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 — sticker price $35,000.
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) Explain in one sentence why a buyer without $32,000 in savings might still rationally take Option B. 1 mark
Stuck? Revisit lesson § Think First and § Car Loans and Dealer Finance.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 credit-card path costs roughly three times the dedicated car loan. 2 marks
Problem 3 — The credit-card minimum-payment trap
A graduate has a $5,000 credit-card balance at 19.99% p.a. compounded monthly. The bank's minimum payment is $150/month.
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 one-paragraph warning (3–4 sentences) addressed to a younger sibling about minimum-payment culture, using your numerical results. 3 marks
Stuck? Revisit lesson § Credit Cards and the Minimum Payment Trap.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) The shopper effectively borrowed $500 for an average of 4 weeks. Convert the $20 surcharge into an annualised effective rate using rann ≈ (surcharge / principal) × (52 / 4). 2 marks
(iii) Compare this annualised rate with the 19.99% credit-card rate in Problem 3, and write one sentence on why BNPL's "no interest" label is misleading. 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
Stuck? Use the same M = Pr / [1 − (1+r)⁻ⁿ] formula from the lesson.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 = 54.89. Trial r = 0.00365/month gives PV factor 54.89 ✓. Nominal annual rate = 0.00365 × 12 = 4.38% p.a. (matches lesson).
(iii) A buyer without $32,000 saved still needs to drive — Option B is rational provided they cannot borrow elsewhere at less than 4.38% p.a. and the car is genuinely needed (not deferred to save up).
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 = 0.00542, n = 60): M = $585.69. Personal loan (r = 0.00792): M = $629.93. Credit card (r = 0.01667): M = $795.07.
(ii) Total interest: car $5,141; personal $7,796; credit card $17,704.
(iii) Ranking by cost: car < personal < credit card. The credit-card path costs roughly 3× the car-loan path because exponential interest at 20% on a 5-year horizon means each month adds about 1.7% to the outstanding balance — more than three times the car loan's 0.54%/month.
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 = 0.01666. Pr/M = 5,000 × 0.01666 / 150 = 0.5553. n = −ln(1 − 0.5553) / ln(1.01666) = 0.8105 / 0.01652 ≈ 49 months. Total interest = 150 × 49 − 5,000 ≈ $2,350.
(ii) At $300/month: Pr/M = 0.2776. n = −ln(0.7224) / 0.01652 = 0.3252 / 0.01652 ≈ 20 months. Total interest = 300 × 20 − 5,000 = $1,000.
(iii) Sample paragraph: "If you only pay the minimum $150 on a $5,000 credit-card balance, you will be in debt for over four years and pay almost $2,400 in interest — nearly half the original purchase. Doubling the payment to $300 clears the card in under two years and cuts interest by $1,350. Banks set minimum payments to maximise their interest income; pay the full balance every month, or as much above the minimum as you can manage."
Problem 4 — BNPL "no interest"
Set up. We are converting a fixed-dollar late fee into an annualised effective rate so we can compare it to a credit card.
(i) Total paid = 4 × 125 + 2 × 10 = $520. Surcharge = $20 on a $500 cash price.
(ii) rann ≈ (20 / 500) × (52 / 4) = 0.04 × 13 = 52% p.a. effective (using a simple-interest annualisation — the true compounded rate is even higher).
(iii) 52% p.a. is more than 2.5 times the 19.99% credit-card rate — BNPL's "no interest" claim is only true if every payment is on time; one missed instalment turns the product into one of the most expensive forms of short-term credit available.
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 = 0.0054167, n = 60. (1.0054167)⁶⁰ = 1.38390. M = 32,000 × 0.0054167 / (1 − 1/1.38390) = 173.33 / 0.27744 = $624.74/month. Total = $37,484.
(ii) Dealer-finance total = $34,980. Bank-loan total = $37,484. The dealer's "0%" pitch is actually $2,504 cheaper than the 6.5% bank loan on the cash price — the dealer's hidden rate (4.38%) is below the bank's 6.5%.
(iii) The general rule: never compare a 0% offer to a "no loan"; compare it to the rate at which you could actually borrow the cash price, and let the lower effective rate win.