Credit card interest, daily compounding and the cost of revolving debt for HSC Maths Standard 2

Syllabus dot point

“Calculate credit card interest using daily compounding, identify the interest-free period and the minimum monthly repayment”

HSCMaths Standard 2Year 12: Financial Mathematics12 min readUpdated 20 Jun 2026

Quick answer

Credit card interest is calculated using daily compounding $A = P(1 + R/365)^n$ where $n$ is the number of days the balance is outstanding; if the full balance is paid by the due date there is no interest, but if any is left unpaid the back-dating rule usually charges interest from the original purchase date on the whole balance, and paying only the minimum keeps the debt alive for years.

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What this dot point is asking
The answer
Exam-style questions
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What this dot point is asking

NESA wants you to do four things. Work out credit card interest using daily compounding (interest added once a day). Explain the interest-free period. Apply the back-dating rule. And explain why paying only the minimum keeps a balance alive for years. Australian credit card rates sit around $15$ - $22\%$ per annum (per RBA statistics). That is far above home-loan rates, which is what makes carrying a balance so expensive.

The answer

How credit card interest works

A credit card charges a high annual interest rate (typically $13$ - $22\%$ in Australia, per RBA statistics). That rate is compounded daily (added to the balance each day) on any balance carried past the due date. Two features make a credit card behave differently from a normal loan: an interest-free period, and a back-dating rule that switches it off.

If you pay the full closing balance by the due date, you are charged no interest on purchases made in that statement period. This is the interest-free period. On most Australian cards it can be up to $44$ to $55$ days. That is because it runs from the purchase date all the way to the due date of the statement the purchase lands in. The timeline below shows how it works.

If even $1 is left unpaid, most cards charge interest from the original purchase date on the entire balance, not just the unpaid part. This is the back-dating rule, and it is the major trap. The interest-free period is a reward for paying in full, and it vanishes entirely the moment you do not.

Daily rate

Annual rate divided by $365$ :

r_{\text{day}} = \frac{R}{365}.

A nominal $19.99\%$ per annum is a daily rate of $\frac{0.1999}{365} \approx 0.0005477$ , or about $0.055\%$ per day.

Key fact

Credit card interest uses daily compounding on the balance carried past the due date: $A = P\left(1 + \dfrac{R}{365}\right)^{n}$ , where $R$ is the annual rate as a decimal and $n$ is the number of calendar days the balance is outstanding. The interest charged is $A - P$ . If the full balance is paid by the due date there is no interest at all; if any is left unpaid, interest is usually back-dated to the purchase date on the whole balance.

Balance after $n$ days

Using compound interest with daily compounding:

A = P(1 + r_{\text{day}})^n.

Interest charged is $A - P$ . Here $n$ is a count of actual calendar days, so read the dates carefully: from $1$ June to $5$ August is $65$ days, not "two months".

For short periods the result is close to simple interest $I = P r_{\text{day}} n$ . This is because over a few weeks the interest-on-interest (interest charged on interest already added) is tiny. But the compound formula is what NESA expects, unless the question clearly says "simple interest". A question that says "calculated daily" is a signal to use daily compounding.

Effective annual rate

A daily-compounded rate has a slightly higher effective annual rate than the nominal rate, because the interest itself earns interest through the year:

r_{\text{eff}} = (1 + r_{\text{day}})^{365} - 1.

A nominal $20\%$ gives an effective annual rate of $(1 + 0.20/365)^{365} - 1 \approx 22.1\%$ . The headline rate understates the true annual cost.

Minimum monthly repayment

Cards usually require a minimum monthly repayment of about $2$ - $3\%$ of the closing balance (or a small fixed amount such as $25, whichever is greater). The trap is just arithmetic. The monthly interest on a typical card is itself close to that same percentage. So the minimum payment barely clears more than the interest, and the balance falls at a crawl.

The chart below tracks a $5000 balance at $19\%$ per annum. Paying only the minimum, the balance is still about $3000 after ten years. Paying a fixed $200 a month instead clears the whole debt in under three years.

