Mathematics Standard • Year 12 • Module 7 • Lesson 8
Reducing Balance Loans — Skill Drill
Build fluency comparing flat rate and reducing balance loans: use I = P × r_flat × n, the loan repayment formula, and the approximation r_reducing ≈ 2·n·r_flat / (n + 1).
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete each definition with "flat" or "reducing".
In a ____________ rate loan, interest is charged on the original principal for the whole term. In a ____________ balance loan, interest each period is charged only on the current balance.
Q1.2 Complete the flat-rate total-interest formula and the approximate-true-rate formula.
I_flat = ____ × r_flat × ____ r_reducing ≈ ____________
Q1.3 Convert each annual nominal rate to a monthly rate r for use in the loan formula.
8% p.a. → r = ____________ 9% p.a. → r = ____________ 10% p.a. → r = ____________
2. Worked example — flat rate total cost and true rate
Follow each line of working. Every step has a reason on the right.
Problem. A $20,000 car loan is offered at 6% flat rate over 5 years. Find the total interest, total cost, monthly repayment and approximate equivalent reducing-balance rate.
Step 1 — Total flat interest = P × r_flat × n.
I = 20,000 × 0.06 × 5 = $6,000
Reason: flat rate charges interest on the whole original principal for the whole term.
Step 2 — Total cost = P + I; monthly repayment = total / months.
Total = $26,000 M = 26,000 / 60 = $433.33
Reason: equal monthly instalments share principal + interest evenly over n months.
Step 3 — Approximate equivalent reducing-balance rate.
r_red ≈ 2 × n × r_flat / (n + 1) = 2 × 60 × 0.06 / 61 = 7.2 / 61 ≈ 0.118
Reason: a flat-rate loan is mathematically closer to a reducing-balance loan at roughly double the rate.
Conclusion. Total interest $6,000; monthly $433.33; "6% flat" ≈ 11.8% reducing balance.
3. Faded example — compare a flat-rate offer with a reducing-balance offer
Dealer A offers $15,000 at 5.5% flat over 4 years. Dealer B offers $15,000 at 8% p.a. compounded monthly, reducing balance over 4 years. Use the given Dealer B repayment M = $366.19. Fill in each blank. 4 marks
Step 1 — Dealer A total interest:
I_A = 15,000 × ________ × ________ = $ ____________
Step 2 — Dealer A total cost and monthly:
Total_A = $ ____________ M_A = ____________ / 48 = $ ____________
Step 3 — Dealer B total cost and interest:
Total_B = $366.19 × 48 = $ ____________ I_B = $ ____________
Step 4 — Approximate true rate of Dealer A:
r_red ≈ 2 × 48 × 0.055 / 49 = ____________ ≈ ________ %
Conclusion. The cheaper option is Dealer ________, saving $ ____________ in total interest. Dealer A's "5.5% flat" is really about ____________ % reducing balance.
4. Graduated practice — flat rate vs reducing balance
Show your working below each part. Keep dollar amounts to 2 dp.
Foundation — single-formula substitution (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | P = $10,000, r_flat = 7%, n = 3 years. Find total flat interest. | |
| 4.2 1 | P = $18,000, r_flat = 5%, n = 4 years. Find total flat interest. | |
| 4.3 1 | Find the monthly rate r used in the reducing-balance loan formula for an 8.5% p.a. loan (monthly compounding). | |
| 4.4 1 | n = 48 months, r_flat = 5%. Find the approximate r_reducing using 2·n·r_flat/(n+1). |
Standard — typical HSC difficulty (6 questions)
Show formulas before substituting and label final answers with units.
4.5 A $25,000 car loan is offered at 6% flat rate over 5 years. Find the total interest, total cost and equal monthly repayment. 2 marks
4.6 A $25,000 car loan at 9% p.a. compounded monthly (reducing balance) over 5 years has monthly repayment $518.96. Find the total amount paid and the total interest. 2 marks
4.7 Using your answers to 4.5 and 4.6, state which deal is cheaper and by how much. 1 mark
4.8 A $18,000 loan at 5% flat rate over 4 years. Find total interest, monthly repayment, and the approximate equivalent reducing-balance rate (to 1 dp). 3 marks
4.9 Calculate the approximate reducing-balance equivalent of an 8% flat rate over 4 years (n = 48). 2 marks
4.10 A $12,000 furniture loan is offered at 4% flat rate over 2 years. Find total interest, monthly repayment, and explain in one sentence why the true rate is much higher than 4%. 3 marks
Extension — full comparison (2 questions)
4.11 $25,000 car loan. Deal X: 6% flat over 5 years. Deal Y: 9% reducing balance over 5 years (monthly M = $518.96). Which costs less total interest, by how many dollars? Also state the approximate reducing-balance equivalent of Deal X. 3 marks
4.12 A payday lender charges 20% flat over 1 month on a $500 loan. (a) State the total to repay after 1 month. (b) Calculate the approximate annualised reducing-balance equivalent using r_red ≈ 2·n·r_flat / (n + 1) with n = 1 (in months) and then multiplying by 12 to express per year. (c) Explain in one sentence why this style of loan is described as predatory. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Definitions
Flat rate: interest on original principal for whole term. Reducing balance: interest on current balance only.
