Mathematics Standard • Year 12 • Module 7 • Lesson 8
Reducing Balance Loans — Past-Paper Style
Practise HSC Mathematics Standard 2-style writing on flat-rate vs reducing-balance loans — three short-answer questions and one longer scenario with marking criteria.
1. Short-answer questions
1.1 Calculate the total flat-rate interest on a $15,000 car loan at 6.4% flat rate over 4 years, and state the equal monthly instalment. 2 marks Band 3
1.2 A $18,000 car loan is offered at 5% flat rate over 4 years.
(a) Find the total interest, total cost and monthly repayment.
(b) Find the approximate equivalent reducing-balance rate (to 1 dp). 3 marks Band 3-4
1.3 A furniture store advertises "12 months interest-free" with a $150 establishment fee and a $10/month account-keeping fee on a $3,000 purchase, paid in 12 equal monthly instalments.
(a) Find the total amount paid and the monthly instalment.
(b) Find the equivalent flat rate p.a. for the extra paid above $3,000.
(c) Briefly explain in one sentence why describing the deal as "interest-free" is misleading. 4 marks Band 4
2. Extended response
2.1 Vincent is buying a $20,000 used car and comparing three offers from different dealers, each over a 5-year term.
Offer A: 6% flat rate over 5 years.
Offer B: 8% p.a. compounded monthly, reducing balance, monthly repayment $405.53.
Offer C: 9% p.a. compounded monthly, reducing balance, monthly repayment $415.17.
(a) Calculate the total interest under Offer A.
(b) Calculate the total interest under Offer B.
(c) Calculate the total interest under Offer C.
(d) Find the approximate equivalent reducing-balance rate for Offer A using r_red ≈ 2·n·r_flat / (n+1).
(e) Recommend the best offer for Vincent, naming the offer and the dollar interest saved compared to Offer A, and explain in one sentence why "lowest advertised rate" was not the cheapest deal. 7 marks Band 5-6
Explicit marking criteria
Part (a) — 1 mark
• 1 mark — correct flat interest using I = P × r_flat × n.
Part (b) — 1 mark
• 1 mark — correct total interest for Offer B = M × n − PV.
Part (c) — 1 mark
• 1 mark — correct total interest for Offer C.
Part (d) — 1 mark
• 1 mark — correct approximate reducing-balance equivalent for Offer A.
Part (e) — 3 marks
• 1 mark — comparison of the three interest totals on the same "total interest over 5 years" basis.
• 1 mark — recommendation sentence naming the chosen offer AND the dollar interest saved compared to Offer A.
• 1 mark — explanation links the flat-rate vs reducing-balance method (e.g. "6% flat is roughly 11.6% reducing, so Offer A is actually more expensive than Offer B's 8% reducing").
Your response:
Stuck on (e)? After comparing the three interest totals, the recommendation must name the offer, quote the dollar saving vs Offer A, AND attribute the result to the flat-vs-reducing method difference.How did this worksheet feel?
What I'll revisit before next class:
1.1 — $15,000 flat at 6.4% over 4 years (2 marks)
Sample response. I = 15,000 × 0.064 × 4 = $3,840.00. Total = $18,840. M = 18,840 / 48 = $392.50/month.
Marking notes. 1 mark — correct flat interest. 1 mark — correct monthly instalment. A common slip is using r = 6.4 (not 0.064), which gives an absurd interest figure and scores 0/2.
1.2 — $18,000 flat at 5% over 4 years (3 marks)
(a) Sample response. I = 18,000 × 0.05 × 4 = $3,600. Total = $21,600. M = 21,600 / 48 = $450.00/month.
(b) Sample response. r_red ≈ 2 × 48 × 0.05 / 49 = 4.8 / 49 ≈ 9.8% reducing.
Marking notes. (a) 1 mark — correct interest. 1 mark — correct M (must divide total by 48, not by 4). (b) 1 mark — correct r_red to 1 dp.
1.3 — "Interest-free" furniture (4 marks)
(a) Sample response. Total = 3,000 + 150 + 12 × 10 = $3,270. Monthly = 3,270 / 12 = $272.50.
(b) Sample response. Extra paid = $270 over 1 year on P = $3,000. r_flat = 270 / (3,000 × 1) = 0.09 = 9% p.a. flat.
(c) Sample response. Although no line item is called "interest", the customer pays $270 above the price tag in fees over 12 months — that is mathematically the same as a 9% flat rate loan, not genuinely interest-free.
Marking notes. (a) 1 mark — correct total. 1 mark — correct monthly. (b) 1 mark — correct equivalent flat rate. (c) 1 mark — explanation links fees to an equivalent interest charge.
2.1 — Vincent's three car-loan offers (7 marks): sample Band-6 response with annotations
Sample Band-6 response.
(a) Offer A interest.
I_A = 20,000 × 0.06 × 5 = $6,000.00. [1 mark — correct flat interest.]
(b) Offer B interest.
Total = 405.53 × 60 = $24,332. Interest = 24,332 − 20,000 = $4,332. [1 mark — correct interest = M × n − PV.]
(c) Offer C interest.
Total = 415.17 × 60 = $24,910. Interest = 24,910 − 20,000 = $4,910. [1 mark — correct interest.]
(d) Approximate true rate of Offer A.
r_red ≈ 2 × 60 × 0.06 / 61 = 7.2 / 61 ≈ 11.8% reducing balance. [1 mark — correct approximation.]
(e) Comparison and recommendation.
Total interest: A = $6,000, B = $4,332, C = $4,910. Offer B is the lowest. [1 mark — comparison on the same total-interest basis.]
Conclusion: Vincent should take Offer B, saving $6,000 − $4,332 = $1,668 in interest compared to Offer A over 5 years. [1 mark — recommendation names the offer and quotes the dollar saving.] Even though Offer A has the lowest advertised rate (6%), it is a flat rate — equivalent to about 11.8% reducing balance — which is much higher than B's 8% reducing rate, so the "low rate" labelling is misleading. [1 mark — links the flat-rate vs reducing-balance method to the result.]
Total: 7/7.
Band descriptors for marker.
Band 3: Correct flat interest for A but treats Offers B/C as flat rather than total = M × n. ≈ 2 marks.
Band 4: Three interest totals correct, true-rate approximation missing or recommendation has no dollar gap. ≈ 4 marks.
Band 5: All numerical parts correct and recommendation names Offer B with the dollar saving, but no flat-vs-reducing explanation. ≈ 6 marks.
Band 6: Complete: three correct totals, true-rate approximation, recommendation with dollar saving AND an explanation that links to the flat-vs-reducing method. 7/7.