Mathematics Standard • Year 12 • Module 7 • Lesson 9
Credit Cards — Skill Drill
Build fluency with credit-card maths: daily interest using Balance × r/365 × days, effective annual rate (1 + r/365)³⁶⁵ − 1, minimum-payment splits, and the interest-free period.
1. Quick recall
Answer each question in the space provided. 1 mark each
Q1.1 Complete the credit-card daily-interest formula and the effective annual rate formula.
Daily interest = Balance × ____________ Effective annual rate = ____________ − 1
Q1.2 Convert each nominal annual rate to a daily rate (r/365) and to a monthly rate (r/12).
19.99% p.a. → daily r ≈ ____________ monthly r ≈ ____________
22.0% p.a. → daily r ≈ ____________ monthly r ≈ ____________
Q1.3 Define each term in one short phrase.
Interest-free period: ____________________________________________
Minimum payment: ____________________________________________
2. Worked example — daily interest and the effective rate
Follow each line of working. Every step has a reason on the right.
Problem. A credit card has a $2,000 balance at 19.99% p.a. with interest compounded daily. Find the daily interest, the monthly interest charge if the balance is held for 30 days, and the effective annual rate.
Step 1 — Daily interest = Balance × r / 365.
I_day = $2,000 × 0.1999 / 365 ≈ $1.0953/day
Reason: card issuers calculate interest each day on the closing balance, not monthly in arrears.
Step 2 — Approximate monthly interest if the balance is held all month.
I_30 ≈ $1.0953 × 30 ≈ $32.86
Reason: 30 days of daily charges add up to the bill the customer sees on their monthly statement.
Step 3 — Effective annual rate = (1 + r/365)³⁶⁵ − 1.
r_eff = (1 + 0.1999/365)³⁶⁵ − 1 ≈ 0.2213 = 22.13%
Reason: daily compounding pushes the true annual cost above the advertised 19.99%.
Conclusion. Daily interest $1.10, monthly ≈ $32.86, effective rate ≈ 22.13% p.a.
3. Faded example — minimum payment split on a $3,000 balance
A card charges 20% p.a. (compounded monthly for this calculation) and the minimum payment is $60/month. The opening balance is $3,000. Fill in each blank for Months 1 and 2. 4 marks
Step 1 — Monthly rate:
r = 0.20 / 12 = ____________
Step 2 — Month 1 interest, principal repaid and closing balance:
I₁ = $3,000 × ____________ = $ ____________
P₁ = $60 − $ ____________ = $ ____________
B₁ = $3,000 − $ ____________ = $ ____________
Step 3 — Month 2 interest and principal:
I₂ = $ ____________ × ____________ = $ ____________
P₂ = $60 − $ ____________ = $ ____________
Comment. After two months, the balance is barely changing because most of each $60 payment is being eaten by interest. This is the minimum-payment trap.
4. Graduated practice — credit-card calculations
Show your working below each part. Round dollars to 2 dp.
Foundation — single-step substitution (4 questions)
| Q | Problem | Answer |
|---|---|---|
| 4.1 1 | Balance $1,500 at 19.99% p.a. Find daily interest (4 dp). | |
| 4.2 1 | Balance $2,500 at 19.99% p.a. Find monthly interest using monthly r = 0.1999/12. | |
| 4.3 1 | Minimum payment is 2.5% of balance with a $25 floor. Find the minimum for a $400 balance. | |
| 4.4 1 | Minimum payment $112.50 (2.5% of $4,500). Monthly interest is $75. Find principal repaid this month. |
Standard — typical HSC difficulty (6 questions)
Show formulas before substituting; label final answers with units.
4.5 A $4,500 balance is held all month on a card at 19.99% p.a. Calculate the monthly interest using daily interest × 30 days. 2 marks
4.6 Calculate the effective annual rate for a card advertised at 21.99% p.a. compounded daily. 2 marks
4.7 A $5,000 balance at 19.99% p.a. (monthly r = 0.1999/12) with minimum payment $125. Find (a) monthly interest, (b) principal repaid in Month 1, (c) closing balance. 3 marks
4.8 A customer makes a $1,000 purchase on day 5 of a billing cycle. The statement is issued at the end of day 30 with a 25-day grace period. If they pay the full balance by the due date, how much interest is charged? 1 mark
4.9 A balance transfer offer is 0% for 12 months with a 2% fee on the transferred amount. Find the fee on transferring $5,000. 1 mark
4.10 A $2,000 balance at 22% p.a. (monthly r = 0.22/12) with $25 minimum payment. Find Month 1 interest and explain in one sentence why the balance grows rather than shrinks. 3 marks
Extension — daily interest with mid-cycle payments (2 questions)
4.11 A card balance is $3,000 for the first 10 days of a 30-day cycle, then $2,000 for the remaining 20 days (after a $1,000 payment). Using daily interest at 18% p.a., find the total interest charged for the cycle. 3 marks
4.12 A balance transfer offer: 0% for 15 months, 1.5% transfer fee, on a $6,000 debt currently sitting at 20% p.a. Calculate (a) the fee, (b) the monthly payment needed to clear the transferred balance in exactly 15 months, and (c) the approximate interest saved versus paying that same monthly payment on the existing card at 20%. 3 marks
5. Self-check the easy 3
Tick the first three once you've checked your method works.
