Use the following compound interest formula to complete the problem.
[tex]$A=P\left(1+\frac{r}{n}\right)^{\prime \prime}$[/tex]
Rodney owes [tex]$1,541.05[/tex] on his credit card. His card has an APR of [tex]$16.29 \%$[/tex], compounded monthly. Assuming that he makes no payments and no purchases, how much will he owe after one year?
A. [tex]$1,811.70[/tex]
B. [tex]$1,792.09[/tex]
C. [tex]$1,541.05[/tex]
Please select the best answer from the choices provided.
Answer (2)
Substitute the given values into the compound interest formula: A = P ( 1 + n r ) n t .
Plug in the values: A = 1541.05 ( 1 + 12 0.1629 ) ( 12 ) ( 1 ) .
Calculate the value inside the parentheses: 1 + 12 0.1629 = 1.013575 .
Calculate the amount A: A = 1541.05 × ( 1.013575 ) 12 = 1811.704698 . Therefore, the best answer is $1 , 811.70 .
Explanation
Understanding the Problem We are given the principal amount Rodney owes, the annual interest rate, and the compounding period. We need to find the amount he will owe after one year, assuming no payments or purchases are made. We will use the compound interest formula: A = P \[ 1 e x ] ( 1 + n r ) n t , 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 (as a decimal) n = the number of times that interest is compounded per year t = the number of years the money is invested or borrowed for
Substituting the Values We are given:
P = $1541.05 r = 16.29% = 0.1629 n = 12 (compounded monthly) t = 1 year
Now, we substitute these values into the formula:
Applying the Formula A = 1541.05 ( 1 + 12 0.1629 ) ( 12 ) ( 1 )
Calculating the Future Value First, we calculate the value inside the parentheses: 12 0.1629 = 0.013575 1 + 0.013575 = 1.013575 Next, we calculate the exponent: ( 12 ) ( 1 ) = 12 Now, we can calculate the amount A :
A = 1541.05 × ( 1.013575 ) 12 A = 1541.05 × 1.175637 A = 1811.704698 Rounding to two decimal places, we get: A = $1811.70
Final Answer The amount Rodney will owe after one year is approximately $1811.70 . Comparing this to the provided options, the closest answer is b. $1 , 811.70 .
Selecting the Best Answer Therefore, the best answer is B.
Examples
Compound interest is a powerful concept that applies to many real-life situations, such as investments, loans, and credit cards. 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 due to the effects of compounding. Similarly, understanding compound interest helps you make informed decisions about loans and credit cards, allowing you to estimate the total cost of borrowing and plan your finances effectively. This knowledge is essential for managing your money and achieving your financial goals.
Rodney will owe approximately $1,811.70 after one year on his credit card debt, using the compound interest formula. The calculation uses his principal debt, interest rate, and compounding frequency. The correct option from the choices given is B. $1,811.70.
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