Answered

Sergei has a credit card that uses the average daily balance method. For the first 12 days of one of his billing cycles, his balance was [tex]\$350[/tex], and for the last 18 days of the billing cycle, his balance was [tex]\$520[/tex]. If his credit card's APR is [tex]14\%[/tex], which of these expressions could be used to calculate the amount Sergei was charged in interest for the billing cycle?

A. [tex]\left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot \[tex]$350 + 18 \cdot \$[/tex]520}{30}\right)[/tex]

B. [tex]\left(\frac{0.14}{365} \cdot 31\right)\left(\frac{18 \cdot \[tex]$350 + 12 \cdot \$[/tex]520}{31}\right)[/tex]

C. [tex]\left(\frac{0.14}{365} \cdot 31\right)\left(\frac{12 \cdot \[tex]$350 + 18 \cdot \$[/tex]520}{31}\right)[/tex]

D. [tex]\left(\frac{0.14}{365} \cdot 30\right)\left(\frac{18 \cdot \[tex]$350 + 12 \cdot \$[/tex]520}{30}\right)[/tex]



Answer :

GinnyAnswer
To determine the correct expression used to calculate the interest charged to Sergei, we'll use the average daily balance method step-by-step.

### 1. Understand the Problem

Sergei's balance during a 30-day billing cycle is:
- For the first 12 days: \[tex]$350 - For the last 18 days: \$[/tex]520

Annual Percentage Rate (APR): 14%

### 2. Calculate the Average Daily Balance

The average daily balance is calculated using the formula:
[tex]\[ \text{Average Daily Balance} = \frac{\text{Total Weighted Balance}}{\text{Total Days}} \][/tex]

The total weighted balance is:
[tex]\[ \text{Total Weighted Balance} = (\text{Balance for the First Period} \times \text{Days in the First Period}) + (\text{Balance for the Second Period} \times \text{Days in the Second Period}) \][/tex]

Plugging in Sergei's data:
[tex]\[ \text{Total Weighted Balance} = (\$350 \times 12) + (\$520 \times 18) \][/tex]

[tex]\[ \text{Total Weighted Balance} = 4200 + 9360 = 13560 \][/tex]

Since the total days in the billing cycle is 30, the average daily balance is:
[tex]\[ \text{Average Daily Balance} = \frac{13560}{30} = 452 \][/tex]

### 3. Calculate the Daily Interest Rate

The daily interest rate is given by:
[tex]\[ \text{Daily Interest Rate} = \frac{\text{APR}}{365} \][/tex]

Using the APR of 14%:
[tex]\[ \text{Daily Interest Rate} = \frac{0.14}{365} \approx 0.00038356 \][/tex]

### 4. Calculate the Interest Charged

The interest charged over the billing cycle (30 days) is:
[tex]\[ \text{Interest Charged} = \text{Daily Interest Rate} \times \text{Total Days} \times \text{Average Daily Balance} \][/tex]

Plugging in the values:
[tex]\[ \text{Interest Charged} = 0.00038356 \times 30 \times 452 \approx 5.183 \][/tex]

### 5. Identify the Correct Expression

Now we will match this method with the given expressions:

#### Expression A:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right)\left(\frac{12 \cdot 350 + 18 \cdot 520}{30}\right) \][/tex]

Simplifying inside the parentheses:

[tex]\[ \frac{12 \cdot 350 + 18 \cdot 520}{30} = 452 \][/tex]

The expression simplifies to:
[tex]\[ \left(\frac{0.14}{365} \cdot 30\right) \times 452 \][/tex]

This matches our step-by-step calculation, confirming that the correct expression is:

### Correct Answer:
[tex]\[ \boxed{1} \][/tex]