Mathematics Standard • Year 12 • Module 7 • Lesson 9
Credit Cards — Problem Set
Apply credit-card maths to realistic Australian scenarios — interest-free purchases, minimum-payment traps, balance transfers and store-card comparisons.
Problem 1 — Using a credit card responsibly
Lara pays for $1,800 of groceries and bills on her credit card each month at 19.99% p.a. (daily compounding) and pays the full statement balance by the due date every time. Her purchases are spread evenly across the cycle.
Set up: What are we solving for?
(i) How much interest does Lara pay each month while operating this way? 1 mark
(ii) Calculate the effective annual rate of her card and explain in one sentence why it is higher than 19.99%. 2 marks
(iii) Lara's card pays 1% cashback on all purchases. Calculate her annual cashback at her current spend, and explain why the cashback only benefits her because she pays in full each month. 2 marks
Stuck? Revisit lesson § Interest-Free Periods — pay full balance → zero interest.Problem 2 — The minimum-payment trap
Sam has run a $4,000 balance on a credit card at 19.99% p.a. (monthly r = 0.1999/12) and pays only the $100 monthly minimum.
Set up: What are we solving for?
(i) Calculate the interest charged in Month 1 and the principal that is actually paid down. 2 marks
(ii) If Sam paid $300/month instead, the card would clear in approximately 15 months (total paid ≈ $4,500). Approximately how many months and dollars does Sam save by paying $300 instead of the minimum (use lesson § Worked Example — ~$5,500 saving at $300/month on $5,000 — and scale)? 2 marks
(iii) In one sentence, explain mathematically why "minimum payment only" leaves Sam paying for decades. 2 marks
Stuck? Revisit lesson § Minimum Payments — the $3,000 at 20% takes over 9 years on minimum payments.Problem 3 — Balance transfer offer
Mei has $8,000 debt on a card at 19.99% p.a. A competitor offers a balance transfer: 0% for 12 months, with a 2% transfer fee. Mei can afford $700/month.
Set up: What are we solving for?
(i) Calculate the transfer fee and the new opening balance on the 0% card. 1 mark
(ii) At $700/month, will Mei clear the transferred balance within the 12-month promotional period? 2 marks
(iii) If instead Mei stays on the existing 19.99% card and pays $700/month, the loan formula gives approximately 12 months to clear with total interest ~$850. Calculate the total dollar saving from accepting the transfer offer (vs staying), and recommend in one sentence whether she should accept. 2 marks
Stuck? Revisit lesson § Balance Transfers — the $5,000 → $900 net saving example.Problem 4 — Store card vs standard credit card
Diego is buying a $1,200 set of tyres. The retailer offers a store card at 25% p.a. with no interest-free period (interest accrues from the day of purchase). Diego could instead use his standard credit card at 20% p.a. with a 55-day interest-free window, and pay $400/month over 3 months.
Set up: What are we solving for?
(i) If Diego uses the store card and pays $400/month, the balance approximately follows: Month 1 → $1,200, Month 2 → ~$825, Month 3 → ~$442 before final payment. Using monthly interest at 25%/12, estimate the total interest charged across the three months. 2 marks
(ii) If Diego uses his standard card, he pays the first $400 within the interest-free window, then the remaining $800 attracts interest at 20% p.a. for about 2 months. Estimate the total interest charged. 2 marks
(iii) Recommend the cheaper option for Diego and explain in one sentence why the headline rate alone is not the whole story. 2 marks
Stuck? Revisit lesson § Activity 2 Q1 — comparing a 25% store card to a 20% card with interest-free period.Problem 5 — Comparing two cards on effective rate
Two competing cards advertise different rates:
Card X: 18.99% p.a. compounded daily.
Card Y: 19.50% p.a. compounded monthly.
Set up: What are we solving for?
