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) Assume interest is applied monthly before each end-of-month payment. Find the continuous estimate using n = −ln(1 − Pr/M) / ln(1+r), then use a monthly balance recurrence to find scheduled payments, the final partial payment and total interest. 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
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 modeled credit-card interest exceeds the car-loan interest under the stated assumptions, 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 ≈ $587, Y ≈ $630, Z ≈ $795) with substitution shown.
• 1 mark correct total interest for all three (using unrounded formula payments: X ≈ $5,219.07, Y ≈ $7,803.35, Z ≈ $17,688.99).
Part (b), 2 marks
• 1 mark correct ranking X < Y < Z by total interest.
• 1 mark correct cost difference Z − X ≈ $12,469.92.
Part (c), 2 marks
• 1 mark identifies the approximately 3.08× 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:
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 = 42.307692. Solving [1−(1+r)⁻⁴⁸]/r = 42.307692 gives r ≈ 0.005274526/month. Effective annual rate = (1+r)¹²−1 ≈ 6.5% p.a. to one decimal place.
Marking notes. 1 mark, correct total. 1 mark, correct surcharge. 1 mark, correct effective annual rate ≈ 6.5% (accept 6.4–6.6% with valid 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 = 0.016658333… per month. (a) The formula gives n = −ln(1 − 4,000r/120)/ln(1+r) ≈ 49.0467 months, a continuous estimate. (b) Under the monthly recurrence, 49 end-of-month payments of $120 leave a final payment of about $5.65 in month 50; total scheduled interest is $1,885.65.
Marking notes. 1 mark, explicit r and Pr/M. 1 mark, correct continuous estimate (about 49.0467 months) or correctly described 50-payment schedule. 1 mark, correct interest with the method stated: about $1,885.60 from the continuous estimate M n − P, or $1,885.65 from the monthly recurrence with a $5.65 final payment. Do not accept a materially different payoff estimate.
1.3, Car loan vs personal loan on $25,000 (4 marks)
Sample response. (i) Car loan: r = 0.06/12; M = 25,000r/[1−(1+r)⁻⁶⁰] = $483.32/month (unrounded M = $483.320038…). Personal loan: r = 0.085/12; M = $512.91/month (unrounded M = $512.913283…). Using unrounded payments, total-cost difference = 60(512.913283…−483.320038…) = $1,775.59. (iii) The car loan is secured by the vehicle, while the personal loan is unsecured; collateral can reduce lender risk and may be reflected in the quoted rates.
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, n = 60. M = 30,000r/[1−(1+r)⁻⁶⁰] = $586.98/month (unrounded M = $586.984447…). Using unrounded M, total = $35,219.07 and interest = $5,219.07. [r/n + M + interest marks]
Product Y. r = 0.095/12, n = 60. M = 30,000r/[1−(1+r)⁻⁶⁰] = $630.06/month (unrounded M = $630.055839…). Using unrounded M, total = $37,803.35 and interest = $7,803.35. [M + interest marks]
Product Z. r = 0.20/12, n = 60. M = 30,000r/[1−(1+r)⁻⁶⁰] = $794.82/month (unrounded M = $794.816511…). Using unrounded M, total = $47,688.99 and interest = $17,688.99. [M + interest marks]
(b) Ranking and cost difference.
By total interest: X < Y < Z. [1 mark]
The difference in modeled interest is Z − X = 17,688.99 − 5,219.07 = $12,469.92. [1 mark]
(c) Comparing the modeled interest amounts. The nominal annual-rate ratio is 20/6.5 ≈ 3.08, while the modeled interest ratio is 17,688.99/5,219.07 ≈ 3.39. Under the stated level-payment model, interest is charged monthly on the remaining balance; the resulting interest difference is $12,469.92 over 60 payments. [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.