Interest on a Credit Card Balance
Learn how interest on a credit card balance is worked out for NSW Year 12 Mathematics Standard 1 (NESA outcome MST-12-S1-03). Credit card interest is compound interest charged every day, so an unpaid balance grows a little more each day it is carried.
You will work out the compound interest added to a purchase or outstanding balance over a set number of days, by changing the annual rate to a daily rate, finding the amount owed and subtracting to find the interest — so you can check the interest on your statement and compare cards with different rates.
Theory
Credit card interest is compound interest charged every day. In this Year 12 Mathematics Standard 1 (NSW) topic you change the annual rate to a daily rate, grow the balance with \(A=P\left(1+\dfrac{r}{100}\right)^{n}\) over the number of days it is carried, and subtract to find the interest charged.
Credit card interest is compound interest charged every day. The daily rate is the annual rate divided by \(365\), e.g. \(18.25\%\) p.a. \(\div\,365=0.05\%\) per day.
The amount owed after \(n\) days is \(A=P\left(1+\dfrac{r}{100}\right)^{n}\), where \(P\) is the balance, \(r\) is the daily percentage rate and \(n\) is the number of days. The interest charged is \(I=A-P\).
Compounded daily means each day's interest is added to the balance, and the next day's interest is charged on that larger balance, so the amount owed keeps growing while it is unpaid. This is how you check the interest on a statement and compare cards with different rates.
Change the annual rate to a daily percentage rate:
Amount owed after \(n\) days (\(r\) is the daily rate as a percentage):
Interest charged:
How to find the interest on a credit card balance
- Find the daily rate: annual rate \(\div\,365\) (skip this if the daily rate is given).
- Substitute into \(A=P\left(1+\dfrac{r}{100}\right)^{n}\) with \(n\) the number of days, and evaluate on your calculator.
- Subtract the balance if the question asks for the interest: \(I=A-P\). Round to the nearest cent.
Divide the annual rate by \(365\).
| \(\text{daily rate}\) | \(=\) | \(\dfrac{20.075}{365}\) |
| \(\) | \(=\) | \(0.055\) |
The daily rate is \(0.055\%\) per day.
Substitute \(P=1300\), \(r=0.035\), \(n=28\).
| \(A\) | \(=\) | \(1300\left(1+\dfrac{0.035}{100}\right)^{28}\) |
| \(\) | \(=\) | \(1300(1.00035)^{28}\) |
| \(\) | \(=\) | \(1312.80\ldots\) |
The amount owed is \(\$1312.80\).
Find the daily rate, compound over \(45\) days, then subtract the balance.
| \(\text{daily rate}\) | \(=\) | \(16.425\div365=0.045\%\) |
| \(A\) | \(=\) | \(2400(1.00045)^{45}\) |
| \(\) | \(=\) | \(2449.08\) |
| \(I\) | \(=\) | \(2449.08-2400\) |
| \(\) | \(=\) | \(49.08\) |
The interest charged is \(\$49.08\).
Find the interest on each card, then subtract.
| \(I_X\) | \(=\) | \(1600(1.00055)^{40}-1600\) |
| \(\) | \(=\) | \(35.58\) |
| \(I_Y\) | \(=\) | \(1600(1.00035)^{40}-1600\) |
| \(\) | \(=\) | \(22.55\) |
| \(\text{difference}\) | \(=\) | \(35.58-22.55=13.03\) |
Card X charges \(\$13.03\) more interest.
Common pitfalls
Frequently asked questions
How is interest calculated on a credit card balance?
Credit card interest is usually compounded daily. Divide the annual rate by 365 to get the daily percentage rate r, then the amount owed after n days is A = P(1 + r/100)^n. The interest charged is A minus P.
How do you find the daily interest rate on a credit card?
Divide the annual percentage rate by 365. For example, 20.075 percent per year divided by 365 is 0.055 percent per day.
What does compounded daily mean?
Each day the interest is added to the balance, and the next day's interest is charged on that larger balance. So the amount owed grows a little faster every day the balance is unpaid.
How much interest will I pay on a credit card for 30 days?
Use A = P(1 + r/100)^n with n = 30. For a 2000 dollar balance at 0.06 percent per day, A is about 2036.31 dollars, so the interest is 36.31 dollars.
Why does a small difference in daily rate matter?
It is compounded over every day the balance is carried. On 1600 dollars for 40 days, 0.055 percent per day costs 35.58 dollars but 0.035 percent per day costs 22.55 dollars, a difference of 13.03 dollars.