Mathematics Standard • Year 12 • Module 7 • Lesson 8
Reducing Balance Loans — Problem Set
Apply flat-rate and reducing-balance reasoning to realistic Australian borrowing — dealer car finance, "interest-free" furniture deals, store cards and payday lending.
Problem 1 — Dealer's "low rate" car finance
A dealer offers a $22,000 used Hyundai i30 on the following terms: 5.9% flat rate over 5 years, with equal monthly instalments.
Set up: What are we solving for?
(i) Calculate the total interest payable over the 5 years. 2 marks
(ii) Calculate the equal monthly instalment. 1 mark
(iii) Find the approximate equivalent reducing-balance rate using r_red ≈ 2·n·r_flat / (n + 1), and explain in one sentence what that tells the buyer. 2 marks
Stuck? Revisit lesson § Finding the True Rate — the 6% flat → 11.8% reducing example.Problem 2 — Two car-loan offers head-to-head
A buyer needs $20,000 for a small SUV over 5 years and is comparing two offers.
Offer A: 6% flat rate over 5 years.
Offer B: 8% p.a. compounded monthly, reducing balance, monthly repayment M = $405.53.
Set up: What are we solving for?
(i) Calculate total interest under Offer A. 1 mark
(ii) Calculate total amount paid and total interest under Offer B. 2 marks
(iii) Recommend which offer the buyer should take, naming the offer and the dollar interest saved. 2 marks
Stuck? Revisit lesson § Reducing Balance — the $20,000 example showing 8% reducing beats 6% flat.Problem 3 — "12 months interest-free" furniture deal
A furniture store advertises a $3,000 lounge on "12 months interest-free" terms, charging a $150 establishment fee and a $10/month account-keeping fee. The customer pays in 12 equal monthly instalments that cover principal + all fees.
Set up: What are we solving for?
(i) Calculate the total amount the customer will pay over 12 months, including all fees. 1 mark
(ii) Calculate the equal monthly instalment, and treat the $270 in extra payments as the "interest" charged. State the equivalent flat rate over 12 months. 2 marks
(iii) Find the approximate equivalent reducing-balance annual rate using r_red ≈ 2·n·r_flat / (n + 1) with n = 12. Comment in one sentence on whether "interest-free" is an honest description. 2 marks
Stuck? Revisit lesson § Activity 2 Q2 — "0% interest" with an establishment fee on a $5,000 purchase.Problem 4 — Payday lender on a small short-term loan
A payday lender offers $1,000 on the following terms: pay back $1,250 after 4 weeks (1 month). This is described as a "25% flat" loan.
Set up: What are we solving for?
(i) State the total interest charged in dollars. 1 mark
(ii) Using n = 1 (in months), apply r_red ≈ 2·n·r_flat / (n + 1) to find the equivalent monthly reducing-balance rate, then multiply by 12 to express it as a simple annual rate. 2 marks
(iii) In one or two sentences, explain mathematically why a typical credit card (≈ 20% p.a.) would be far cheaper if the customer can roll the debt onto it. 2 marks
Stuck? Revisit lesson § Activity 1 Q3 — the $500 payday lender at 20% over 1 month.Problem 5 — When "low" and "high" swap places
For an $8,000 loan over 3 years:
Loan P: 7% flat rate.
Loan Q: 10% p.a. compounded monthly, reducing balance, monthly repayment M = $258.14.
Set up: What are we solving for?
(i) Calculate total interest under Loan P. 1 mark
(ii) Calculate total interest under Loan Q. 1 mark
(iii) Identify which loan is actually cheaper and explain in one or two sentences why the "lower" advertised rate ends up being the more expensive option. 2 marks
Stuck? Revisit lesson § Key insight — always ask whether a quoted rate is flat or reducing.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Dealer car finance
Set up. Compute total flat interest, then split into equal monthly instalments, then approximate the equivalent reducing rate.
(i) I = 22,000 × 0.059 × 5 = $6,490.00.
(ii) Total = $22,000 + $6,490 = $28,490. Monthly = 28,490 / 60 = $474.83.
(iii) r_red ≈ 2 × 60 × 0.059 / 61 = 7.08 / 61 ≈ 0.116 = 11.6%. The "5.9% flat" is roughly equivalent to 11.6% reducing balance — nearly double the advertised number.
Problem 2 — Offer A vs Offer B
Set up. Compute total interest for each loan and compare directly.
(i) Offer A: I = 20,000 × 0.06 × 5 = $6,000.00.
(ii) Offer B: Total = 405.53 × 60 = $24,332 (to nearest dollar). Interest = 24,332 − 20,000 = $4,332.
(iii) Recommend Offer B — even though 8% sounds higher than 6%, the reducing-balance method saves $6,000 − $4,332 = $1,668 in interest over the life of the loan.
Problem 3 — "Interest-free" furniture
Set up. Add all fees to find total cost, then back-out the equivalent flat and reducing rates.
(i) Total = 3,000 + 150 + 12 × 10 = $3,270. Extra paid = $270.
(ii) Monthly = 3,270 / 12 = $272.50. Equivalent flat rate over 1 year: r_flat = 270 / (3,000 × 1) = 9% p.a. flat.
(iii) r_red ≈ 2 × 12 × 0.09 / 13 = 2.16 / 13 ≈ 0.166 = 16.6% reducing balance. "Interest-free" is misleading — the fees act exactly like a 9% flat / ~16.6% reducing-balance loan.
Problem 4 — Payday lender
Set up. Find the dollar cost, then express the rate annually.
(i) Interest charged = $1,250 − $1,000 = $250.
(ii) With n = 1: r_red ≈ 2 × 1 × 0.25 / 2 = 0.25 per month. Simple annual ≈ 0.25 × 12 = 3.00 = 300% p.a.
(iii) Even at the high end, a credit card charges ~20% p.a. so on $1,000 the monthly interest would be about $1,000 × 0.20 / 12 ≈ $16.67 — compared to $250 from the payday lender. The customer pays roughly 15× more interest for the same one-month loan.
Problem 5 — "Low" advertised rate, "high" true cost
Set up. Compute total interest for each method, then compare.
(i) Loan P (flat): I = 8,000 × 0.07 × 3 = $1,680.00.
(ii) Loan Q (reducing): Total = 258.14 × 36 = $9,293 (to nearest dollar). Interest = $1,293.
(iii) Loan Q (the "higher-rate" 10% reducing balance) is cheaper by $387. Because Loan P charges 7% on the full $8,000 for the whole 3 years, but Loan Q charges 10%/12 per month only on the steadily shrinking balance — by year 2 the balance is well under half, so most of Q's monthly interest is tiny while P's stays anchored to $8,000.