Module 7 · L8 of 12~25 minMST-12-S2-02⚡ +50 XP available
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.
Today's hook, 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.
0/5QUESTS
Get oriented
Recall what you already know, meet the key ideas and settle the terms.
Worksheets
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
A $20,000 car loan: Option A is 8% reducing balance over 5 years. Option B is 6% flat rate over 5 years. Which costs less total interest? Predict before calculating.
Before reading on write your gut feeling. We will revisit this at the end of the lesson.
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Key ideas for this lesson
reference
Two fundamentally different loan types, understanding the difference can save you thousands of dollars.
The problem: After 2.5 years, you have repaid half the principal ($\$10,000$), yet you are still paying interest on the full $\$20,000$ as though you owe every dollar. This is fundamentally unfair and disguises the true cost of borrowing.
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.
A flat-rate loan charges simple interest on the full original principal for the whole term. True monthly repayment = (P + I)/n. Because you owe less as you pay down the loan but interest stays fixed, the effective rate is much higher.
Pause, copy the flat-rate repayment formula: monthly repayment = (P + I_total) / n, and explain why the effective rate is higher: you still pay interest on the original principal even after repaying most of it into your book.
Quick check: A $20,000 loan at 6% flat rate over 5 years. The total interest charged is:
Charge interest on what is owed
Follow how a reducing balance loan charges interest only on the outstanding amount.
Beyond the syllabus. Comparing reducing-balance and flat-rate costs, and the effect of fees, is core. Deriving a flat-to-reducing conversion formula and annualising payday-loan charges are extension — the exam asks you to compare supplied products, not derive conversion formulas.
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Reducing balance loans, fair interest on what you owe
core concept
A flat-rate loan charges simple interest on the full original principal for the entire term, you keep paying as if you owe the whole amount even after repaying half. A reducing balance loan charges interest only on the outstanding balance each period: interest = r × current balance. As you repay principal, the interest component falls each period, so total interest paid over the life of the loan is lower than a flat-rate loan at the same stated rate.
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:
Total repaid $= 405.53 \times 60 = \$24{,}332$. Total interest $= \$4{,}332$.
Counter-intuitive result: The 6% flat rate loan costs $6,000 in interest. The 8% reducing balance loan costs only $4,332 in interest, $1,668 less, despite the higher advertised rate. The 8% reducing balance loan is cheaper.
A reducing balance loan charges interest only on the outstanding balance each period. As the principal falls, so does the interest component, this is how mortgages and most bank loans actually work.
Pause, copy the reducing balance interest formula: interest for each period = r × opening balance for that period, and note the consequence: as the balance falls, the interest component of each repayment falls, making the loan fairer than a flat-rate loan into your book.
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.
Find the true rate
Convert a flat rate into the reducing balance rate it is really equivalent to.
Worked examples · reveal each step
PROBLEM 1 · COMPARING LOAN TYPES
$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.
For compound comparison, raise the periodic factor to the power of periods per year
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Even the simple annual rate of 650% is far beyond any fair cost of borrowing. This mathematically justifies why payday lending is considered predatory.
The mathematical evidence demonstrates the predatory nature of the product
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Finding the true rate, what is the flat rate really costing?
core concept
Reducing balance loans charge interest = r × current balance each period, so the true rate equals the stated rate. For flat-rate loans, the stated rate understates the true cost. To find the true effective rate of a flat-rate loan: either use the approximation (true rate ≈ 2 × flat rate), or solve for r by setting the present value of all repayments equal to the original loan amount using the PV annuity formula.
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).
The 6% flat rate is equivalent to approximately 11.8% reducing balance nearly double the advertised rate.
Why does this formula work? For a loan repaid in $n$ equal instalments, the average outstanding balance is approximately $\frac{P(n+1)}{2n}$. Setting the total interest equal under both methods and solving gives $r_{\text{red}} \approx \frac{2n \cdot r_{\text{flat}}}{n+1}$.
To find the true annual rate of a flat-rate loan: use the reducing balance formula or table to find the rate r such that the present value of repayments equals the original loan. The true rate is typically close to twice the flat rate.
