Ali has a credit card which has no interest-free period. Interest is charged at 13.5% per annum, compounding daily, on the amount owing. During the month, Ali made only one purchase of $450 using the credit card. The full amount owing was repaid 21 days later. (a) Calculate the amount of interest charged on the purchase, assuming that interest is charged for the 21 days. (b) What percentage of the full amount repaid is the interest? Give the answer to two decimal places.

SSCE HSC Mathematics Standard Annuities: Ali has a credit card which has

Ali has a credit card which has no interest-free period - HSC - SSCE Mathematics Standard - Question 32 - 2023 - Paper 1

Question 32

Ali has a credit card which has no interest-free period. Interest is charged at 13.5% per annum, compounding daily, on the amount owing. During the month, Ali made ... show full transcript

Worked Solution & Example Answer:Ali has a credit card which has no interest-free period - HSC - SSCE Mathematics Standard - Question 32 - 2023 - Paper 1

Step 1

Calculate the amount of interest charged on the purchase, assuming that interest is charged for the 21 days.

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Answer

To calculate the total amount owed after 21 days of interest compounding, we use the formula for compound interest:

$A = P \left(1 + \frac{r}{n}\right)^{nt}$

Where:

$A$ is the amount of money accumulated after n days, including interest.
$P$ is the principal amount (the initial amount of money), which is $450.
$r$ is the annual interest rate (decimal), so for 13.5% it is $0.135$ .
$n$ is the number of times that interest is compounded per year, which is 365 for daily compounding.
$t$ is the time the money is invested for in years, which for 21 days is $\frac{21}{365}$ .

Substituting the values:

$A = 450 \left(1 + \frac{0.135}{365}\right)^{365 \times \frac{21}{365}}$

Calculating this:

$A \approx 450 \left(1 + 0.000369863$ \right)^{21} \approx 450 \times 1.0026 \approx 453.51$$

The interest charged is thus:

$\text{Interest charged} = A - P = 453.51 - 450 = 3.51$

Step 2

What percentage of the full amount repaid is the interest? Give the answer to two decimal places.

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Answer

To find the percentage of the total repayment that was the interest, we use the formula:

$\text{Percentage} = \left( \frac{\text{Interest}}{\text{Total Amount Repaid}} \right) \times 100$

In this case:

Interest = $3.51
Total Amount Repaid = $453.51

Substituting the values:

$\text{Percentage} = \left( \frac{3.51}{453.51} \right) \times 100 \approx 0.77\%$

Thus, the percentage of the full amount repaid that is interest is approximately $0.77\%$ .

Ali has a credit card which has no interest-free period - HSC - SSCE Mathematics Standard - Question 32 - 2023 - Paper 1

Worked Solution & Example Answer:Ali has a credit card which has no interest-free period - HSC - SSCE Mathematics Standard - Question 32 - 2023 - Paper 1

Calculate the amount of interest charged on the purchase, assuming that interest is charged for the 21 days.

What percentage of the full amount repaid is the interest? Give the answer to two decimal places.

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