Credit Cards
Learn credit cards for NSW Year 12 Mathematics Standard 2. A credit card is a short-term reducing-balance loan: pay the closing balance in full within the interest-free period and you pay no interest, but carry a balance and interest is compounded daily at the annual rate divided by 365.
You will calculate the interest charged on a balance carried for a number of days, work out the minimum monthly repayment, see why paying only the minimum clears the debt so slowly, and read a credit-card statement — core borrowing skills for Standard 2 financial mathematics.
Theory
Credit cards are part of the loans topic in Year 12 Standard 2 (NSW). A card works like a short-term reducing-balance loan: pay the closing balance in full within the interest-free period and you pay no interest, but carry a balance and interest is compounded daily. This guide shows how to find the daily interest, work out the minimum repayment, and read a statement.
A credit card lets you borrow up to a limit and works like a short-term reducing-balance loan. Each statement period it offers an interest-free period (often up to 55 days): if you pay the closing balance in full by the due date, no interest is charged on purchases.
Carry any of the balance past the due date and interest is charged, usually compounded daily. The daily rate is the annual rate divided by \(365\), and the amount owed after \(n\) days is \(A=P\left(1+\dfrac{r}{365}\right)^{n}\), so the interest is \(I=A-P\). Cash advances usually get no interest-free days — interest runs from day one.
The minimum monthly payment is the greater of a fixed dollar amount or a set percentage of the closing balance. In this Year 12 Standard 2 (NSW) topic you also read a statement (opening balance, purchases, payments, interest, closing balance, minimum due) and see why paying only the minimum clears the debt very slowly.
Convert the annual rate to a daily rate:
With daily compounding, the amount owed after \(n\) days on a balance \(P\) (annual rate \(r\) as a decimal) is:
The minimum monthly payment is the greater of a fixed amount \(m\) or a percentage \(k\) of the closing balance \(B\):
How to work with a credit card
- Check the interest-free period. If the closing balance is paid in full by the due date, no interest is charged on purchases and you can stop.
- Find the daily rate \(=\) annual rate \(\div\,365\), and count the number of days \(n\) the balance is carried.
- Amount owed: \(A=P\left(1+\dfrac{r}{365}\right)^{n}\); the interest is \(I=A-P\). (For a rate quoted per month on the unpaid balance, compound monthly instead.)
- Minimum payment: take the greater of the fixed amount or the percentage of the closing balance, and interpret the statement figures.
Compare the payment day with the \(55\)-day interest-free period.
| \(\text{day }47\) | \(\le\) | \(55 \Rightarrow \text{in time},\ I=\$0\) |
| \(\text{day }62\) | \(>\) | \(55 \Rightarrow \text{charged over }62\text{ days}\) |
| \(A\) | \(=\) | \(2400\left(1+\dfrac{0.189}{365}\right)^{62}=\$2478.28\) |
| \(I\) | \(=\) | \(2478.28-2400=\$78.28\) |
Paying in time costs \(\$0\); paying late costs \(\$78.28\).
Use the daily rate \(\dfrac{0.2045}{365}\) in \(A=P\left(1+\dfrac{r}{365}\right)^{n}\).
| \(A\) | \(=\) | \(1850\left(1+\dfrac{0.2045}{365}\right)^{24}\) |
| \(\) | \(=\) | \(1850(1.000560)^{24}=\$1875.04\) |
| \(I\) | \(=\) | \(1875.04-1850=\$25.04\) |
(i) \(\$1875.04\) owed, (ii) \(\$25.04\) interest.
Take the greater of the two amounts, then subtract this month's interest.
| \(3\% \text{ of }1650\) | \(=\) | \(0.03\times1650=\$49.50\) |
| \(\text{minimum}\) | \(=\) | \(\max(\$30,\ \$49.50)=\$49.50\) |
| \(\text{interest}\) | \(=\) | \(0.018\times1650=\$29.70\) |
| \(\text{off the debt}\) | \(=\) | \(49.50-29.70=\$19.80\) |
Minimum \(\$49.50\), but only \(\$19.80\) comes off the balance — the debt falls slowly.
Interest on the opening balance, then closing \(=\) opening \(+\) purchases \(+\) interest \(-\) payments.
| Opening balance | $1200.00 |
| Purchases | $360.00 |
| Payments / credits | $500.00 |
| Interest (\(1.5\%\) of opening) | ? |
| Closing balance | ? |
| \(\text{interest}\) | \(=\) | \(0.015\times1200=\$18.00\) |
| \(\text{closing}\) | \(=\) | \(1200+360+18-500\) |
| \(\) | \(=\) | \(\$1078.00\) |
Interest \(\$18.00\), closing balance \(\$1078.00\).
Common pitfalls
Frequently asked questions
How is credit-card interest calculated?
If you do not pay the closing balance in full, interest is usually compounded daily. Divide the annual rate by 365 to get the daily rate, then the amount owed after n days is A = P(1 + r/365)^n, where P is the balance and r is the annual rate as a decimal. The interest is A minus P.
What is the interest-free period on a credit card?
It is a window, often up to 55 days, in which purchases are charged no interest as long as you pay the closing balance in full by the due date. If you carry any of the balance past the due date, interest is charged on purchases from the date you bought them.
Do cash advances get an interest-free period?
No. Cash advances (and usually balance transfers) normally have no interest-free days, so interest is charged from the day you take the cash. That is why cash advances on a credit card are expensive.
How is the minimum payment worked out?
The minimum monthly payment is the greater of a fixed dollar amount, such as $30, or a set percentage of the closing balance, such as 3%. You compare the two and pay whichever is larger.
Why does paying only the minimum take so long to clear the debt?
Because interest is added every month before your payment. On a $1650 balance at 1.8% per month, about $29.70 of interest is added, so a $49.50 minimum payment only reduces the debt by about $19.80. The balance falls very slowly and you pay a lot of interest overall.
How do you read a credit-card statement?
A statement shows the opening balance, any payments or credits, new purchases and cash advances, interest and fees, and the closing balance, plus the minimum payment due and the due date. The closing balance equals the opening balance plus purchases plus interest and fees minus payments.