Mathematics Advanced • Year 12 • Module 7 • Lesson 17
Car Loans, Personal Loans and Credit Cards
Practise HSC-style writing on effective rates, total cost and consumer-loan comparison.
1. Short-answer questions
1.1 A $22,000 car can be bought outright for $22,000 or financed at "0% interest" for $520/month over 48 months. (a) State the total cost of the finance option. (b) Find the dollar surcharge over cash. (c) Find the effective annual interest rate, to one decimal place. 3 marks Band 3
1.2 A $4,000 credit-card debt is paid down at $120/month. The card rate is 19.99% p.a. compounded monthly. (a) Find the number of months to clear the balance using n = −ln(1 − Pr/M) / ln(1+r). (b) Find the total interest paid. 3 marks Band 3-4
1.3 A student is offered $25,000 over 5 years on (a) a car loan at 6.0% p.a. monthly, or (b) a personal loan at 8.5% p.a. monthly. (i) Find the monthly repayment on each. (ii) Find the difference in total cost. (iii) State, with one line of justification, why the car-loan rate is lower than the personal-loan rate. 4 marks Band 4
Stuck on 1.3(iii)? Think about what collateral the bank can repossess if the borrower defaults.2. Extended response
2.1 A consumer wants to fund a $30,000 purchase over 5 years. Three offers are on the table:
Product X (Car loan): 6.5% p.a. compounded monthly, 60 months.
Product Y (Personal loan): 9.5% p.a. compounded monthly, 60 months.
Product Z (Credit card): 20.0% p.a. compounded monthly, level repayments over 60 months.
(a) Use M = Pr / [1 − (1+r)⁻ⁿ] to find the monthly payment, total cost and total interest for each product. Show working.
(b) Rank the three products by total interest and quantify the cost difference between best and worst.
(c) Explain, in 2–3 sentences, why the credit-card pathway is mathematically catastrophic relative to the car loan, referencing both the rate gap and the way compounding magnifies it over 60 periods. 8 marks Band 5-6
Explicit marking criteria
Part (a) — 4 marks
• 1 mark — correct periodic rate and n for all three products (rX = 0.005417, rY = 0.007917, rZ = 0.016667; n = 60).
• 2 marks — correct M for all three (X ≈ $586, Y ≈ $630, Z ≈ $795) with substitution shown.
• 1 mark — correct total interest for all three (X ≈ $5,141, Y ≈ $7,796, Z ≈ $17,704).
Part (b) — 2 marks
• 1 mark — correct ranking X < Y < Z by total interest.
• 1 mark — correct cost difference Z − X ≈ $12,500.
Part (c) — 2 marks
• 1 mark — identifies the 3× rate ratio (20% vs 6.5%).
• 1 mark — explains that interest is charged on the outstanding balance each month, so the gap compounds 60 times rather than scaling linearly with the rate.
Your response:
Stuck on (c)? Quote a specific dollar figure (e.g. $12,500 extra interest) and connect it to the monthly compounding multiplier.How did this worksheet feel?
What I'll revisit before next class:
1.1 — "0% finance" on $22,000 (3 marks)
Sample response. (a) Total = 520 × 48 = $24,960. (b) Surcharge = 24,960 − 22,000 = $2,960. (c) Required annuity factor = 22,000 / 520 = 42.308. Trial r = 0.00432/month: factor = (1 − (1.00432)⁻⁴⁸) / 0.00432 = 0.1879 / 0.00432 ≈ 43.5 — too high; trial r = 0.0050: factor = 42.58 — closer; trial r = 0.0052: factor ≈ 42.30 ✓. Annual rate = 0.0052 × 12 ≈ 6.2% p.a.
Marking notes. 1 mark — correct total. 1 mark — correct surcharge. 1 mark — correct effective rate (accept 5.5–6.5% p.a. with valid trial-and-error working). A bald "0% means 0%" scores 0/3.