Paying only the minimum means the balance reduces extremely slowly while interest keeps compounding, so the debt can persist for decades. Cards now legally have to print a minimum-repayment warning on every statement for exactly this reason.

Statement period

A typical statement period is one month. Interest is worked out daily on the balance and added to the account at the end of the statement period. That added interest then earns interest itself the next period. This is what makes it compounding rather than simple.

How exam questions ask about credit card interest

Each wording maps to one method:

"Calculate the interest charged on a balance of $X outstanding for $n$ days." Daily compounding: $I = P\left((1 + R/365)^n - 1\right)$ .
"Find the new balance after $n$ days." $A = P(1 + R/365)^n$ ; quote both $A$ and the interest $A - P$ if asked.
"A purchase is made on (date) and paid on (date)." Count the calendar days between the two dates for $n$ , and apply the back-dating rule if the balance was not paid in full by the due date.
"Identify / explain the interest-free period." From purchase date to the statement due date (up to about $55$ days), provided the closing balance is paid in full; no interest on those purchases.
"Find the minimum monthly repayment." A percentage (usually $2$ - $3\%$ ) of the closing balance, or a fixed floor, whichever is greater.
"Explain why paying only the minimum is expensive" or "how long to repay?" Compare the monthly interest with the minimum payment; most of the payment is interest, so the balance barely falls.
"What is the effective annual rate?" $(1 + R/365)^{365} - 1$ , slightly above the nominal rate.

Credit card interest: late payment, carried balance, and the minimum repayment trap

Three problems illustrate how daily compounding and minimum payments interact to make credit card debt expensive.

Single purchase paid late

A purchase of $800 is made on $1$ June. The statement closes on $30$ June and the due date is $25$ July ( $25$ days of grace). The cardholder does not pay until $5$ August, $66$ days after the purchase. The annual rate is $20\%$ .

Convert the annual rate to a daily rate: Credit card interest accrues daily, so divide the annual rate by $365$ to find the rate applied each day.

\text{Daily rate} = \frac{0.20}{365} \approx 0.0005479

Apply compound interest from the purchase date: Under the back-dating rule, interest runs from the purchase date (not the due date), because the balance was not paid in full by the due date. So all $66$ days are charged. Substitute into the compound interest formula.

A = 800 \times (1.0005479)^{66} \approx 800 \times 1.03682 \approx 829.45

i.e. a balance of $829.45. Interest charged: $29.45.

Final answer: The cardholder is charged approximately $29.45 in interest, giving a total of $829.45.

Carrying a balance (Australian context, 2025)

A cardholder has a balance of $3500 at $18.99\%$ per annum daily compounding. They make no purchases and no repayments for $30$ days.

Convert the annual rate to a daily rate: Divide $18.99\%$ by $365$ to obtain the daily compounding rate.

\text{Daily rate} = \frac{0.1899}{365} \approx 0.0005203

Compound the balance over 30 days: Substitute the daily rate and number of days into the compound interest formula to find the new balance.

A = 3500 \times (1.0005203)^{30} \approx 3500 \times 1.01573 \approx 3555.04

i.e. a balance of $3555.04. Interest accrued: $55.04 in one month.

Final answer: The balance grows to approximately $3555.04, with $55.04 in interest charged over $30$ days.

The minimum repayment trap

A $5000 balance at $19\%$ per annum with a $2\%$ minimum monthly payment. No further purchases are made and only the minimum payment is paid each month.

Calculate interest, minimum payment, and net balance reduction in month 1: The monthly interest rate is $\frac{0.19}{12}$ . The minimum payment is $2\%$ of the opening balance. The key is to compare the interest charge with the repayment to see how little of the balance is actually cleared.

\text{Monthly interest} = 5000 \times \frac{0.19}{12} \approx 79.17

\text{Minimum repayment} = 2\% \times 5000 = 100.00

\text{Net reduction in balance} = 100.00 - 79.17 = 20.83

\text{New balance} = 5000 - 20.83 = 4979.17

In dollars: the monthly interest is about $79.17, the minimum repayment is $100.00, the net reduction is $20.83, and the new balance is $4979.17.

