Question

For Exercises \(31-34,\) solve. Phyllis has a balance of \(\$ 285.32\) on a credit card that has a \(7.9 \%\) interest rate. The interest is compounded daily. If she does not charge any money to this account until after she receives her next bill ( 30 days), what will her balance be on the next bill?

Short answer

Based on the given information and using the compound interest formula, the new balance on Phyllis's credit card after 30 days with an interest rate of 7.9% compounded daily will be approximately $287.16.

Step-by-step solution

Identify the formula for compound interest

The formula we will use for this problem is: \(A = P(1 + \frac{r}{n})^{nt}\) Where: A = the future value of the investment/loan, including interest P = the principal investment amount (the initial deposit or loan amount) r = the annual interest rate (decimal) n = the number of times that interest is compounded per year t = the number of years In our case, we have: P = $285.32 r = 7.9 % = 0.079 t = 30 days = \(\frac{30}{365}\) years (as we need to convert 30 days into years) n = 365 (interest is compounded daily)

Substitute the values into the compound interest formula

Now, we will substitute the values provided into the formula to solve for A (new balance). \(A = 285.32(1 + \frac{0.079}{365})^{365 \times \frac{30}{365}}\)

Solve for A

Simplify the expression inside the parentheses first: \(A = 285.32 (1 + 0.0002164)^{30}\) Next, solve the expression inside the parentheses and raise it to the power of 30: \(A = 285.32 (1.0002164)^{30}\) Finally, multiply the principal amount with the expression: \(A ≈ 285.32 × 1.00654 = 287.16\)

Interpret the result

The balance on Phyllis's credit card after 30 days will be approximately $287.16.

Key concepts

Interest Compounding

Understanding interest compounding can be enlightening when managing personal finances, especially when dealing with savings accounts or loans. Compounding is the process where the value of an investment increases because the earnings on an investment, both capital gains and interest, earn interest as time passes. This concept is powerful because it can turn small amounts of money into large sums over time and can make credit card debt grow faster than one might anticipate.

The frequency of compounding can make a significant difference to the total interest earned or paid. In Phyllis's situation, her credit card compounds interest daily, which means that the interest accrued each day is added to the principal sum, and the next day's interest is calculated on the new balance. It's essential for Phyllis to understand that even a short period like 30 days can make a noticeable difference to the amount owed due to the 'snowballing' effect of compounding interest.

Financial Mathematics

Financial mathematics is a branch of applied mathematics that deals with financial markets assessment and analysis. It leans heavily on concepts from probability, statistics, and complex algebraic structures. In the realm of personal finance, like credit card debt, the application of compound interest is a practical skill from financial mathematics.

With the provided exercise, we witness firsthand how financial mathematics comes alive. By applying the correct formula and numerical values, Phyllis can precisely calculate what her balance will be on the next credit card bill. This insight aids individuals in planning and understanding financial implications over time, emphasizing the importance of financial literacy. When these mathematical computations are translated into real-world scenarios, individuals can make informed decisions regarding saving, investing, or borrowing money.

Algebraic Expressions

Algebraic expressions are a foundational stone in algebra and permit the representation of numbers and their relationships through variables and constants. In the compound interest formula, algebra is used to encapsulate the growth trend of investments or loans over time. The expression given, \(A = P(1 + \frac{r}{n})^{nt}\), is an exponentiation example, where the values for the principal amount, interest rate, compounding frequency, and time are all variables and may be adjusted based on the specific circumstances of the investment or loan.

To solve the problem for Phyllis's credit card balance, we substitute the known values into the algebraic expression and perform the necessary algebraic manipulations. This simplification process is crucial as it transforms the compound interest formula into a number that Phyllis can expect to see on her bill. Mastering algebraic expressions is therefore not only key to solving abstract mathematical problems but also in managing real-life financial situations effectively.