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Elizabeth went on a fabulous vacation in May and racked up a lot of charges on her credit card. When it came time to pay her June credit card bill, she left a balance of \(\$ 1200\). Elizabeth's credit card billing cycle runs from the nineteenth of each month to the eighteenth of the next month, and her interest rate is \(19.5 \% .\) She started the billing cycle June \(19-\) July 18 with a previous balance of \(\$ 1200 .\) In addition, she made three purchases, with the dates and amounts shown in Table 10-11. On July 15 she made an online payment of \(\$ 500.00\) that was credited to her balance the same day. (a) Find the average daily balance on the credit card account for the billing cycle June 19 -July 18 . (b) Compute the interest charged for the billing cycle June 19-July 18 . (c) Find the new balance on the account at the end of the June 19 -July 18 billing cycle. $$ \begin{array}{|c|c} \hline \text { Date } & \text { Amount of purchase/payment } \\ \hline 6 / 21 & \$ 179.58 \\ \hline 6 / 30 & \$ 40.00 \\ \hline 7 / 5 & \$ 98.35 \\ \hline 7 / 15 & \text { Payment } \$ 500.00 \end{array} $$

Step by step solution

01

Calculate the total balance for each day

Given a previous balance of \$1200 and the additional amounts spent on certain days which are: \$179.58 on day 3 (June 21), \$40 on day 12 (June 30), and \$98.35 on day 17 (July 5), calculate the total balance for each day until the payment of \$500 on day 27 (July 15). For example, from day 1 to day 3, the total balance remains \$1200. On day 3, she adds \$179.58 resulting in a balance of \$1379.58. This continues until the payment on day 27.

02

Calculate the total daily balance

Sum up the balance of all the days up until the payment. From day 1 to day 3, it's 3 * 1200 = \$3600. From day 3 to day 12, it's 9 * 1379.58 = \$12416.22. From day 12 to day 17, it's 5 * 1419.58 = \$7097.9. From day 17 to day 27, it's 10 * 1517.93 = \$15179.3. So, the total daily balance from day 1 to day 27 is: \$3600 + \$12416.22 + \$7097.9 + \$15179.3 = \$38293.42. Then add the daily balance from day 27 to day 30, which is 4 * 1017.93 = \$4071.72. So the total daily balance for the entire billing cycle is \$38293.42 + \$4071.72 = \$42365.14

03

Calculate the average daily balance

To calculate the average daily balance, divide the total daily balance by the number of days in the billing cycle, which is 30 days in this case. Hence, the average daily balance is \$42365.14 / 30 = \$1412.17.

04

Compute the credit card interest

The interest rate is 19.5%. However, this is likely the annual rate so you have to divide this by 12 to get the monthly rate. Hence, the monthly interest rate is 19.5/12 = 1.625%. To get the interest charged for the billing cycle compute 1.625% of the average daily balance: 0.01625 * \$1412.17 = \$22.94.

05

Compute the new balance

The new balance for the next billing cycle is simply the balance at the end of the current cycle, \$1017.93, plus the interest charged, \$22.94. Hence, the new balance at the end of this cycle is \$1017.93 + \$22.94 = \$1040.87.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Average Daily Balance

The average daily balance is an essential concept in understanding credit card billing cycles. It represents the average amount you owe on your credit card each day during a billing cycle, and is used primarily for calculating interest. To find it, you'll first need to determine the balance for each day.

Follow these steps:

• List all transactions, including purchases and payments, within the billing cycle.
• Calculate the outstanding balance for each day. Adjust the balance on the days transactions occur.
• Sum the daily balances for all days in the cycle to get a total daily balance.
• Finally, divide this total daily balance by the number of days in the billing cycle.

For example, if you have daily balances over 30 days that total to $42,365.14, the average daily balance would be $42,365.14 divided by 30, which equals $1412.17. Understanding this helps in predicting interest and managing finances better.

Interest Calculation

Interest calculation is where the average daily balance is put into practical use. Credit card companies often state an annual interest rate, which must be converted to a monthly rate for billing purposes.

Here's how you can calculate credit card interest:

• Divide the annual interest rate by 12 to find the monthly rate. For example, an annual rate of 19.5% results in a monthly rate of 1.625%.
• Apply the monthly rate to the average daily balance over the billing cycle to find the interest charged. In our case, multiply $1412.17 by 0.01625 to calculate an interest of $22.94.

Interest calculation is crucial for understanding how much extra you might pay if you carry a balance on your credit card. It’s important to manage your balance to minimize interest payments.

Financial Problem-Solving

Financial problem-solving involves identifying practical solutions to various credit and cash flow issues, such as high interest charges or mounting debt. By understanding the mechanisms of average daily balances and interest calculation, you empower yourself to make informed financial decisions.

To tackle such problems effectively:

• Keep track of your transactions within each billing cycle.
• Analyze spending patterns to highlight problematic habits, like frequent small purchases that add up.
• Consider making payments before the billing cycle ends to reduce daily balances early.

Problem-solving requires continual learning and application of financial concepts, paving the way toward better money management and reduced debt loads.

Mathematics Education

Mathematics education plays a key role in understanding and utilizing concepts like average daily balance and interest calculation, which are essential for personal finance management. Learning to solve real-world problems, like calculating credit card balances, highlights the importance of math in everyday life.

Education emphasizes:

• Basic arithmetic skills, such as addition, subtraction, and division, necessary for calculating balances and interest.
• Understanding percentages, which are vital for interest rate calculations.
• Developing problem-solving skills to apply mathematical concepts to financial scenarios.

By boosting math skills, students gain the confidence and competence needed to manage personal finance, fostering financial literacy early and enabling informed decision-making as adults.