Reducing Balance Loans
A car dealer offers two loans: Option A at 8% reducing balance, Option B at 6% flat rate. Most people choose Option B because 6% sounds better than 8%. They are making a costly mistake. A flat rate loan charges interest on the original principal for the entire term — even as you pay it down. The true rate of a 6% flat rate loan is often 10–12%, nearly double the advertised rate.
Practise this lesson
Three printable worksheets that build from foundations to mastery — or build your own from any module’s questions.
A $20,000 car loan: Option A is 8% reducing balance over 5 years. Option B is 6% flat rate over 5 years. Which costs less total interest? Predict before calculating.
Before reading on — write your gut feeling. We will revisit this at the end of the lesson.
Two fundamentally different loan types — understanding the difference can save you thousands of dollars.
Flat rate interest: $\text{Total Interest} = P \times r_{\text{flat}} \times n$
Reducing balance: Interest charged on current balance only. Use $M = PV \times \dfrac{r}{1-(1+r)^{-n}}$
True rate conversion: $r_{\text{reducing}} \approx \dfrac{2 \times n \times r_{\text{flat}}}{n + 1}$
Rule of thumb: True reducing rate $\approx 2 \times$ flat rate
Key facts
- Reducing balance vs flat rate
- True rate approximation formula
- Where flat rates appear in practice
Concepts
- Why flat rates are deceptive
- How to compare loan types fairly
- The mathematics behind each method
Skills
- Compare flat and reducing balance loans
- Estimate the true interest rate of a flat rate loan
- Make informed borrowing decisions
A flat rate loan calculates interest on the original principal for the entire loan term, regardless of repayments made.
$$\text{Total Interest} = P \times r_{\text{flat}} \times n$$Example: $20,000 car loan at 6% flat rate over 5 years.
Total interest $= 20000 \times 0.06 \times 5 = \$6{,}000$. Total repayment $= \$26{,}000$. Monthly $= 26000 \div 60 = \$433.33$.
Flat rates are common in car finance, personal loans, and consumer credit. They are banned or heavily regulated in many countries because they mislead borrowers.
What to write in your book
- Flat rate: Total Interest = P × r_flat × n (interest on original principal for full term).
- Monthly repayment for flat rate = (P + Total Interest) ÷ n.
- Flat rates appear in car loans, personal loans, and store credit. Always ask if a rate is flat or reducing.
Quick check: A $20,000 loan at 6% flat rate over 5 years. The total interest charged is:
A reducing balance loan charges interest only on the outstanding balance each period. As you repay the principal, your interest bill falls.
Same example at 8% reducing balance (compounded monthly) over 5 years:
$r = 0.08 \div 12 = 0.00\overline{6}$, $n = 60$.
$$M = 20000 \times \frac{0.00\overline{6}}{1 - (1.00\overline{6})^{-60}} = \$405.53 \text{ per month}$$Total repaid $= 405.53 \times 60 = \$24{,}332$. Total interest $= \$4{,}332$.
What to write in your book
- Reducing balance: interest each period = Balance × r (periodic rate).
- Use the repayment formula: M = PV × r / [1 − (1+r)^−n].
- Total interest = M × n − PV.
- A higher reducing balance rate can still be cheaper than a lower flat rate.
True or false: An 8% reducing balance loan can cost less total interest than a 6% flat rate loan on the same principal and term.
Worked examples · reveal each step
$15,000 car loan. Dealer A: 5.5% flat rate over 4 years. Dealer B: 8% reducing balance over 4 years (compounded monthly). Which is cheaper? Find the true rate of Dealer A.
A $1000 loan at 25% flat rate over 2 weeks. Find total repayment and approximate annual equivalent reducing rate.
To compare a flat rate loan fairly with a reducing balance loan, convert the flat rate to an approximate reducing balance rate:
$$r_{\text{reducing}} \approx \frac{2 \times n \times r_{\text{flat}}}{n + 1}$$where $n$ is the number of repayment periods.
Example: 6% flat rate over 5 years (60 monthly periods).
$$r_{\text{reducing}} \approx \frac{2 \times 60 \times 0.06}{61} = \frac{7.2}{61} \approx 11.8\%$$The 6% flat rate is equivalent to approximately 11.8% reducing balance — nearly double the advertised rate.
What to write in your book
- True reducing rate $\approx (2n \times r_{\text{flat}}) \div (n+1)$, where n = number of periods.
- Rule of thumb: true rate ≈ double the flat rate for long-term loans.
- Always check: is this flat or reducing? If flat, apply the conversion before comparing.
Fill the gap: A 6% flat rate loan over 5 years (60 monthly periods) has an approximate equivalent reducing balance rate of %.
Common errors · the 3 traps that cost marks
What to write in your book
- Lower headline rate does NOT guarantee a cheaper loan — always determine if it is flat or reducing balance.
- Flat rate: Total Interest = P × r × n (simple). Reducing: use M = PV × r / [1 − (1+r)^−n].
- In the conversion formula, n = number of repayment periods (e.g. 48 for 4-year monthly loan).
Match each loan feature to its description:
Quick-fire practice · 2 activities
A $25,000 car loan: Option A is 6% flat rate over 5 years. Option B is 9% reducing balance (compounded monthly) over 5 years. Which is cheaper? By how much? Also find the approximate true reducing rate for Option A.
