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hscscience Maths Adv · Y12
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Module 7 · L17 of 20 ~40 min ⚡ +95 XP available

Car Loans, Personal Loans & Credit Cards

A car dealer says "0% finance", sounds like free money. But the sticker price is $3,000 higher than cash. A credit card rewards you with points while quietly charging 20% interest. By the end of this lesson you will calculate the true cost of every consumer loan and never be fooled by marketing maths again.

Today's hook, A dealer offers "$\$0$ down, 0% interest, $\$583$/month for 5 years" on a $\$35,000$ car, while the cash price is $\$32,000$. Before you sign, you need the maths to expose what "free" finance really costs.
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.

Build Foundations & guided practice Apply Application practice Master Mastery challenge Build custom Build your own from any module question
01
Recall, your gut answer first
+5 XP warm-up

A car dealership offers two options for a $35,000 car:

Option A: Cash price $32,000. Pay today.

Option B: $\$0$ down, 0% interest, $\$35,000$ over 5 years = $583/month.

Without calculating which is the better deal? What is your gut feeling and why?

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02
The key formula
+5 XP to read

Every consumer loan in this lesson uses one core formula the present value of an annuity. Lock it in and every loan comparison becomes a rearrangement.

The present value (loan amount) equals the monthly payment times the annuity factor. The annuity factor converts future payments into today's dollars. Solve for $M$ to find repayments; solve for $r$ to find the true rate.

The loan (present value) formula and what each symbol means.
$$P = M \times \dfrac{1 - (1+r)^{-n}}{r}$$
Total cost
Total paid $= M \times n$. Subtract the principal $P$ to find total interest paid.
Finding true rate
If cash price $\ne$ finance total, substitute both and solve for $r$ by trial and error or technology.
Monthly vs annual rate
Always divide annual rate by 12 for $r$. Multiply monthly $r$ by 12 to get the annual effective rate.
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What you'll master
Know

Key Facts

  • How dealer finance, personal loans, and credit cards work
  • The true cost of 0% finance and BNPL
  • How to calculate effective interest rates
Understand

Concepts

  • Why 0% finance is rarely truly 0%
  • How minimum payments keep you in debt for decades
  • The psychology of deferred payment products
Can Do

Skills

  • Calculate total cost of any consumer loan
  • Find the effective interest rate of dealer finance
  • Compare cash vs finance options mathematically
  • Evaluate BNPL offers using effective rates
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Key terms
Annuity factor$\frac{1-(1+r)^{-n}}{r}$, converts periodic payments to a present value.
Effective annual rateThe true annual cost of a loan, accounting for compounding: $(1+r_m)^{12} - 1$.
Dealer financeFinance arranged by the car dealer, often with an inflated purchase price to hide interest.
Minimum paymentThe smallest payment a credit card requires; designed to maximise interest revenue.
BNPLBuy Now Pay Later, instalment products (Afterpay, Zip) with late fees instead of declared interest.
Cash priceThe up-front price if you pay in full today; the baseline for comparing finance options.

Price a car loan

See what dealer finance really charges once the fees are counted.

05
Car Loans and Dealer Finance
core concept

Car dealers use two common tricks to make finance look attractive:

  • 0% finance with an inflated price: The car costs $\$35,000$ on finance but $\$32,000$ cash. The $\$3,000$ difference is hidden interest. You are borrowing $\$32,000$ and paying back $35,000, that is not 0%.
  • Low monthly payments with long terms: A 7-year car loan has low payments but you pay interest long after the car has lost most of its value (most cars depreciate 40–60% in 3 years).
Finding the true rate. If cash price = $\$32,000$ and finance payments = $\$583$/month for 60 months, what is the actual interest rate? Substitute into $P = M \times \frac{1-(1+r)^{-n}}{r}$: $32{,}000 = 583 \times \frac{1-(1+r)^{-60}}{r}$. By trial and error (or technology): $r \approx 0.00365$ per month = 4.38% p.a. So "0% finance" is actually 4.38%, not terrible, but definitely not zero.
Comparison table for the 0% finance trap: cash vs finance

The $3,000 price difference between cash and finance is the hidden interest cost.

Key formula: $P = M \times \dfrac{1-(1+r)^{-n}}{r}$ where $P$ = loan, $M$ = payment, $r$ = monthly rate, $n$ = payments; Total cost = $M \times n$. Interest paid = Total cost $- P$

Pause, copy the PV annuity formula $P = M \times \dfrac{1-(1+r)^{-n}}{r}$ (max loan for a given repayment) and the total-interest formula: Interest $= M \times n - P$ into your book.

Did you get this? True or false: when a dealer offers "0% finance" with an inflated purchase price, you still effectively pay interest on the loan.

Work the examples

Follow three worked problems across car loans, personal loans and cards.

Worked examples · 3 in a row, reveal as you go

PROBLEM 1 · TOTAL LOAN COST

A $\$25,000$ car. Cash price: $\$23,500$. Dealer finance: $\$0$ down, $\$475$/month for 60 months. Find (a) total finance cost and (b) effective annual rate.

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$(a)\; \text{Total} = 475 \times 60 = \$28{,}500$
Multiply monthly payment by number of payments. Interest = $28{,}500 - 23{,}500 = \$5{,}000$.
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PROBLEM 2 · CREDIT CARD MINIMUM PAYMENT

$\$5,000$ balance at 19.99% p.a. Minimum payment $\$150$/month. Find (a) months to repay and (b) total interest.

