over Option A. 2 marks
(ii) Use 32,000 = 583 × [1 − (1+r)⁻⁶⁰] / r to find the implied monthly rate r, and quote it as a nominal annual percentage. 3 marks
(iii) State one assumption needed to use the stated payment stream as a fixed-rate annuity model. 1 mark
Problem 2, Three ways to fund a $30,000 car
A student needs $30,000 over 5 years. Three lenders quote:
Car loan: 6.5% p.a. monthly.
Personal loan: 9.5% p.a. monthly.
Credit card: 20% p.a. monthly (level personal-loan-style payments).
Set up: What are we solving for?
(i) Using M = Pr / [1 − (1+r)⁻ⁿ], find the monthly payment for each product. 3 marks
(ii) Find the total interest paid under each product. 2 marks
(iii) Rank the products and explain in one sentence why the modeled credit-card interest is about 3.39 times the dedicated car-loan interest under these assumptions. 2 marks
Problem 3, The credit-card minimum-payment trap
A $5,000 balance accrues interest at 19.99% p.a. compounded monthly. Payments are made at the end of each month after interest is applied: $150 in part (i), then $300 in part (ii).
Set up: What are we solving for?
(i) Use n = −ln(1 − Pr/M) / ln(1+r) to find how many months it takes to clear the balance, and the total interest paid. 3 marks
(ii) Now suppose the graduate doubles the payment to $300/month. Recompute n and the total interest. 2 marks
(iii) Write a 3–4 sentence explanation for a younger sibling, using your numerical results to compare the formula estimate with the scheduled payments in this model. 3 marks
Problem 4, Buy Now Pay Later, the "no interest" claim
A shopper buys a $500 item on a BNPL platform: 4 equal fortnightly payments of $125. The platform claims "no interest". Two payments are late, triggering $10 late fees each.
Set up: What are we solving for?
(i) State the total paid and the dollar surcharge over the cash price. 1 mark
(ii) For a simple four-week comparison only, calculate the surcharge index (surcharge / principal) × (52 / 4). State why this index is not the effective annual rate of the four-installment cash flow. 2 marks
(iii) State what the $20 in late fees changes about the total cash paid. Do not infer an effective annual rate without specifying the cash-flow timing and annualization model. 2 marks
Problem 5, Take the dealer cash discount, finance separately
The same $35,000 sticker car can be bought for $32,000 cash. A bank will lend $32,000 at 6.5% p.a. monthly over 5 years.
Set up: What are we solving for?
(i) Find the monthly bank-loan payment and the total paid over 5 years. 2 marks
(ii) Compare with dealer-finance total of $35,000 (the lesson's Option B). Which path costs less in absolute dollars, and by how much? 2 marks
(iii) Write one sentence explaining the general consumer-finance rule the calculation supports. 1 mark
How did this worksheet feel?
What I'll revisit before next class:
Problem 1, Cash vs "0% finance"
Set up. We are computing the total cost of each option, the dollar premium, and the hidden monthly rate that turns the $32,000 cash price into the $583 payment.
(i) Option B total = 583 × 60 = $34,980 (≈ $35,000 sticker). Premium over cash = $2,980.
(ii) Required annuity factor = 32,000 / 583 = 54.8885. Solving [1−(1+r)⁻⁶⁰]/r = 54.8885 gives r ≈ 0.00296689/month. Nominal annual rate = 12r ≈ 3.56% p.a.; effective annual rate = (1+r)¹²−1 ≈ 3.62% p.a..
(iii) This model assumes 60 equal end-of-month payments, a fixed monthly rate, and no additional fees.
Problem 2, Three lenders for a $30,000 car
Set up. We are computing M for three rates on the same principal and horizon, then ranking by total interest.
(i) Car loan (r = 0.065/12, n = 60): M = $586.98. Personal loan (r = 0.095/12): M = $630.06. Credit card (r = 0.20/12): M = $794.82.
(ii) Using unrounded formula payments, total interest is: car $5,219.07; personal $7,803.35; credit card $17,688.99.
(iii) Ranking by cost: car < personal < credit card. Under these fixed-rate, no-fee assumptions, the modeled credit-card interest is about 3.39 times the car-loan interest. The corresponding monthly rates are 0.20/12 and 0.065/12.
Problem 3, Minimum-payment trap on $5,000
Set up. We are timing the payoff at minimum vs doubled payments and translating the numbers into advice.
(i) r = 0.1999/12 = 0.016658333… per month. Formula estimate: n ≈ 49.0467 months. In the monthly recurrence, 49 payments of $150 leave about $7.06 for month 50, giving $2,357.06 interest.
(ii) At $300/month, the formula estimate is n ≈ 19.6857 months. The monthly recurrence has 19 payments of $300 and a final payment of about $206.25 in month 20, giving $906.25 interest.
(iii) Sample explanation: "For this $5,000 balance, the formula gives a continuous estimate of 49.0467 months at $150 per month, while the month-by-month schedule needs 50 payments, with a final payment of about $7.06 and $2,357.06 interest. At $300 per month the formula estimate is 19.6857 months, and the schedule ends in month 20 with a final payment of about $206.25 and $906.25 interest. The last payment is smaller because the balance is cleared during that final period."
Problem 4, BNPL "no interest"
Set up. We are calculating a simple four-week surcharge index, which is distinct from an effective annual rate for the stated installment cash flow.
(i) Total paid = 4 × 125 + 2 × 10 = $520. Surcharge = $20 on a $500 cash price.
(ii) The simple four-week comparison index is (20/500) × (52/4) = 52%. It is not an effective annual rate for the four-installment cash flow. If, hypothetically, a 4% surcharge repeated as a compounded charge every four weeks for 13 periods, the equivalent rate would be (1.04)¹³ − 1 ≈ 66.51%; that is a different assumption from the stated fee schedule.
(iii) Two $10 late fees raise the stated total paid from $500 to $520. The fee changes the total cash outflow; an annual rate comparison would require a fully specified timing model.
Problem 5, Take the discount, finance with a bank
Set up. We are comparing the bank-financed cash-price option against the dealer's 0% pitch on the inflated sticker.
(i) r = 0.065/12, n = 60. M = 32,000r/[1−(1+r)⁻⁶⁰] = $624.74/month (unrounded M = $624.7372…). Using unrounded M, total paid = $37,484.23.
(ii) The two modeled payment totals are $34,980 and $37,484.23, a difference of $2,504.23. This calculation uses the stated payment streams and excludes other contract terms or fees.
(iii) This is a comparison of the specified payment streams; no recommendation follows without the full contract terms.