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

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".

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.

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

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

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.

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:

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.

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.

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

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

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.

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

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 $n = 30$:

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.

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:

Convert the annual rate to a daily rate.

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

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.

Convert the annual rate to a daily rate.

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

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.

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.

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

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.

Convert the annual rate to a daily rate.

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

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:

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.

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

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:

Convert the annual rate to a daily rate.

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

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.

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

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.

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

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

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.

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

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:

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:

Convert the annual rate to a daily rate.

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

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.

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

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.

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

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

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.