1
$r = 19.99\%/12 = 0.01666$/month
Convert annual rate to monthly. This is the rate applied each month to the outstanding balance.
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PROBLEM 3 · BNPL EFFECTIVE RATE

A $\$500$ BNPL purchase: 4 fortnightly instalments of $\$125$. One missed payment triggers a $15 late fee. Find the effective annualised rate if you miss one payment.

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$\text{Extra cost} = \$15$ over a loan of $\$500$
The $15 fee is the "interest" charged on this BNPL product.
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Quick check: A $20,000 personal loan at 9% p.a. over 4 years (48 payments). What is the monthly interest rate used in the annuity formula?

Avoid the traps

Fix the rate-period and fee slips that cost marks.

Common errors · the 3 traps that cost marks

Trap 01
Using the wrong "P"
When finding the true effective rate of dealer finance, use the cash price as $P$, not the finance amount. The cash price is what you're actually borrowing; the higher financed amount already includes the hidden interest.
Trap 02
Annual vs monthly rate mix-up
Always convert to monthly before using the formula. $r = \text{annual rate}/12$. Substituting 9% directly instead of 0.75% will give a wildly wrong answer and lose all method marks.
Trap 03
Forgetting the minimum payment trap
If $M \le P \times r$ (your payment equals or falls below the monthly interest charge), the balance never decreases, it grows forever. Always check this condition for credit card problems.

Fill in the gap: To find the effective rate of dealer finance where cash price = $P$ and finance payments = $M$ for $n$ months, you substitute the as $P$ in the annuity formula and solve for $r$ by .

Drill it and revisit

Run the drill, then return to your opening answer and name what has changed.

Quick-fire practice · 4 calculations

1

A $\$3,000$ laptop. Cash: $\$2,800$. Finance: $140/month for 24 months. Find the total finance cost and interest paid.

2

Car loan: $\$30,000$ at 6.5% p.a. over 5 years (60 months). State the monthly rate $r$ and give the annuity factor.

3

$\$5,000$ credit card at 20% p.a. Paying $\$300$/month. Show that the balance shrinks (monthly interest $< $300).

4

A BNPL purchase of $\$800$. Two $\$15$ late fees over 6 weeks. Calculate the effective annual rate.

Odd one out: Which of these is NOT a consumer lending product discussed in this lesson?

Two truths, one lie: Identify the FALSE statement about consumer credit.

12
Revisit your thinking

Earlier you were asked: Option A (cash $\$32,000$) vs Option B (0% finance $\$583$/month for 5 years).

Option B total = $583 × 60 = $35,000. The extra $\$3,000$ represents hidden interest. Using the annuity formula with the cash price as $P$: $32{,}000 = 583 \times \frac{1-(1+r)^{-60}}{r}$ gives $r \approx 0.00365$/month = 4.38% p.a.

Option A is mathematically better if you have the cash. However, if you cannot afford $32,000 today, Option B is a legitimate path, the key is knowing the true cost before signing.

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Practise consumer loans

Write full short-answer responses, then check them against the model answers.

01
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. Each retry pulls a fresh mix from the bank.

02
Short answer
ApplyBand 43 marks

Q1. A car costs $\$22,000$ cash. Dealer finance: $\$475$/month for 48 months. (a) Calculate the total finance cost. (b) Calculate the extra paid compared to cash. (c) Find the effective monthly interest rate by trial and error. (3 marks)

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ApplyBand 43 marks

Q2. A credit card has a $\$4,000$ balance at 19.99% p.a. Monthly minimum payment is $\$120$. (a) Calculate the monthly interest charge. (b) Determine the number of months to repay. (c) Find total interest paid. (3 marks)

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AnalyseBand 54 marks

Q3. Compare two options to finance a $30,000 car: (a) Car loan at 6.5% p.a. for 5 years; (b) Personal loan at 9.5% p.a. for 5 years. For each: calculate the monthly repayment and total interest. Explain why the rates differ. (4 marks)

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Comprehensive answers (click to reveal)

Drill 1: Total = $\$3,360$; Interest = $\$560$. 2: $r = 0.5417\%$/month; annuity factor = $50.50$. 3: Monthly interest = $83.33 < 300$ ✓ balance shrinks. 4: Rate = $30/800 = 3.75\%$ over $6/52$ yr; annualised = $3.75 \div 6/52 = 32.5\%$ p.a.

Q1 (3 marks): (a) $475 \times 48 = \$22{,}800$ [1]. (b) Extra = $22{,}800 - 22{,}000 = \$800$ [1]. (c) Solve $22{,}000 = 475 \times [1-(1+r)^{-48}]/r$; by trial $r \approx 0.00122$/month $= 1.46\%$ p.a. [1].

Q2 (3 marks): (a) $4{,}000 \times 0.01666 = \$66.64$/month [1]. (b) $n = -\ln(1-4{,}000 \times 0.01666/120)/\ln(1.01666) = 43.5$ months [1]. (c) Interest $= 120 \times 43.5 - 4{,}000 = \$1{,}220$ [1].

Q3 (4 marks): Car loan: $r = 0.005417$, $M = 30{,}000 \times 0.005417/[1-(1.005417)^{-60}] = \$584$/month, total interest $= \$5{,}040$ [1+1]. Personal: $r = 0.007917$, $M = \$631$/month, total interest $= \$7{,}860$ [1]. Car loans are secured (car as collateral) so lower risk to lender = lower rate [1].