Interpret the long-run consequence: Because nearly all of the minimum payment is consumed by interest, the balance falls by only about $20 in the first month, and as the balance and the minimum payment both shrink the progress only slows further. As the chart above shows, after ten years of minimum-only payments the balance is still around $3000. Paying a fixed $200 each month instead would clear the same debt in under three years, because the extra $100 goes straight onto the principal.

Final answer: After the first month the balance is $4979.17; paying only the minimum, the debt persists for decades, whereas a fixed $200 a month clears it in under three years.

Edge case: interest-free period versus a carried balance, side by side

Take the same $800 purchase from Part A, but suppose the cardholder pays the full closing balance by the $25$ July due date. Then the interest-free period applies and the interest charged is exactly $0, even though the money was effectively borrowed for over a month. But pay $799 and leave just $1 unpaid, and the back-dating rule kicks in. Interest is charged from $1$ June on the whole $800, not on the $1 shortfall. So leaving a single dollar unpaid can change the interest from $0 to tens of dollars. This all-or-nothing feature is the single most important thing to understand about credit cards, and a favourite exam "explain why" prompt.

Exam technique

State the daily rate $R/365$ as your first line and keep it to at least $6$ or $7$ decimal places; markers reward the per-day rate and the compound form even if the final figure is slightly out. Count calendar days carefully from the dates given (a "what if paid on (date)" question is testing exactly this), and remember February has $28$ days. Use the compound formula whenever the question says "calculated daily" or "compounded daily"; only use simple interest if the question says so. Quote both the new balance and the interest as a separate figure when asked. For "explain" questions about the interest-free period or the minimum repayment, the marks are for the back-dating rule (any unpaid amount triggers interest from the purchase date on the whole balance) and for the fact that the minimum barely exceeds the monthly interest.

Common traps

Using simple interest: Credit cards compound daily. Use $A = P(1 + R/365)^n$ , not $I = Prn$ , unless the question explicitly says simple.
Forgetting the back-dating rule: If the balance is not paid in full by the due date, interest typically applies from the original purchase date on the entire balance, including the part already paid.
Wrong number of days: Count actual calendar days, not weeks or months. February has $28$ days, not $30$ .
Annual rate substituted directly: Use the daily rate $R/365$ for daily compounding, then raise to the number of days.
Misreading minimum payment: $2\%$ of the balance is usually barely above the monthly interest charge. Paying the minimum barely reduces the balance, and the balance can persist for decades.
Thinking interest-free means free credit always: The interest-free period only applies if the closing balance is paid in full by the due date. Any carried balance loses it entirely.

Note

A credit card lets you buy things with the bank's money for a while. If you pay back the whole bill by the due date, the bank charges you nothing for that loan. But if you leave even a tiny bit unpaid, the bank goes back to the day you bought each thing and charges interest on the full amount, a little bit added on every single day. That daily charge keeps stacking up the longer you take to pay. Each month the bank lets you pay just a small "minimum" amount, but that is barely more than the interest, so the debt shrinks very slowly. The lesson is simple: pay the whole bill on time, or it can quietly cost you for years.

Exam-style questions

Questions in the style of NESA exam questions on this dot point, each with a worked answer. They are written by ExamExplained unless tagged "Past paper"; the year shows the paper a question is modelled on.

2023 HSC-style3 marks

A credit card charges

19.99\%

per annum interest, calculated daily and added monthly. Find the interest charged on a balance of $1500 outstanding for the full

30

days of a statement period.

Show worked answer →

Daily rate: $r = \frac{0.1999}{365} \approx 0.0005477$ .

Interest over $30$ days using daily compounding:

$I = 1500((1.0005477)^{30} - 1) \approx 1500(1.01656 - 1) \approx 1500 \times 0.01656 \approx 24.84$ , i.e. $24.84.

Markers reward the per-day rate, the compound formula (not simple interest), and the answer to cents. Half marks if simple interest is used, since the question says "calculated daily".

2021 HSC-style4 marks

A credit card has an annual interest rate of

21.5\%

compounded daily. On 1 May, the opening balance is $2400. No payments are made for

40

days. Find the new balance and the interest charged.

Show worked answer →

Daily rate: $r = \frac{0.215}{365} \approx 0.0005890$ .