Pause, copy the true-rate rule of thumb (true effective rate ≈ 2 × stated flat rate) and note when to use the full PV method: when exact precision is required, set PV = M × [1 − (1+r)^(−n)] / r and solve for r into your book.
Fill the gap: A 6% flat rate loan over 5 years (60 monthly periods) has an approximate equivalent reducing balance rate of %.
Avoid the advertised-rate traps
Stop assuming the lower advertised rate is the cheaper loan.
Common errors · the 3 traps that cost marks
Trap 01
Assuming lower advertised rate = cheaper loan
A 6% flat rate is often more expensive than an 8% or even 10% reducing balance rate. Always convert to the same basis before comparing. The advertised rate alone tells you nothing.
Trap 02
Using the wrong formula for flat rate loans
For flat rate loans, Total Interest = P × r × n (simple interest on original principal). Do NOT use the reducing balance repayment formula for a flat rate loan, you will get the wrong answer.
Trap 03
Forgetting n = number of periods, not years
In the true rate formula, n is the number of repayment periods. For a 4-year monthly loan, n = 48, not 4. Using n = 4 gives the wrong conversion.
Match each loan feature to its description:
Flat rate
Reducing balance
True rate formula
Rule of thumb
r_reducing = 2n x r_flat / (n+1)
True reducing rate is approximately double the flat rate
Interest charged on original principal for entire term
Interest charged only on the outstanding balance each period
Drill it and revisit
Run the drill, model the loan in a spreadsheet, then name what has changed.
Quick-fire practice · 2 activities
1
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.
2
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.
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Modelling a reducing balance loan in a spreadsheet
core concept
A reducing balance loan cannot be done in one calculation, because each month's interest depends on the balance the previous month left behind. That is exactly the shape a spreadsheet is for: one row per month, each row referring to the row above.
Take $\$20\,000$ borrowed at $6\%$ p.a. compounded monthly, so $0.5\%$ a month, with a repayment of $\$600$.
Cell
What goes in it
Fill down?
B2
opening balance, $20000$
no, this is the starting figure
C2
interest for the month, $=$B2*0.005
yes
D2
principal repaid, $=$600-C2
yes
E2
closing balance, $=$B2-D2
yes
B3
next opening balance, $=$E2
yes, and this is the link that makes it work
Filling down four months gives:
Month
Opening
Interest
Principal
Closing
1
$\$20\,000.00$
$\$100.00$
$\$500.00$
$\$19\,500.00$
2
$\$19\,500.00$
$\$97.50$
$\$502.50$
$\$18\,997.50$
3
$\$18\,997.50$
$\$94.99$
$\$505.01$
$\$18\,492.49$
4
$\$18\,492.49$
$\$92.46$
$\$507.54$
$\$17\,984.95$
The repayment never changes, but the split does. Interest falls every month and the principal repaid rises to match, because the balance the interest is charged on keeps shrinking. Graph the closing balance against the month to see the debt curve away toward zero.
Change one cell and answer a different question. This is what the dot-point means by a variety of contexts. Raise the repayment to $\$700$ and the loan clears sooner; raise the rate to $0.6\%$ a month and every later row changes on its own. A model you can re-point at a new question beats a calculation you have to redo.
To model a reducing balance loan, use one row per month: interest $=$ opening balance $\times$ monthly rate, principal repaid $=$ repayment $-$ interest, closing balance $=$ opening $-$ principal repaid, and the next opening balance refers to the previous closing balance. The repayment stays fixed while the interest falls and the principal repaid rises.
Pause, copy the four cell formulas and the link from one row's closing balance to the next row's opening balance, into your book.
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Revisit your thinking
Most people predict Option B (6% flat) is cheaper because 6% < 8%. But Option A (8% reducing balance) actually costs less total interest.
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?
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Practise reducing balance loans
Answer the question bank, then write full short-answer responses.
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Multiple choice
+5 XP per correct · +25 XP all-correct
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:
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Short answer
ApplyBand 42 marks
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)
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ApplyBand 42 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)
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AnalyseBand 53 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)
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].
Finish the lesson
Take the module quiz, mark the lesson complete and move on.
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Take the full module quiz
quiz
A full module quiz covering every lesson in this module, not just this one. Set aside a decent block of time and treat it like a real assessment.