A furniture store advertises "0% interest for 12 months" on a $3,000 purchase but charges a $150 establishment fee and $10/month account-keeping fee. Is this really interest-free? What is the effective annual rate? Would a 10% p.a. reducing balance loan (compounded monthly) be cheaper?
Top 3 list: Name THREE real-world situations where flat rate loans appear. For each, explain one mathematical fact a borrower should check before signing.
Most people predict Option B (6% flat) is cheaper because 6% < 8%. But Option A (8% reducing balance) actually costs less total interest.
Flat rate: Interest $= 20000 \times 0.06 \times 5 = \$6{,}000$.
Reducing balance: Monthly $= \$405.53$, total $= \$24{,}332$, interest $= \$4{,}332$.
The reducing balance loan saves $1,668 despite the higher advertised rate. This is the flat rate trap — always convert to comparable terms before choosing a loan.
What has changed in your understanding? What surprised you most?
Pick your answer, then rate your confidence — that tells the system what to drill next.
Q1. A $20,000 loan at 6% flat rate over 5 years. The total interest charged is:
Q2. The approximate true reducing balance rate for a 6% flat rate loan repaid monthly over 5 years (n = 60) is closest to:
Q3. In a reducing balance loan, the interest charged in each period is calculated on:
Q4. A car dealer advertises a loan at "5% flat rate". Before accepting, a smart borrower should:
Q5. Flat rate loans are considered deceptive because:
SA 1. A $18,000 car loan is offered at 5% flat rate over 4 years. (a) Find total interest and monthly repayment. (b) Find the approximate true reducing balance rate. (c) Compare to an 8.5% reducing balance loan (compounded monthly) over 4 years — which is cheaper? (2 marks)
SA 2. A furniture store offers "12 months interest free" with a $150 establishment fee and $10/month account-keeping fee on a $3,000 purchase. (a) What is the total cost? (b) If paid in 12 equal monthly payments, what is the effective interest rate? (c) Would a 10% p.a. reducing balance loan (compounded monthly) be cheaper? (2 marks)
SA 3. (a) Derive the formula for converting a flat rate to an approximate reducing balance rate. (b) A payday lender charges 25% flat rate over 2 weeks on a $1,000 loan. Calculate the total repayment and the approximate equivalent annual reducing rate. (c) Explain, using mathematical evidence, why payday lending is considered predatory. (3 marks)
Comprehensive answers (click to reveal)
MC 1 — A: Flat: Interest = 20000 × 0.06 × 5 = $6,000.
MC 2 — C: r_red = (2 × 60 × 0.06) / 61 = 7.2/61 ≈ 11.8%.
MC 3 — B: Reducing balance — interest each period is calculated on the outstanding balance, not the original principal.
MC 4 — D: Always convert flat rate to equivalent reducing rate before comparing with other loan products.
MC 5 — C: Flat rates charge interest on the full original principal throughout the term, even as the balance falls, making the true rate much higher than advertised.
SA 1 (2 marks): (a) Flat: Interest = 18000 × 0.05 × 4 = $3,600. Total = $21,600. Monthly = 21600/48 = $450.00 [0.5 mark]. (b) True rate = (2 × 48 × 0.05)/49 = 4.8/49 ≈ 9.8% reducing [0.5 mark]. (c) Reducing 8.5%: r = 0.085/12 = 0.007083, n = 48. M = 18000 × 0.007083/[1−(1.007083)^−48] ≈ $443.52. Total = 443.52 × 48 = $21,289. Interest = $3,289. Reducing balance is cheaper by $311 [1 mark].
SA 2 (2 marks): (a) Total = 3000 + 150 + 12×10 = $3,270. Extra = $270 [0.5 mark]. (b) M = 3270/12 = $272.50. Solve: 3000 = 272.50 × [1−(1+r)^−12]/r. By trial/iteration: r ≈ 1.5%/month = 18% p.a. effective [0.5 mark]. (c) 10% reducing: r = 0.00833, n = 12. M = 3000 × 0.00833/[1−(1.00833)^−12] ≈ $263.34. Total = $3,160. Yes, the 10% reducing balance loan is cheaper than the "interest free" deal [1 mark].
SA 3 (3 marks): (a) For n equal repayments, average balance ≈ P(n+1)/(2n). Total interest: P × r_flat × n = P(n+1)/(2n) × r_red × n. Divide both sides by P × n: r_flat = (n+1)/(2n) × r_red. Therefore r_red = 2n × r_flat/(n+1) [1 mark]. (b) Total = 1000 + 250 = $1,250. Simple annual = 25% × 26 = 650% p.a. Compound: (1.25)^26 − 1 ≈ 32,300% p.a. [1 mark]. (c) Annual rates of 650–32,000% are vastly beyond any reasonable cost of borrowing. Borrowers unable to repay see debt multiply rapidly. The flat rate structure disguises the true cost. These facts constitute mathematical evidence of predatory intent [1 mark].
Five timed questions on flat rate loans, reducing balance loans, true rate conversion and loan comparisons. Beat the boss to bank a tier — gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering questions on reducing balance vs flat rate loans. Pool: lesson 8.
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