Balance after $40$ days: $A = 2400(1.0005890)^{40}$ .

$(1.0005890)^{40} \approx 1.02383$ .

$A \approx 2400 \times 1.02383 \approx 2457.20$ , i.e. $2457.20.

Interest: $2457.20 - 2400 = 57.20$ , i.e. $57.20.

Markers reward the daily rate, the compound formula over the right number of days, and both the new balance and the interest as a separate figure.

Practice questions

Original practice questions graded from foundation to exam level, each with a full worked solution. Try them before revealing the solution.

foundation2 marks

A credit card charges

22\%

per annum, calculated daily and added monthly. A balance of $1200 is carried for the full

28

days of a statement period. (a) Find the daily interest rate as a decimal, to

7

decimal places. (b) Find the interest charged over the

28

days.

Show worked solution →

Find the daily rate. Credit card interest accrues daily, so divide the annual rate by $365$ :

r_{\text{day}} = \frac{0.22}{365} \approx 0.0006027.

Apply daily compounding for $28$ days. Use $A = P(1 + r_{\text{day}})^n$ with $P = 1200$ and $n = 28$ :

A = 1200 \times (1.0006027)^{28} \approx 1200 \times 1.01701 \approx 1220.42.

The interest is $A - P = 1220.42 - 1200 = 20.42$ , i.e. $20.42.

Check. Over only $28$ days the simple-interest estimate $1200 \times 0.0006027 \times 28 \approx 20.25$ is very close, confirming the figure of about $20.42 is the right size.

foundation2 marks

Mia buys a $540 textbook bundle on her credit card on

8

March. The statement closes on

31

March and the due date is

25

April. The card has an annual rate of

19.99\%

. Mia pays the full closing balance on

25

April and makes no other purchases. (a) How many days pass between the purchase and the due date? (b) How much interest is she charged on the purchase, and why?

Show worked solution →

Count the days from purchase to due date. From $8$ March to $31$ March is $23$ days, then a further $25$ days to $25$ April:

23 + 25 = 48 \text{ days}.

So the purchase sits on the card for $48$ days.

Apply the interest-free rule. Because Mia pays the full closing balance by the due date, the interest-free period applies to purchases in that statement period. No interest is charged, even though the $540 was effectively borrowed for $48$ days.

\text{Interest charged} = 0.

Check. The interest-free period runs from the purchase date to the due date, here $48$ days, which is within the usual $44$ to $55$ day window, so paying in full means $0 interest.

foundation2 marks

A credit card has an annual interest rate of

18.49\%

, calculated daily. A balance of $

980

is left unpaid and carried for

14

days. Find the interest charged over those

14

days.

Show worked solution →

Find the daily rate. Credit card interest accrues daily, so divide the annual rate by $365$ :

r_{\text{day}} = \frac{0.1849}{365} \approx 0.0005066.

Apply daily compounding for $14$ days. Use $A = P(1 + r_{\text{day}})^n$ with $P = 980$ and $n = 14$ :

A = 980 \times (1.0005066)^{14} \approx 980 \times 1.00712 \approx 986.97.

The interest is $A - P = 986.97 - 980 = 6.97$ , i.e. $6.97.

Check. Over just $14$ days the simple-interest estimate $980 \times 0.0005066 \times 14 \approx 6.95$ is very close, confirming about $6.97 is the right size.

Answer: The interest charged is approximately $6.97.

foundation2 marks

A store credit card charges

23.99\%

per annum, calculated daily. A balance of $

760

is carried for the full

30

days of a statement period with no repayment. (a) Find the daily interest rate as a percentage, to

2

decimal places. (b) Find the interest charged over the

30

days.

Show worked solution →

Find the daily rate. Divide the annual rate by $365$ :

r_{\text{day}} = \frac{0.2399}{365} \approx 0.0006573.

As a percentage this is about $0.07\%$ per day.

Apply daily compounding for $30$ days. Use $A = P(1 + r_{\text{day}})^n$ with $P = 760$ and $n = 30$ :

A = 760 \times (1.0006573)^{30} \approx 760 \times 1.01991 \approx 775.13.