(i) Calculate the effective annual rate for Card X using (1 + r/365)³⁶⁵ − 1. 1 mark
(ii) Calculate the effective annual rate for Card Y using (1 + r/12)¹² − 1. 1 mark
(iii) State which card is actually cheaper to carry a balance on, by what percentage points, and explain in one sentence why the advertised "lower" rate may not be the lower-cost card. 2 marks
Stuck? Revisit lesson § Interest Calculation — daily compounding pushes 19.99% to 22.13%.How did this worksheet feel?
What I'll revisit before next class:
Problem 1 — Responsible use
Set up. Pay-in-full users avoid all interest; we need to confirm zero interest and analyse cashback.
(i) Interest = $0/month — she pays the full balance by the due date, so the 55-day interest-free benefit applies.
(ii) r_eff = (1 + 0.1999/365)³⁶⁵ − 1 ≈ 0.2213 = 22.13% p.a. Daily compounding means interest is charged on interest, so the realised rate is higher than the advertised 19.99%.
(iii) Annual spend = $1,800 × 12 = $21,600. Cashback = $21,600 × 0.01 = $216/year. This is profit only because she pays no interest; if she carried a $4,000 balance for a year she would pay ≈ $880 in interest, wiping out the cashback four times over.
Problem 2 — Minimum-payment trap
Set up. Compute Month 1 interest and principal; then estimate the saving from a $300/month plan.
(i) r = 0.1999/12 ≈ 0.01666. I₁ = 4,000 × 0.01666 ≈ $66.62. P₁ = $100 − $66.62 = $33.38. The balance drops by only $33 of the $100 paid.
(ii) At $300/month, the card clears in approximately 15 months with total paid ≈ $4,500 (about $500 interest). At $100/month, the term is approximately 200+ months with total ≈ $7,000+. Saving ≈ 15-16 years and roughly $2,500 in interest (a smaller version of the $5,000 → $5,500 lesson example).
(iii) Most of the $100 minimum is consumed by monthly interest, so only a tiny amount of principal is paid down each month — over time interest compounds faster than the principal shrinks.
Problem 3 — Balance transfer
Set up. Find the fee, check whether the transferred balance clears in 12 months, and quantify the saving vs staying.
(i) Fee = 8,000 × 0.02 = $160. New balance = $8,160.
(ii) 8,160 / 700 ≈ 11.66, so she clears in about 12 months — just barely. Strictly she would need $680/month minimum, and $700 gives a small safety buffer.
(iii) Saving = (Total on existing card $700 × 12 + extra interest after 12 mo ≈ $8,400+) − Total on transfer ($8,160) ≈ $200-$400 saved over the year (depending on exact payoff month on the 20% card). Recommendation: accept the transfer — it locks in a guaranteed pay-down within the 0% window.
Problem 4 — Store card vs standard card
Set up. Compute total monthly interest on each card; compare directly.
(i) Store card monthly r = 25%/12 ≈ 0.02083. Approx interest: Month 1 on $1,200 ≈ $25; Month 2 on $825 ≈ $17.18; Month 3 on $442 ≈ $9.21. Total ≈ $51.39 interest.
(ii) First $400 paid within 55 days = $0 interest. Remaining $800 attracts interest at 20% for ~2 months: 800 × 0.20/12 × 2 ≈ $26.67 interest.
(iii) The standard card is cheaper — about $25 less interest. Even though the store card's headline rate (25%) is only slightly higher than the credit card's (20%), losing the 55-day interest-free window is what tips the balance.
Problem 5 — Effective-rate comparison
Set up. Convert each nominal rate to an effective annual rate; compare.
(i) r_eff(X) = (1 + 0.1899/365)³⁶⁵ − 1 ≈ 0.2092 = 20.92% p.a.
(ii) r_eff(Y) = (1 + 0.1950/12)¹² − 1 ≈ 0.2134 = 21.34% p.a.
(iii) Card X is cheaper, by about 0.42 percentage points. Card Y's lower compounding frequency (monthly vs daily) does not save enough to overcome its higher nominal rate; the advertised rate alone is misleading.