Q1.2 — Formulas
I_flat = P × r_flat × n. r_reducing ≈ 2 × n × r_flat / (n + 1).
Q1.3 — Monthly rates
8% p.a. → r = 0.00667. 9% p.a. → r = 0.0075. 10% p.a. → r = 0.00833.
Q3 — Faded example (Dealer A flat vs Dealer B reducing)
I_A = 15,000 × 0.055 × 4 = $3,300.
Total_A = $18,300. M_A = 18,300 / 48 = $381.25.
Total_B = $366.19 × 48 = $17,577 (to nearest dollar). I_B = 17,577 − 15,000 = $2,577.
r_red ≈ 2 × 48 × 0.055 / 49 = 5.28 / 49 ≈ 0.1078 = 10.8%.
Conclusion: cheaper is Dealer B, saving $3,300 − $2,577 = $723. Dealer A's "5.5% flat" is really about 10.8% reducing balance.
Q4.1 — Flat interest
I = 10,000 × 0.07 × 3 = $2,100.00.
Q4.2 — Flat interest
I = 18,000 × 0.05 × 4 = $3,600.00.
Q4.3 — Monthly r
r = 0.085 / 12 ≈ 0.00708.
Q4.4 — Approximate reducing rate
r_red ≈ 2 × 48 × 0.05 / 49 = 4.8 / 49 ≈ 0.0980 = 9.8%.
Q4.5 — $25,000 flat at 6% for 5 yr
I = 25,000 × 0.06 × 5 = $7,500. Total = $32,500. M = 32,500 / 60 = $541.67. Interest = $7,500.
Q4.6 — $25,000 reducing at 9% for 5 yr
Total = 518.96 × 60 = $31,138 (to nearest dollar). Interest = 31,138 − 25,000 = $6,138.
Q4.7 — Which deal wins
Reducing 9% is cheaper by $7,500 − $6,138 = $1,362, even though 9% sounds higher than 6%.
Q4.8 — $18,000 flat at 5% for 4 yr
I = 18,000 × 0.05 × 4 = $3,600. Total = $21,600. M = 21,600 / 48 = $450.00/month. Approximate r_red ≈ 2 × 48 × 0.05 / 49 = 9.8% reducing.
Q4.9 — 8% flat over 4 years
r_red ≈ 2 × 48 × 0.08 / 49 = 7.68 / 49 ≈ 0.157 = 15.7%.
Q4.10 — $12,000 flat at 4% for 2 yr
I = 12,000 × 0.04 × 2 = $960. Total = $12,960. M = 12,960 / 24 = $540.00. Interest = $960. The true rate is much higher than 4% because customers keep paying interest on the original $12,000 even after most of the loan has been repaid — equivalent reducing rate ≈ 2 × 24 × 0.04 / 25 ≈ 7.7%.
Q4.11 — Full comparison Deal X vs Deal Y
From 4.5 and 4.6: Deal X interest = $7,500; Deal Y interest = $6,138. Deal Y is cheaper by $1,362. Equivalent reducing rate for Deal X ≈ 2 × 60 × 0.06 / 61 ≈ 11.8% — i.e. Deal X's "6% flat" is actually worse than Deal Y's 9% reducing balance.
Q4.12 — Payday loan
(a) Total repaid = $500 + $500 × 0.20 = $600 after 1 month. (b) With n = 1: r_red ≈ 2 × 1 × 0.20 / 2 = 0.20 per month, so annualised ≈ 0.20 × 12 = 240% p.a. (simple-style annualisation). (c) An annualised rate above 200% is roughly 10× a typical credit card and far above any reasonable cost of borrowing — borrowers who cannot repay quickly see balances grow much faster than they can pay them down.