Question

For Exercises \(31-34,\) solve. Kevin has a balance of \(\$ 860.20\) on a credit card that has a \(16.9 \%\) interest rate. The interest is compounded daily. If he does not charge any money to this account until after he receives his next bill ( 30 days), what will his balance be on the next bill?

Short answer

Answer: The balance on Kevin's credit card after 30 days will be approximately $882.79.

Step-by-step solution

Understand the Given Values

We are given the following values: 1. Initial balance: $860.20 2. Interest rate: 16.9% 3. Compounding period: daily (30 days) We need to find the balance after 30 days.

Convert Interest Rate to Decimal Value

To calculate the compound interest, we need to convert the given interest rate from percentage to a decimal value. Divide the given interest rate by 100. \[16.9\% = \frac{16.9}{100} = 0.169\]

Find Daily Interest Rate

Next, find the daily interest rate by dividing the annual interest rate by the number of days in a year: \[Daily \ Interest \ Rate = \frac{0.169}{365} \approx 0.000463\]

Calculate the Compound Interest

Now, we will apply the compound interest formula: \[ A = P \cdot (1 + r)^n \] Where: - \(A\) is the final balance after interest compounding - \(P\) is the principal (initial balance) - \(r\) is the daily interest rate - \(n\) is the number of days the interest is compounded Substitute the values: \[A = 860.20 \cdot (1 + 0.000463)^{30}\]

Solve for the Final Balance

Calculate the compounded balance: \[A \approx 860.20 \cdot (1.000463)^{30}\] \[A \approx 882.79\] The balance on Kevin's credit card after 30 days will be approximately $882.79.

Key concepts

Credit Card Balance Calculation

When calculating a credit card balance with daily compounded interest, it's crucial to understand how initial amounts grow over time due to interest. Each day adds a small amount of interest onto the previous day's balance. Over time, this compounding effect can significantly increase the total balance.

This calculation starts with identifying the principal balance, which is the amount owed before interest is added. For Kevin, this is $860.20. The next step is to apply the daily interest rate to this principal. Over a period of 30 days, each day's interest increases the amount slightly more for the next day's calculation. This compounding effect is what leads to the final balance.It's important to use the correct formula for compound interest, which is: \[ A = P \times (1 + r)^n \] where:

• \(A\) is the amount on the credit card after compounding interest.
• \(P\) is the principal balance.
• \(r\) is the daily interest rate.
• \(n\) refers to the number of days the interest is compounded.

This equation ensures that the compounding effect over the specified time period is accurately reflected in the balance.

Daily Interest Rate

The daily interest rate plays a crucial role in calculating compounded interest on credit card balances. It represents the fraction of the annual interest rate that applies to a single day. To find it, you divide the annual interest rate, expressed as a decimal, by the number of days in the year (usually 365 days).For Kevin's credit card, with a 16.9% annual interest rate, the daily interest becomes: \[ Daily \ Interest\ Rate = \frac{0.169}{365} \approx 0.000463 \] Converting to scientific terms, this means about 0.0463% is added to the balance each day.

This tiny daily addition may seem negligible, but over 30 days, it can culminate in a noticeable increase in the overall balance. Understanding how this number is derived is key to managing and projecting credit card costs effectively. Knowing the rate helps predict the future balance and adjust spending or repayment strategies accordingly.

Percentage to Decimal Conversion

Understanding percentage to decimal conversion is essential when dealing with interest rate calculations. This process involves changing the percentage figure into a form that's easier to work with in formulas and equations.

To convert a percentage to a decimal, you divide the percentage by 100. The general rule is: \[ Decimal \ Form = \frac{Percentage}{100} \] In Kevin's scenario, with an interest rate of 16.9%, the conversion process becomes: \[ 16.9\% = \frac{16.9}{100} = 0.169 \] This decimal form is necessary for use in the compound interest formula and in calculating daily interest rates. It simplifies the mathematical calculations needed to assess how interest affects the principal balance in question.

Recognizing when and how to convert percentages into decimals can streamline solving financial equations and ensure accuracy in financial planning and predictions.