1.2 — Credit-card payoff time on $4,000 (3 marks)
Sample response. r = 0.1999/12 = 0.01666. Pr/M = 4,000 × 0.01666 / 120 = 0.5553. (a) n = −ln(1 − 0.5553) / ln(1.01666) = 0.8105 / 0.01652 ≈ 49 months. (b) Total interest = 120 × 49 − 4,000 ≈ $1,880.
Marking notes. 1 mark — explicit r and Pr/M. 1 mark — correct n (accept 47–50). 1 mark — correct total interest. Accept the lesson's rounded values 43.5 months / $1,220 if students use M = 120 × 43.5 = 5,220 with rounding-down monthly rate; either calculation paths earns full marks if internally consistent.
1.3 — Car loan vs personal loan on $25,000 (4 marks)
Sample response. (i) Car loan: r = 0.005, n = 60. (1.005)⁶⁰ = 1.34885. M = 25,000 × 0.005 / (1 − 1/1.34885) = 125 / 0.25864 = $483.32/month. Personal loan: r = 0.007083. (1.007083)⁶⁰ = 1.52762. M = 25,000 × 0.007083 / (1 − 1/1.52762) = 177.08 / 0.34538 = $512.65/month. (ii) Difference in total = (512.65 − 483.32) × 60 = $1,759.80. (iii) The car loan is secured by the vehicle — the lender can repossess if the borrower defaults — so the lender's risk (and therefore the borrower's rate) is lower than for an unsecured personal loan.
Marking notes. 1 mark — correct M for car loan. 1 mark — correct M for personal loan. 1 mark — correct cost difference. 1 mark — secured-vs-unsecured justification. A bald "personal loans are riskier" without naming collateral scores 0/1 on (iii).
2.1 — Three loan products on $30,000 over 5 years (8 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Monthly payment, total cost and total interest.
Product X. r = 0.065/12 = 0.005417, n = 60. (1.005417)⁶⁰ = 1.38390. M = 30,000 × 0.005417 / (1 − 1/1.38390) = 162.51 / 0.27744 = $585.69/month. Total = $35,141.66. Interest = $5,141.66. [r/n + M + interest marks]
Product Y. r = 0.095/12 = 0.007917. (1.007917)⁶⁰ = 1.60548. M = 30,000 × 0.007917 / (1 − 1/1.60548) = 237.50 / 0.37714 = $629.78/month. Total = $37,786.80. Interest = $7,786.80. [M + interest marks]
Product Z. r = 0.20/12 = 0.016667. (1.016667)⁶⁰ = 2.71040. M = 30,000 × 0.016667 / (1 − 1/2.71040) = 500.00 / 0.63103 = $792.51/month. Total = $47,550.52. Interest = $17,550.52. [M + interest marks]
(b) Ranking and cost difference.
By total interest: X < Y < Z. [1 mark]
Best-vs-worst gap: Z − X interest = 17,550.52 − 5,141.66 ≈ $12,409 extra paid by routing the purchase through a credit card instead of a car loan. [1 mark]
(c) Why the credit-card path is mathematically catastrophic. The credit-card rate (20%) is roughly 3.08 times the car-loan rate (6.5%), but the cost ratio (Z interest / X interest = 17,550 / 5,142 ≈ 3.41) exceeds the rate ratio because each month the unpaid balance attracts interest, and that interest itself attracts interest the following month — compounding 60 times. The same payment structure that lets the car-loan principal shrink rapidly leaves the credit-card balance bloated by accrued interest, so the gap is wider than a simple "3× the rate ⇒ 3× the cost" intuition would suggest. [2 marks]
Total: 8/8.
Band descriptors for marker.
Band 3: Two of three Ms correct, total interest shown but inconsistencies in r or n. ≈ 3–4 marks.
Band 4: All Ms correct, ranking stated, but (c) is a one-line "credit cards are bad" with no numeric reference. ≈ 5–6 marks.
Band 5: All calculations correct including total interest, ranking justified, (c) mentions either the rate ratio or compounding but not both. ≈ 6–7 marks.
Band 6: Full calculations to the cent, ranking with explicit dollar gap, (c) connects the rate ratio to the higher cost ratio via monthly compounding with a numeric illustration. 8/8.