Ted has a credit card that uses the average daily balance method. For the first 9 days of one of his billing cycles, his balance was $[tex]$2030$[/tex], and for the last 21 days of the billing cycle, his balance was $[tex]$1450$[/tex]. If his credit card's APR is [tex]$23 \%$[/tex], which of these expressions could be used to calculate the amount Ted was charged in interest for the billing cycle?
A. [tex]$\left(\frac{0.23}{365} \cdot 30\right)\left(\frac{9 \cdot \$ 2030+21 \cdot \$ 1450}{30}\right)$[/tex]
B. [tex]$\left(\frac{0.23}{365} \cdot 31\right)\left(\frac{21 \cdot \$ 2030+9 \cdot \$ 1450}{31}\right)$[/tex]
C. [tex]$\left(\frac{0.23}{365} \cdot 30\right)\left(\frac{21 \cdot \$ 2030+9 \cdot \$ 1450}{30}\right)$[/tex]
D. [tex]$\left(\frac{0.23}{365} \cdot 31\right)\left(\frac{9 \cdot \$ 2030+21 \cdot \$ 1450}{31}\right)$[/tex]
Answer (2)
The total interest charged to Ted is calculated using the average daily balance method, resulting in the correct formula being option A: ( 365 0.23 ⋅ 30 ) ( 30 9 ⋅ 2030 + 21 ⋅ 1450 ) . This expression accounts for both the average daily balance and the daily interest rate for the billing period.
;
Calculate the average daily balance: 30 ( 9 × $2030 ) + ( 21 × $1450 ) .
Determine the daily interest rate: 365 0.23 .
Calculate the interest charged: Average Daily Balance × Daily Interest Rate × 30 .
The correct expression is ( 365 0.23 ⋅ 30 ) ( 30 9 ⋅ $2030 + 21 ⋅ $1450 ) .
Explanation
Understanding the Problem We are given that Ted's credit card uses the average daily balance method. We need to determine which expression correctly calculates the interest charged for the billing cycle. The APR is 23%, the balance was $2030 for the first 9 days, and $1450 for the last 21 days.
Calculating the Average Daily Balance First, we need to calculate the average daily balance. The average daily balance is calculated by multiplying each daily balance by the number of days it was maintained, summing these values, and dividing by the total number of days in the billing cycle. The total number of days in the billing cycle is 9 + 21 = 30 days.
Determining the Average Daily Balance The average daily balance is calculated as follows: Average Daily Balance = 30 ( 9 × $2030 ) + ( 21 × $1450 ) = 30 $18270 + $30450 = 30 $48720 = $1624 So, the average daily balance is $1624.
Calculating the Daily Interest Rate Next, we need to calculate the daily interest rate. The daily interest rate is the APR divided by 365 (the number of days in a year): Daily Interest Rate = 365 0.23
Calculating the Interest Charged Now, we can calculate the interest charged for the billing cycle. The interest charged is the average daily balance multiplied by the daily interest rate and the number of days in the billing cycle (30 days): Interest = Average Daily Balance × Daily Interest Rate × Number of Days Interest = $1624 × 365 0.23 × 30 Interest = ( 365 0.23 × 30 ) × $1624 Interest = ( 365 0.23 × 30 ) × ( 30 9 × $2030 + 21 × $1450 )
Matching the Expression Comparing the expression we derived with the given options, we see that option A matches our expression: ( 365 0.23 ⋅ 30 ) ( 30 9 ⋅ $2030 + 21 ⋅ $1450 )
Final Answer Therefore, the correct expression to calculate the amount Ted was charged in interest for the billing cycle is: ( 365 0.23 ⋅ 30 ) ( 30 9 ⋅ $2030 + 21 ⋅ $1450 )
Stating the Answer The correct answer is A.
Examples
Understanding average daily balance is crucial for managing credit card debt. For instance, if you carry a balance of $1000 for 10 days and then pay it off, but then charge $500 again for the next 20 days, your average daily balance isn't just $500. It's calculated as 30 ( 10 × $1000 ) + ( 20 × $500 ) = $666.67 . This average is used to calculate the interest you'll be charged. Knowing this helps you plan payments to minimize interest charges, especially if you can't pay off the full balance each month. This concept is also applicable in other financial scenarios, such as calculating average investment values over time.