The interest is $A - P = 775.13 - 760 = 15.13$ , i.e. $15.13.

Check. The simple-interest estimate $760 \times 0.0006573 \times 30 \approx 14.99$ is close to $15.13, so the figure is the right size. Store cards charge a high rate, so a small balance still costs over $ $15$ in a month.

Answer: The daily rate is about $0.07\%$ and the interest charged is approximately $15.13.

core3 marks

A $1450 purchase is made on a credit card on

3

February 2026. The due date is

28

March and the cardholder does not pay until

12

April 2026, having paid nothing by the due date. The annual rate is

19.49\%

compounded daily. (

2026

is not a leap year.) Find the interest charged and the amount finally owing.

Show worked solution →

Apply the back-dating rule to find the number of days. Because nothing was paid by the due date, interest runs from the original purchase date of $3$ February. Counting calendar days, $3$ to $28$ February is $25$ days (February has $28$ days in $2026$ ), plus $31$ days in March, plus $12$ days in April:

25 + 31 + 12 = 68 \text{ days}.

Convert the annual rate to a daily rate.

r_{\text{day}} = \frac{0.1949}{365} \approx 0.0005340.

Compound the balance over $68$ days. Use $A = P(1 + r_{\text{day}})^n$ :

A = 1450 \times (1.0005340)^{68} \approx 1450 \times 1.03697 \approx 1503.60.

The interest is $1503.60 - 1450 = 53.60$ , i.e. $53.60.

Check. February has $28$ days in $2026$ (not $30$ ), so the day count is $68$ ; the amount owing is about $1503.60 with $53.60 of interest.

core3 marks

A cardholder carries a balance of $2750 at

20.99\%

per annum compounded daily. No purchases and no repayments are made for

31

days. Find the new balance and state the interest charged as a separate figure.

Show worked solution →

Convert the annual rate to a daily rate.

r_{\text{day}} = \frac{0.2099}{365} \approx 0.0005751.

Compound the balance over $31$ days. Use $A = P(1 + r_{\text{day}})^n$ with $P = 2750$ and $n = 31$ :

A = 2750 \times (1.0005751)^{31} \approx 2750 \times 1.01798 \approx 2799.45.

State both figures. The new balance is approximately $2799.45, and the interest charged is $2799.45 - 2750 = 49.45$ , i.e. $49.45.

Check. A simple-interest estimate $2750 \times 0.0005751 \times 31 \approx 49.04$ is close to $49.45, confirming the new balance of about $2799.45 is right.

core3 marks

A credit card advertises a nominal annual rate of

17.99\%

compounded daily. (a) Explain why the effective annual rate is higher than the nominal rate. (b) Find the effective annual rate, correct to two decimal places.

Show worked solution →

Explain the difference. With daily compounding the interest charged each day is itself added to the balance, so it earns interest for the rest of the year. This interest-on-interest means the true yearly cost, the effective annual rate, is a little above the headline nominal rate.

Write the daily rate.

r_{\text{day}} = \frac{0.1799}{365} \approx 0.0004929.

Apply the effective-rate formula over a full year. Use $r_{\text{eff}} = (1 + r_{\text{day}})^{365} - 1$ :

r_{\text{eff}} = (1.0004929)^{365} - 1 \approx 1.1970 - 1 = 0.1970.

As a percentage this is about $19.70\%$ .

Check. The effective rate $19.70\%$ is slightly above the nominal $17.99\%$ , as expected for daily compounding, so the headline rate understates the real annual cost.

core3 marks

A cardholder carries a balance of $

3180

19.74\%

per annum compounded daily and makes no payments for

45

days. (a) Find the new balance and the interest charged. (b) What percentage of the new balance is interest, to one decimal place?

Show worked solution →

Convert the annual rate to a daily rate.

r_{\text{day}} = \frac{0.1974}{365} \approx 0.0005408.

Compound the balance over $45$ days. Use $A = P(1 + r_{\text{day}})^n$ with $P = 3180$ and $n = 45$ :

A = 3180 \times (1.0005408)^{45} \approx 3180 \times 1.02463 \approx 3258.32.

