Answer :
To solve this problem, we need to calculate the interest accrued on a credit card balance with an annual percentage rate (APR) of 22.79% over one day and thirty days. Here are the steps to find out how much you would save by paying 1 day late compared to paying 30 days late:
1. Convert APR to Daily Interest Rate:
The credit card interest compounds daily. Therefore, we need to convert the APR to a daily interest rate.
[tex]\[ \text{Daily Interest Rate} = \frac{\text{APR}}{365} \][/tex]
Given the APR is 22.79%:
[tex]\[ \text{Daily Interest Rate} = \frac{22.79\%}{365} \approx 0.000624 \][/tex]
2. Calculate Amount Owed After 1 Day:
To find out how much you owe after one day of interest:
[tex]\[ \text{Amount After 1 Day} = \text{Balance} \times \left(1 + \text{Daily Interest Rate}\right)^1 \][/tex]
Given the balance is [tex]$4,750: \[ \text{Amount After 1 Day} = 4750 \times (1 + 0.000624) \approx 4752.97 \] 3. Calculate Amount Owed After 30 Days: To find out how much you owe after thirty days of interest: \[ \text{Amount After 30 Days} = \text{Balance} \times \left(1 + \text{Daily Interest Rate}\right)^{30} \] \[ \text{Amount After 30 Days} = 4750 \times (1 + 0.000624)^{30} \approx 4839.78 \] 4. Calculate the Savings: Finally, to find out how much you save by paying 1 day late compared to paying 30 days late, subtract the amount owed after one day from the amount owed after thirty days: \[ \text{Savings} = \text{Amount After 30 Days} - \text{Amount After 1 Day} \] \[ \text{Savings} = 4839.78 - 4752.97 \approx 86.82 \] So, by paying 1 day late instead of 30 days late, you would save approximately $[/tex]86.82.
1. Convert APR to Daily Interest Rate:
The credit card interest compounds daily. Therefore, we need to convert the APR to a daily interest rate.
[tex]\[ \text{Daily Interest Rate} = \frac{\text{APR}}{365} \][/tex]
Given the APR is 22.79%:
[tex]\[ \text{Daily Interest Rate} = \frac{22.79\%}{365} \approx 0.000624 \][/tex]
2. Calculate Amount Owed After 1 Day:
To find out how much you owe after one day of interest:
[tex]\[ \text{Amount After 1 Day} = \text{Balance} \times \left(1 + \text{Daily Interest Rate}\right)^1 \][/tex]
Given the balance is [tex]$4,750: \[ \text{Amount After 1 Day} = 4750 \times (1 + 0.000624) \approx 4752.97 \] 3. Calculate Amount Owed After 30 Days: To find out how much you owe after thirty days of interest: \[ \text{Amount After 30 Days} = \text{Balance} \times \left(1 + \text{Daily Interest Rate}\right)^{30} \] \[ \text{Amount After 30 Days} = 4750 \times (1 + 0.000624)^{30} \approx 4839.78 \] 4. Calculate the Savings: Finally, to find out how much you save by paying 1 day late compared to paying 30 days late, subtract the amount owed after one day from the amount owed after thirty days: \[ \text{Savings} = \text{Amount After 30 Days} - \text{Amount After 1 Day} \] \[ \text{Savings} = 4839.78 - 4752.97 \approx 86.82 \] So, by paying 1 day late instead of 30 days late, you would save approximately $[/tex]86.82.
Answer:
$86.00
Step-by-step explanation:
To determine the amount of money saved by paying one day late instead of 30 days late on a credit card balance with an APR of 22.79%, we need to compare the interest accrued over 1 day and 30 days.
The Average Daily Balance (ADB) method is the most widely used method by credit card issuers to calculate interest payments. This method calculates interest by taking the sum of the daily balances over the billing cycle, dividing it by the number of days in the cycle to find the average balance, and then applying the daily periodic rate, which is derived from the Annual Percentage Rate (APR).
Since no additional payments were made between the due date and the late payment dates, the average daily balance (ADB) remains $4,750.
The daily periodic rate (DPR) is calculated by dividing the annual periodic rate APR by 365, which is the number of days in a year:
[tex]DPR = \dfrac{APR}{365}=\dfrac{0.2279}{365}[/tex]
To find the interest accrued if the payment was one day late, multiply the DPR by the ADB and by 1 day:
[tex]\text{Interest}=\dfrac{0.2279}{365} \times 4750 \times 1=\$2.97[/tex]
To find the interest payment if the payment was 30 days late, multiply the DPR by the ADB and by 30 days:
[tex]\text{Interest}=\dfrac{0.2279}{365} \times 4750 \times 30=\$88.97[/tex]
The amount of money saved by paying one day late instead of 30 days late can be calculated by finding the difference between the interest accrued over the two different periods:
[tex]\text{Money saved}=\$88.97-\$2.97=\$86.00[/tex]
Therefore, the amount of money saved by paying 1 day late instead of 30 days late is:
[tex]\LARGE\boxed{\boxed{\$86.00}}[/tex]
