Find the interest on a credit-card debt compounding daily practice questions | General Mathematics (Year 11, VCE Units 1 & 2)

Credit-card interest compounded daily

In this model, a credit-card debt compounds daily, so convert the annual percentage rate to a daily rate. Interest is charged only after the stated interest-free period ends

A=P\left(1+\frac{r}{36500}\right)^n,\quad I=A-P

Count the interest-free days Add the remaining days in the statement cycle to the further interest-free allowance $\displaystyle (30-20)+7=17\text{ interest-free days}$
Find charged days Subtract the total interest-free days from the whole span between purchase and repayment $\displaystyle n=35-17=18\text{ charged days}$
Form the daily multiplier Divide the annual percentage rate by 365 days and by 100 before adding one $\displaystyle R=1+\frac{19.2}{36{,}500}$
Separate amount and interest Raise the daily multiplier to the charged-day count, then subtract the original debt from the amount owed $\displaystyle A=P R^n,\quad I=A-P$

Worked example

A person puts a $2,400 purchase on a credit card charging 19.2% p.a. compounded daily. The purchase is made on day 20 of a 30-day statement cycle, the card allows a further 7 interest-free days, and the debt is repaid 35 days after purchase. Find the amount owed and the interest charged

P=\$2{,}400,\quad r=19.2\%,\quad \text{elapsed}=35\text{ days}

$\displaystyle (30-20)+7=17\text{ interest-free days}$Add the 10 days remaining in the statement cycle to the further 7 free days
$\displaystyle n=35-17=18\text{ charged days}$Subtract the interest-free period from the whole span
$\displaystyle A=\$2{,}400\left(1+\frac{19.2}{36{,}500}\right)^{18}\approx\$2{,}422.83$Apply the exact daily multiplier for the 18 charged days and round the amount once to cents
$\displaystyle I=\$2{,}422.83-\$2{,}400=\$22.83$Subtract the original debt from the amount owed to find the interest

Common mistakes

Using the annual rate as the daily rate

✕$\displaystyle A=\$1{,}800\left(1+\frac{15}{100}\right)^{18}\approx\$22{,}275.82$✓$\displaystyle A=\$1{,}800\left(1+\frac{15}{36{,}500}\right)^{18}\approx\$1{,}813.36$

Daily compounding uses the annual percentage rate divided by 365 days and by 100 to form the daily multiplier

Charging interest during the interest-free period

✕$\displaystyle n=35\text{ charged days}$✓$\displaystyle n=35-17=18\text{ charged days}$

Only the days after the interest-free period ends carry credit-card interest

Rounding the balance after every day

✕$\displaystyle A=\text{daily rounding applied 18 times}=\$2{,}422.82$✓$\displaystyle A=\$2{,}400\left(1+\frac{19.2}{36{,}500}\right)^{18}\approx\$2{,}422.83$

Carry the exact daily calculation through the chain and round the amount only when it is asked for

Difficulty levels

introUse a stated number of charged days with no interest-free period to find interest to the nearest cent

standardCount remaining statement-cycle days and a further free allowance before finding the amount and interest

challengeCount charged days from two calendar dates, handle a long free period and compare with no free period

examChoose the interest from full-span, annual-rate and intermediate-rounding errors

Questions about this topic

Why is the annual rate divided by 36500?+

Dividing by 365 changes an annual percentage to a daily percentage, and dividing by 100 changes the percentage to a decimal rate used in the multiplier

How are charged days found when there is an interest-free period?+

Find the whole elapsed span, subtract the remaining statement-cycle days and any further free days, and use the remainder as the charged-day count

How is credit-card interest separated from the amount owed?+

Subtract the original debt from the amount owed after daily compounding. The difference is the interest charged

Can I use a calculator?+

A calculator is allowed unless a question is marked no calculator

Which year level and chapter is this?+

Year 11, VCE Units 1 & 2, General Mathematics, chapter 3: Sequences and finance

In this chapter