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
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
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 Mei stays on the existing card, interest is added monthly at 19.99%/12 and she pays $700 at the end of each month. Calculate the total amount she pays before the balance clears and the total interest, then compare this cash outlay with the amount repaid on the 0% transfer card (including its fee). State the saving and recommend an option. 2 marks
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
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
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 transfer fee and compare both plans using the actual payments required to clear each balance; include the payoff month and distinguish the interest charge from the transfer fee.
(i) Fee = 8,000 × 0.02 = $160. New balance = $8,160.
(ii) After 11 payments of $700, the $8,160 transfer balance has $460 left. Mei pays $460 in month 12, so it clears within the 12-month 0% period. Her total transfer-plan outlay is $7,700 + $460 = $8,160, including the $160 fee and no interest.
(iii) On the existing card, using the stated monthly recurrence, the balance after month 12 is $540.17, so the final month-13 payment is $549.17. Total repayments are $700 × 12 + $549.17 = $8,949.17; interest is $8,949.17 − $8,000 = $949.17. The transfer plan costs $8,160 in total, so the saving is $8,949.17 − $8,160 = $789.17. Accepting the transfer saves $789.17, provided Mei pays it off within the 0% period.
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.