The interest is $3258.32 - 3180 = 78.32$ , i.e. $78.32.

Find the interest as a percentage of the new balance. Divide the interest by the new balance:

\frac{78.32}{3258.32} \times 100 \approx 2.4\%.

Check. The simple-interest estimate $3180 \times 0.0005408 \times 45 \approx 77.39$ is close to $78.32, confirming the new balance of about $3258.32.

Answer: The new balance is approximately $3258.32 with $78.32 of interest, about $2.4\%$ of the balance.

exam5 marks

Sam spends $2200 on a credit card on

1

May 2026. The statement closes on

31

May and the due date is

25

June. The annual rate is

20.74\%

compounded daily. (a) Find the interest if Sam pays the full closing balance by

25

June. (b) Find the interest if Sam instead pays nothing by the due date, assuming interest is back-dated to the purchase date and charged up to

25

June. (c) Use the two answers to explain the back-dating trap.

Show worked solution →

Part (a): apply the interest-free rule. Paying the full closing balance by the due date triggers the interest-free period on purchases in that statement period, so

\text{Interest} = 0.

Part (b): count the days from the purchase date. With nothing paid by the due date, interest is back-dated to $1$ May. From $1$ May to $31$ May is $30$ days, plus $25$ days to $25$ June:

30 + 25 = 55 \text{ days}.

Convert the annual rate to a daily rate.

r_{\text{day}} = \frac{0.2074}{365} \approx 0.0005682.

Compound the full balance over $55$ days. The back-dating rule charges interest on the whole $2200:

A = 2200 \times (1.0005682)^{55} \approx 2200 \times 1.03174 \approx 2269.82.

The interest is $2269.82 - 2200 = 69.82$ , i.e. $69.82.

Part (c): explain the trap. Paying in full costs $0, but missing the due date costs about $69.82 because interest is charged from the purchase date on the entire balance, not just on any unpaid part. The interest-free period is all-or-nothing: it is a reward for paying in full and vanishes the moment any amount is left unpaid.

Check. The day count $55$ lies in the usual $44$ to $55$ day window, and the difference between the two scenarios is the full $69.82, which is the whole interest charge.

exam5 marks

A $4200 credit card balance has an annual rate of

21.99\%

. Interest is charged monthly at

\frac{R}{12}

and the minimum repayment is the greater of

2.5\%

of the opening balance or $30. No further purchases are made and only the minimum is paid each month. Each month, add the interest then subtract the payment. Find the closing balance after

3

months and comment on what this shows.

Show worked solution →

Find the monthly interest rate. Divide the annual rate by $12$ :

\frac{0.2199}{12} \approx 0.018325.

Work through month $1$ . Interest is $4200 \times 0.018325 \approx 76.97$ . The minimum is $2.5\%$ of $4200, i.e. $0.025 \times 4200 = 105.00$ , which is more than $30, so the payment is $105.00.

\text{Closing}_1 = 4200 + 76.97 - 105.00 = 4171.97.

Work through month $2$ . Interest is $4171.97 \times 0.018325 \approx 76.45$ . The minimum is $0.025 \times 4171.97 \approx 104.30$ .

\text{Closing}_2 = 4171.97 + 76.45 - 104.30 = 4144.12.

Work through month $3$ . Interest is $4144.12 \times 0.018325 \approx 75.94$ . The minimum is $0.025 \times 4144.12 \approx 103.60$ .

\text{Closing}_3 = 4144.12 + 75.94 - 103.60 = 4116.46.

Comment. After three months the balance has fallen from $4200 to only about $4116.46, a drop of roughly $84, because most of each payment is swallowed by interest. Paying only the minimum keeps the debt alive for years.

Check. The month $1$ net reduction is $105.00 - 76.97 = 28.03$ , so $28.03; three roughly equal monthly reductions of about $28 give a total near $84, matching the closing balance of about $4116.46.

exam5 marks

Priya buys a $

1750

laptop on her credit card on

12

August 2026. The statement closes on

31

August and the due date is

25

September. The annual rate is

22.49\%

compounded daily. (a) State the interest charged if Priya pays the full closing balance by

25

September, and give the length of the interest-free period in days. (b) Instead, Priya pays nothing by the due date and clears the card on

10

October, so interest is back-dated to the purchase date. Find the interest charged. (c) Explain what part (b) shows about leaving any balance unpaid.

