Credit Cards
You buy a $2000 television on your credit card. The minimum payment is $60/month. The interest rate is 19.99% p.a. How long will it take to pay off? How much will you pay in total? The answer shocks most people: over 9 years and more than $6600 total, more than triple the original price. Credit cards are one of the most expensive forms of borrowing, yet their marketing makes them seem convenient and harmless. This lesson reveals the mathematics behind minimum payments, interest-free periods, and balance transfers.
Practise this lesson
Three printable worksheets that build from foundations to mastery, or build your own from any module’s questions.
You have $3000 on a credit card at 20% p.a. compounded monthly. You pay only the minimum ($60/month). Roughly how long until it's paid off?
Before reading onwrite your gut prediction and reasoning. We will revisit this at the end of the lesson.
Credit cards charge daily interest on the outstanding balance. Two formulas underpin almost every credit card question.
Daily interest: $I = \text{Balance} \times \dfrac{r}{365} \times \text{days}$, where $r$ is the annual rate as a decimal.
Effective annual rate: $\left(1 + \dfrac{r}{365}\right)^{365} - 1$. Daily compounding makes the effective rate higher than the advertised nominal rate.
Key facts
- How credit card interest is calculated daily
- What an interest-free period is
- How effective annual rate differs from nominal rate
Concepts
- Why minimum payments are a debt trap
- The true cost of credit card debt
- When a balance transfer saves money
Skills
- Calculate daily and monthly credit card interest
- Compare repayment strategies using iteration
- Analyse balance transfer offers mathematically
Credit cards typically charge daily interest on the outstanding balance:
$$\text{Daily interest} = \text{Balance} \times \frac{r}{365}$$where $r$ is the annual interest rate as a decimal.
Worked example: $2000 balance at 19.99% p.a.
- Daily interest $= 2000 \times \dfrac{0.1999}{365} = \$1.095$/day
- Monthly interest $\approx 1.095 \times 30 = \$32.85$/month
- Or directly: $2000 \times \dfrac{0.1999}{12} = \$33.32$/month
Interest-free period rule:
- Purchases at the start of a billing cycle earn up to 55 days interest-free
- Purchases just before the statement get approximately 25 days
- If you pay the full balance by the due date, zero interest is charged
- If you pay less than the full balance, interest is charged from the original purchase date
Credit card interest is calculated daily on the outstanding balance (daily rate = annual rate ÷ 365). Most cards offer an interest-free period (typically 44–55 days) only if the full balance is paid by the due date.
Pause, copy the daily rate formula (daily rate = annual rate ÷ 365) and the interest-free period condition: no interest is charged only if the entire statement balance is paid in full by the due date into your book.
Quick check: A credit card has a nominal rate of 19.99% p.a. with daily compounding. The effective annual rate is:
Credit card interest is calculated daily at daily rate = annual rate ÷ 365, compounding on the outstanding balance. The interest-free period (typically 44–55 days) only applies if the full balance is paid by the due date. If only the minimum payment is made, usually 2–3% of the balance, almost all of that payment is swallowed by interest, the principal barely decreases, and a $3,000 debt can take over 20 years to clear.
Minimum payments are typically 2–3% of the balance or a fixed floor ($25–$60), whichever is higher. This sounds manageable, but the mathematics is brutal.
Example: $3000 debt at 20% p.a. (compounded monthly), minimum payment $60/month.
| Month | Balance | Interest | Payment | Principal reduced |
|---|---|---|---|---|
| 1 | $3000.00 | $50.00 | $60 | $10.00 |
| 2 | $2990.00 | $49.83 | $60 | $10.17 |
| 3 | $2979.83 | $49.66 | $60 | $10.34 |
Using the loan repayment formula $PV = PMT \times \dfrac{1-(1+r)^{-n}}{r}$, solving gives $n \approx 111$ months = 9.25 years. Total paid $= 60 \times 111 = \$6660$. Interest paid $= \$3660$, more than the original debt!
Why do companies set minimum payments so low? Low minimums maximise interest income for the lender. The longer the debt persists, the more the lender earns.
The minimum payment trap: paying only the minimum each month means most of the payment is interest, and the balance falls very slowly. A $3,000 debt at 20% p.a. paying minimum 2% can take over 20 years to repay.
Pause, copy the minimum payment trap: minimum ≈ 2% of balance means most of each payment is interest and the balance barely falls, a $3,000 debt at 20% p.a. with only minimum payments can take 20+ years to repay and cost more than the original debt in interest into your book.
True or false: If the minimum monthly payment equals the monthly interest charge, the balance will be paid off within a few years.
Worked examples · reveal each step
$5000 credit card debt at 19.99% p.a. compounded monthly. Minimum payment = $125 or 2.5% of balance, whichever is higher. Find the first month's interest and principal reduction. Then compare paying $300/month instead.
$5000 debt at 20% p.a. Balance transfer offer: 0% for 12 months with a 2% fee. Is it worthwhile if you can pay $400/month?
Paying only the minimum means the balance barely falls and total interest cost multiplies dramatically over years. A balance transfer moves existing debt to a low (often 0%) promotional rate, which can save a significant amount of interest. But it is only advantageous if (1) the transfer fee is less than the interest saved, and (2) the full balance is repaid before the promotional rate expires and reverts to a standard high rate.
A balance transfer moves debt to a new card with a low or 0% promotional rate.
Benefits: Pause interest accumulation; pay down principal faster.
