Answer :
Let's solve the problem step-by-step to determine the correct expression for calculating the interest charged to Norma's credit card for the given billing cycle.
1. Understand the balances over the billing cycle:
- For the first 10 days: Balance = [tex]$1850 - For the next 10 days after making a purchase: Balance = $[/tex]2010 (after a [tex]$160 purchase) - For the last 10 days after making a payment: Balance = $[/tex]1080 (after a [tex]$930 payment) 2. Calculate the average daily balance for the 30-day billing cycle: The formula to find the average daily balance over the entire billing cycle is: \[ \text{Average Daily Balance} = \frac{(\text{Balance}_1 \times \text{Days}_1) + (\text{Balance}_2 \times \text{Days}_2) + (\text{Balance}_3 \times \text{Days}_3)}{\text{Total Days}} \] Substituting the values given: \[ \text{Average Daily Balance} = \frac{(1850 \times 10) + (2010 \times 10) + (1080 \times 10)}{30} \] Simplifying step-by-step: \[ (1850 \times 10) = 18500 \] \[ (2010 \times 10) = 20100 \] \[ (1080 \times 10) = 10800 \] Adding them together: \[ 18500 + 20100 + 10800 = 49400 \] Now, divide by the total number of days: \[ \text{Average Daily Balance} = \frac{49400}{30} = 1646.6667 \] 3. Convert annual percentage rate (APR) to daily interest rate: The APR given is 29%, so the daily interest rate is: \[ \text{Daily Interest Rate} = \frac{APR}{365} = \frac{0.29}{365} = 0.0007945205 \] 4. Calculate the total interest charged over the 30-day cycle: Multiply the daily interest rate by the number of days in the cycle and the average daily balance calculated above: \[ \text{Interest Charged} = \text{Daily Interest Rate} \times \text{Days in Cycle} \times \text{Average Daily Balance} \] Substituting the values we have: \[ \text{Interest Charged} = 0.0007945205 \times 30 \times 1646.6667 = 39.2493 \] 5. Identify the correct expression: From the given options, the expression closest to our derived expression is: \[ \left(\frac{0.29}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 1850 + 10 \cdot \[tex]$ 2010 + 10 \cdot \$[/tex] 1080}{30}\right)
\]
This matches with Option A. Thus, the correct expression to calculate the interest Norma was charged is:
A.
[tex]\[ \left(\frac{0.29}{365} \cdot 30\right)\left(\frac{10 \cdot \$ 1850 + 10 \cdot \$ 2010 + 10 \cdot \$ 1080}{30}\right) \][/tex]
1. Understand the balances over the billing cycle:
- For the first 10 days: Balance = [tex]$1850 - For the next 10 days after making a purchase: Balance = $[/tex]2010 (after a [tex]$160 purchase) - For the last 10 days after making a payment: Balance = $[/tex]1080 (after a [tex]$930 payment) 2. Calculate the average daily balance for the 30-day billing cycle: The formula to find the average daily balance over the entire billing cycle is: \[ \text{Average Daily Balance} = \frac{(\text{Balance}_1 \times \text{Days}_1) + (\text{Balance}_2 \times \text{Days}_2) + (\text{Balance}_3 \times \text{Days}_3)}{\text{Total Days}} \] Substituting the values given: \[ \text{Average Daily Balance} = \frac{(1850 \times 10) + (2010 \times 10) + (1080 \times 10)}{30} \] Simplifying step-by-step: \[ (1850 \times 10) = 18500 \] \[ (2010 \times 10) = 20100 \] \[ (1080 \times 10) = 10800 \] Adding them together: \[ 18500 + 20100 + 10800 = 49400 \] Now, divide by the total number of days: \[ \text{Average Daily Balance} = \frac{49400}{30} = 1646.6667 \] 3. Convert annual percentage rate (APR) to daily interest rate: The APR given is 29%, so the daily interest rate is: \[ \text{Daily Interest Rate} = \frac{APR}{365} = \frac{0.29}{365} = 0.0007945205 \] 4. Calculate the total interest charged over the 30-day cycle: Multiply the daily interest rate by the number of days in the cycle and the average daily balance calculated above: \[ \text{Interest Charged} = \text{Daily Interest Rate} \times \text{Days in Cycle} \times \text{Average Daily Balance} \] Substituting the values we have: \[ \text{Interest Charged} = 0.0007945205 \times 30 \times 1646.6667 = 39.2493 \] 5. Identify the correct expression: From the given options, the expression closest to our derived expression is: \[ \left(\frac{0.29}{365} \cdot 30\right)\left(\frac{10 \cdot \$[/tex] 1850 + 10 \cdot \[tex]$ 2010 + 10 \cdot \$[/tex] 1080}{30}\right)
\]
This matches with Option A. Thus, the correct expression to calculate the interest Norma was charged is:
A.
[tex]\[ \left(\frac{0.29}{365} \cdot 30\right)\left(\frac{10 \cdot \$ 1850 + 10 \cdot \$ 2010 + 10 \cdot \$ 1080}{30}\right) \][/tex]