How did this worksheet feel?
What I'll revisit before next class:
Q1.1 — Formulas
Daily interest = Balance × r / 365. Effective annual rate = (1 + r/365)³⁶⁵ − 1.
Q1.2 — Daily and monthly rates
19.99% → daily ≈ 0.000548, monthly ≈ 0.01666. 22% → daily ≈ 0.000603, monthly ≈ 0.01833.
Q1.3 — Definitions
Interest-free period: up to 55 days where no interest is charged, provided the full statement balance is paid by the due date. Minimum payment: the smallest amount the cardholder must pay each month (typically 2-3% of the balance or a $25-$50 floor, whichever is higher).
Q3 — Faded example (Months 1 and 2 of the $3,000 card at 20%)
Step 1: r = 0.20 / 12 = 0.01667.
Step 2: I₁ = 3,000 × 0.01667 = $50.00. P₁ = 60 − 50 = $10.00. B₁ = 3,000 − 10 = $2,990.00.
Step 3: I₂ = 2,990 × 0.01667 = $49.83. P₂ = 60 − 49.83 = $10.17.
Q4.1 — Daily interest on $1,500
I_day = 1,500 × 0.1999 / 365 ≈ $0.8215/day.
Q4.2 — Monthly interest on $2,500
I_month = 2,500 × 0.1999 / 12 ≈ $41.65.
Q4.3 — Minimum payment on $400
2.5% of $400 = $10. Floor = $25. Minimum = $25 (the larger of the two).
Q4.4 — Principal repaid
P = $112.50 − $75.00 = $37.50.
Q4.5 — Monthly interest on $4,500 (daily method)
I_day = 4,500 × 0.1999 / 365 ≈ $2.4644. I_30 ≈ 2.4644 × 30 ≈ $73.93.
Q4.6 — Effective annual rate for 21.99% daily
r_eff = (1 + 0.2199/365)³⁶⁵ − 1 ≈ 0.2459 = 24.59% p.a.
Q4.7 — Minimum payment trap on $5,000
r = 0.1999/12 ≈ 0.01666. (a) I₁ = 5,000 × 0.01666 ≈ $83.29. (b) P₁ = 125 − 83.29 = $41.71. (c) B₁ = 5,000 − 41.71 = $4,958.29.
Q4.8 — Interest-free period
If the full balance is paid by the due date, no interest is charged at all. The grace period preserves the interest-free benefit.
Q4.9 — Transfer fee
Fee = $5,000 × 0.02 = $100.
Q4.10 — $2,000 at 22%, $25 minimum
r = 0.22/12 ≈ 0.01833. I₁ = 2,000 × 0.01833 ≈ $36.67. Because the $25 minimum is less than the $36.67 monthly interest, the balance actually grows by about $11.67 — the debt cannot be cleared at this payment level.
Q4.11 — Mid-cycle payment
Daily r = 0.18 / 365 ≈ 0.000493. First 10 days: I_A = 3,000 × 0.000493 × 10 ≈ $14.79. Last 20 days: I_B = 2,000 × 0.000493 × 20 ≈ $19.73. Total interest ≈ $34.52 for the cycle.
Q4.12 — Balance transfer
(a) Fee = 6,000 × 0.015 = $90. New balance = $6,090. (b) Monthly payment to clear in 15 months at 0% = 6,090 / 15 = $406. (c) On the 0% card, total paid = $6,090. On the existing 20% card, paying $406 for 15 months covers only about $6,090 of payments while interest continues to accrue (monthly interest at start ≈ $100), so the balance is not cleared in 15 months. Net saving ≈ several hundred dollars in interest avoided by using the transfer offer — provided the customer truly pays it down within 15 months.