Question

For Exercises \(31-34,\) solve. Scott has a balance of \(\$ 1248.56\) on a credit card that has a \(21.9 \%\) 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 Scott's credit card after 30 days will be approximately $1335.75.

Step-by-step solution

Find the values for the variables in the formula

First, we will assign values for each variable in the compound interest formula: - P = 1248.56 (Initial balance) - r = 0.219 (Annual interest rate - Converted from 21.9% to a decimal by dividing by 100) - n = 365 (Daily compounding) - t = 30 / 365 (Time in years)

Substitute the values into the formula

Now, we will substitute the values of P, r, n, and t into the compound interest formula: \(A = 1248.56(1 + \frac{0.219}{365})^{(365)(\frac{30}{365})}\)

Perform the calculations

Evaluate the formula to find the balance after 30 days: \(A = 1248.56(1 + \frac{0.219}{365})^{(365)(\frac{30}{365})}\) \(A = 1248.56(1 + 0.0006)^{109.5}\) \(A \approx 1248.56(1.0006)^{109.5}\) \(A \approx 1248.56 * 1.0701\) \(A \approx 1335.747\)

Round the result and interpret

Finally, round the result to two decimal places and provide the final answer: \(A \approx 1335.75\) So, the balance on Scott's credit card after 30 days will be approximately \( \$ 1335.75\).

Key concepts

Credit Card Balance

Credit card balance refers to the amount of money owed on a credit card account. It includes all previous charges, fees, and any accumulated interest. In Scott's case, he owed $1248.56 at the start. Understanding your credit card balance is crucial because it determines how much interest you'll pay if you don't pay it off each month.
A high balance can lead to higher interest charges, which can accumulate over time. Being aware of your credit card terms, such as interest rates and payment cycles, helps you manage your finances effectively.

Daily Compounding

Daily compounding refers to the method by which interest is calculated and added to the account balance each day. This means that the interest is not only calculated on the initial amount but also on any interest that has been added to the account. Therefore, it results in paying interest on interest.
In the context of Scott's credit card balance, the interest rate of 21.9% is applied daily. This can increase the balance more quickly than other compounding frequencies, such as monthly or annually. Understanding daily compounding helps in anticipating the growth of debt and its future implications.

Interest Rate Calculation

Interest rate calculation is key to understanding how interest affects credit card balances. It involves converting the annual interest rate into a daily rate since credit card balances often use daily compounding.

• First, convert the annual interest rate percentage into a decimal by dividing by 100. For example, 21.9% becomes 0.219.
• Then, divide by the number of days in a year (365) to get the daily rate.
• In Scott's case, the daily rate is 0.219/365, which will be used in the compound interest formula.

This interest rate is then used in calculations to predict how a balance will grow over time when no payments are made.

Prealgebra Problem Solving

Prealgebra problem solving involves finding unknowns, using formulas, and interpreting results. It is essential for tackling real-world problems like calculating credit card balances with compound interest.
For instance, the compound interest formula \( A = P(1 + \frac{r}{n})^{nt} \) is a key tool. By substituting known values such as the initial balance (\(P\)), the daily interest rate (\(\frac{r}{365}\)), the number of compounding periods per year (\(n\)), and the length of time in days (\(t\)), you can solve for the future balance \(A\).
Problem-solving in prealgebra requires breaking the problem into manageable steps. Understand the formula, substitute the values, compute the result, and then interpret it in context. This approach allows for confident and accurate results in financial situations like Scott's credit card balance.