Show worked solution →

Part (a): apply the interest-free rule. Paying the full closing balance by the due date triggers the interest-free period on purchases in that statement period, so

\text{Interest} = 0.

The interest-free period runs from the purchase date to the due date. From $12$ August to $31$ August is $19$ days, plus $25$ days to $25$ September:

19 + 25 = 44 \text{ days}.

Part (b): count the days from the purchase date. With nothing paid by the due date, interest is back-dated to $12$ August. From $12$ August to $31$ August is $19$ days, plus $30$ days in September, plus $10$ days in October:

19 + 30 + 10 = 59 \text{ days}.

Convert the annual rate to a daily rate.

r_{\text{day}} = \frac{0.2249}{365} \approx 0.0006162.

Compound the full balance over $59$ days. The back-dating rule charges interest on the whole $ $1750$ :

A = 1750 \times (1.0006162)^{59} \approx 1750 \times 1.03701 \approx 1814.77.

The interest is $1814.77 - 1750 = 64.77$ , i.e. $64.77.

Part (c): explain the trap: Paying in full costs $0, but missing the due date costs about $64.77 because interest is charged from the purchase date on the entire balance, not just on any unpaid part. The interest-free period is a reward for paying in full and disappears the moment any amount is carried.
Check: The interest-free window of $44$ days lies inside the usual $44$ to $55$ day range, and the $64.77 charge is the whole cost of losing it.
Answer: (a) $0 interest, with a $44$ day interest-free period. (b) about $64.77. (c) leaving any balance unpaid back-dates interest to the purchase date on the full amount.

exam5 marks

A $

3600

credit card balance has an annual rate of

20.49\%

. Interest is charged monthly at

\frac{R}{12}

and the minimum repayment is the greater of

2\%

of the opening balance or $

25

. No further purchases are made and only the minimum is paid each month. Each month, add the interest then subtract the payment. (a) Find the closing balance after

2

months. (b) Find the net reduction in the balance in each month and explain why the balance falls so slowly.

Show worked solution →

Find the monthly interest rate. Divide the annual rate by $12$ :

\frac{0.2049}{12} \approx 0.017075.

Work through month $1$ . Interest is $3600 \times 0.017075 \approx 61.47$ . The minimum is $2\%$ of $ $3600$ , i.e. $0.02 \times 3600 = 72.00$ , which is more than $ $25$ , so the payment is $72.00.

\text{Closing}_1 = 3600 + 61.47 - 72.00 = 3589.47.

Work through month $2$ . Interest is $3589.47 \times 0.017075 \approx 61.29$ . The minimum is $0.02 \times 3589.47 \approx 71.79$ .

\text{Closing}_2 = 3589.47 + 61.29 - 71.79 = 3578.97.

Part (b): find each net reduction. Subtract the interest from the payment each month:

\text{Month } 1: 72.00 - 61.47 = 10.53, \qquad \text{Month } 2: 71.79 - 61.29 = 10.50.

So the balance falls by only about $10.53 then $10.50, a total near $21 over two months.

Explain: Almost all of each minimum payment is swallowed by interest, leaving barely $ $10$ to clear the actual debt. Because the balance and the $2\%$ minimum both shrink together, the progress only slows, so paying the minimum keeps the debt alive for years.
Check: Closing after two months is about $3578.97, a drop of $3600 - 3578.97 = 21.03$ , matching the two net reductions of about $ $10.50$ each.
Answer: (a) the closing balance is approximately $3578.97. (b) net reductions of about $10.53 and $10.50, because most of each payment covers interest.

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Mathematics Standard Stage 6 Syllabus (2017) — NESA
Mathematics Standard HSC exam papers — NESA

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Daily rate

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Effective annual rate

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Single purchase paid late

Carrying a balance (Australian context, 2025)

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Edge case: interest-free period versus a carried balance, side by side

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Balance after $n$ days