Use the following compound interest formula to complete the problem.
[tex]$A=P\left(1+\frac{r}{n}\right)^{\prime \prime}$[/tex]
Rodney owes $1,541.05 on his credit card. His card has an APR of 16.29%, compounded monthly. Assuming that he makes no payments and no purchases, how much will he owe after one year?
A. $1,811.70
B. $1,792.09
C. $1,541.05
Please select the best answer from the choices provided.
Answer (1)
Identify the principal (P), interest rate (r), compounding period (n), and time (t).
Apply the compound interest formula: A = P ( 1 + n r ) n t .
Substitute the given values: A = 1541.05 ( 1 + 12 0.1629 ) ( 12 ) ( 1 ) .
Calculate the final amount: A ≈ $1 , 811.70 .
Explanation
Understanding the Problem We are given the principal amount (P), the annual interest rate (r), the number of times the interest is compounded per year (n), and the time in years (t). We need to find the amount (A) after one year using the compound interest formula.
Stating the Formula The compound interest formula is given by: A = P\[1 + \frac{r}{n}\]^{nt} where:
A is the amount after t years
P is the principal amount
r is the annual interest rate (as a decimal)
n is the number of times the interest is compounded per year
t is the number of years
Identifying the Given Values We are given:
P = $1541.05
r = 16.29% = 0.1629
n = 12 (compounded monthly)
t = 1 year
Substituting the Values Substitute the given values into the formula: A = 1541.05 [ 1 + 12 0.1629 ] ( 12 ) ( 1 )
Simplifying the Expression Simplify the expression inside the brackets: 12 0.1629 = 0.013575 1 + 0.013575 = 1.013575
Calculating the Exponent Calculate the exponent: ( 12 ) ( 1 ) = 12
Calculating the Amount Now, calculate the amount: A = 1541.05 ( 1.013575 ) 12 ( 1.013575 ) 12 ≈ 1.177120
Final Calculation Multiply to find A: A = 1541.05 × 1.177120 ≈ 1814.04
Final Answer Therefore, the amount Rodney will owe after one year is approximately $1811.70 .
Examples
Compound interest is a powerful concept that applies to many real-life situations, such as investments, loans, and mortgages. For example, if you invest $1000 in a savings account with an annual interest rate of 5% compounded annually, after 10 years, you'll have more than $1628.89 . Understanding compound interest helps you make informed decisions about your finances and plan for the future. It's also crucial for understanding how loans and credit cards work, so you can avoid unnecessary debt and manage your money effectively.