Risks to watch:
- Transfer fees (typically 1–3%)
- High "revert rate" after the promotional period ends (often 20%+)
- Temptation to spend on the new card, growing total debt
- Missing a single payment can cancel the promotional rate immediately
A balance transfer moves debt to a lower (often 0%) rate for a promotional period. It is advantageous only if: (1) the transfer fee is less than the interest saved, and (2) the full balance is repaid before the promotional rate expires.
Pause, copy the two conditions for a balance transfer to be advantageous: (1) transfer fee < total interest saved during the promotional period; (2) full balance is repaid before the promotional low rate expires into your book.
Fill the gap: A $4000 balance is transferred with a 2% fee. The transfer fee is $ and the new balance is $.
Common errors · the 3 traps that cost marks
Match each term to its correct description:
Quick-fire practice · 2 activities
Calculate: (a) Daily interest on $2500 at 19.99% p.a. (b) Monthly interest on the same balance. (c) Effective annual rate for 21.99% compounded daily.
A balance transfer offers 0% for 12 months with a 2% fee. You have $8000 debt at 19.99%. Is it worthwhile if you can pay $700/month? Show calculations to justify your answer.
Top 3 list: Name THREE strategies a person could use to reduce their credit card debt faster. For each, explain the mathematical reason why it works.
Most people guess 2–3 years. The reality is much worse. At $60/month on $3000 at 20%, month 1 interest $= \$50$, so only $10 reduces the principal. Using the loan formula:
$$3000 = 60 \times \frac{1-(1+0.20/12)^{-n}}{0.20/12}$$Solving gives $n \approx 111$ months $= 9.25$ years. Total paid $= \$6660$. Total interest $= \$3660$, more than the original debt. This is the minimum payment trap.
What has changed in your understanding? What did you get right? What surprised you?
Pick your answer, then rate your confidencethat tells the system what to drill next.
Q1. A credit card has a balance of $3000 at 20% p.a. compounded monthly. The monthly interest charge is closest to:
Q2. A minimum payment of $60/month is made on a $3000 debt at 20% p.a. The amount of principal reduced in month 1 is:
Q3. An interest-free period on a credit card applies only if:
Q4. A $6000 balance is transferred to a 0% card with a 2% fee. The new balance after the transfer is:
Q5. A card advertises 21.99% p.a. with daily compounding. The effective annual rate is:
SA 1. A credit card has a $4500 balance at 19.99% p.a. (a) Calculate the daily and monthly interest. (b) If the minimum payment is $112.50 (2.5%), how much principal is paid off in month 1? (c) Approximately how many months to clear the debt paying only the minimum? (2 marks)
SA 2. A balance transfer offers 0% for 15 months with a 1.5% fee on a $6000 debt currently at 20% p.a. (a) Calculate the transfer fee. (b) If you pay $400/month, will the debt be cleared before the promotional period ends? (c) Calculate the total saving compared to staying on the current card at $400/month. (2 marks)
SA 3. (a) A credit card charges 22% p.a. compounded monthly. The minimum payment is $25. For a $2000 balance, show that the debt can never be cleared with this minimum payment. (b) The same card offers 1% cashback on purchases. A customer spends $1000/month and carries a $3000 balance. Does the cashback exceed the monthly interest cost? (c) Explain mathematically why credit card debt at 20%+ p.a. is one of the most expensive forms of borrowing. (3 marks)
Comprehensive answers (click to reveal)
MC 1, B: $3000 × 0.20/12 = $50.00/month.
MC 2, C: Interest = $50. Payment = $60. Principal = $60 − $50 = $10.
MC 3, A: Interest-free period requires the full balance to be paid by the due date every month.
MC 4, D: Fee = $6000 × 0.02 = $120. New balance = $6000 + $120 = $6120.
MC 5, B: EAR = (1 + 0.2199/365)365 − 1 ≈ 24.6%, higher than the nominal rate.
SA 1 (2 marks): (a) Daily = $2.46, Monthly = $74.96 [0.5]. (b) Principal = 112.50 − 74.96 = $37.54 [0.5]. (c) n ≈ 280 months (~23 years) [1].
SA 2 (2 marks): (a) Fee = $90. New balance = $6090 [0.5]. (b) $6090/$400 = 15.2 months, just over 15, barely not cleared [0.5]. (c) Without transfer at $400/month on 20%: n ≈ 18 months, total = $7200. With transfer: $6090 total. Saving ≈ $1110 [1].
SA 3 (3 marks): (a) Monthly interest = $2000 × 0.22/12 = $36.67. Min payment = $25 < $36.67, balance grows by $11.67/month, debt never cleared [1]. (b) Cashback = $10/month. Interest = 3000 × 0.22/12 = $55/month. Net loss = $45/month [1]. (c) At 22% compounded daily, effective rate ≈ 24.6%, exceeding average investment returns (7–10%), wage growth (2–3%), and inflation (2–3%). Exponential growth works against the borrower; high rates and low minimums create a trap where balances persist for decades [1].
Drill 1: (a) $1.37/day. (b) $41.63/month. (c) EAR = (1+0.2199/365)365 − 1 ≈ 24.6%.
Drill 2: Fee = $160. New balance = $8160. Months = 8160/700 = 11.7, cleared before 12 months. Interest on original card at $700/month over ~13 months ≈ $600. Net saving ≈ $440. Transfer is worthwhile.
Five timed questions on credit card interest, minimum payments, effective rates, and balance transfers. Beat the boss to bank a tier, gold (90% + speed), silver (75%), or bronze (50%). Replays welcome.
⚔ Enter the arenaClimb platforms by answering questions on credit cards. Pool: lesson 9.
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