Answer :
Sure, let's break down the solution step-by-step:
Felipe transferred a balance of [tex]$3,700 to a new credit card at the beginning of the year. The card offers an introductory APR (Annual Percentage Rate) of 5.9% for the first 4 months, and a standard APR of 17.2% for the remaining months of the year. The interest compounds monthly.
Here’s how we can calculate Felipe’s balance at the end of the year:
1. Introductory Period (4 months):
- The introductory APR is 5.9%, which is applied for the first 4 months.
- Since the interest compounds monthly, we convert the annual rate to a monthly rate: \( \frac{0.059}{12} \).
- The balance growth factor for the 4 months is \((1 + \frac{0.059}{12})^4\).
2. Standard Period (8 months):
- After the introductory period, the standard APR of 17.2% applies.
- Convert the annual rate to a monthly rate: \( \frac{0.172}{12} \).
- The balance growth factor for the 8 months is \((1 + \frac{0.172}{12})^8\).
3. Calculating the Final Balance:
- Start with the initial balance of $[/tex]3,700.
- Multiply by the growth factor for the introductory period to account for the interest accrued during the first 4 months.
- Then, multiply by the growth factor for the standard period to account for the interest during the remaining 8 months.
The expression for the final balance at the end of the year is:
[tex]\[
(\$ 3700) \times \left(1 + \frac{0.059}{12}\right)^4 \times \left(1 + \frac{0.172}{12}\right)^8
\][/tex]
This matches option D in the provided choices. Therefore, the correct expression representing Felipe's balance at the end of the year is:
D. [tex]\((\$ 3700)\left(1+\frac{0.059}{12}\right)^4\left(1+\frac{0.172}{12}\right)^8\)[/tex]
Felipe transferred a balance of [tex]$3,700 to a new credit card at the beginning of the year. The card offers an introductory APR (Annual Percentage Rate) of 5.9% for the first 4 months, and a standard APR of 17.2% for the remaining months of the year. The interest compounds monthly.
Here’s how we can calculate Felipe’s balance at the end of the year:
1. Introductory Period (4 months):
- The introductory APR is 5.9%, which is applied for the first 4 months.
- Since the interest compounds monthly, we convert the annual rate to a monthly rate: \( \frac{0.059}{12} \).
- The balance growth factor for the 4 months is \((1 + \frac{0.059}{12})^4\).
2. Standard Period (8 months):
- After the introductory period, the standard APR of 17.2% applies.
- Convert the annual rate to a monthly rate: \( \frac{0.172}{12} \).
- The balance growth factor for the 8 months is \((1 + \frac{0.172}{12})^8\).
3. Calculating the Final Balance:
- Start with the initial balance of $[/tex]3,700.
- Multiply by the growth factor for the introductory period to account for the interest accrued during the first 4 months.
- Then, multiply by the growth factor for the standard period to account for the interest during the remaining 8 months.
The expression for the final balance at the end of the year is:
[tex]\[
(\$ 3700) \times \left(1 + \frac{0.059}{12}\right)^4 \times \left(1 + \frac{0.172}{12}\right)^8
\][/tex]
This matches option D in the provided choices. Therefore, the correct expression representing Felipe's balance at the end of the year is:
D. [tex]\((\$ 3700)\left(1+\frac{0.059}{12}\right)^4\left(1+\frac{0.172}{12}\right)^8\)[/tex]
