To determine which expression correctly represents Clayton's balance at the end of the year, we'll follow a step-by-step method to calculate the compounded balance considering the two different APR rates within the year.
### Initial Balance and APRs
- Initial Balance: \[tex]$4125
- Introductory APR: 7.9% (applied for the first 5 months)
- Standard APR: 25.7% (applied thereafter, for the remaining 7 months)
### Notes on Compounding Interest Monthly
Since interest compounds monthly, the following monthly interest rates apply:
- Introductory monthly rate \( = \frac{7.9\%}{12} \approx 0.6583\% = 0.006583 \)
- Standard monthly rate \( = \frac{25.7\%}{12} \approx 2.1417\% = 0.021417 \)
### Step-by-Step Calculation
1. Balance after the Introductory Period (First 5 months):
\[
\text{Balance after 5 months} = 4125 \times \left(1 + \frac{0.079}{12}\right)^5
\]
After these 5 months, the balance \(\approx 4262.58\).
2. Balance after the Standard Period (Next 7 months):
\[
\text{Balance after remaining 7 months} = \text{Balance after 5 months} \times \left(1 + \frac{0.257}{12}\right)^7
\]
Continuing from the balance after 5 months, the balance \(\approx 4944.17\).
### Intermediate Values
- After 5 months with the introductory APR:
\[
4262.58 \text{ dollars}
\]
- After the next 7 months with the standard APR:
\[
4944.17 \text{ dollars}
\]
### Expression Form
Combining these processes into one single expression to represent Clayton's balance at the end of the year, we have:
\[
(\$[/tex]4125)\left(1+\frac{0.079}{12}\right)^5\left(1+\frac{0.257}{12}\right)^7
\]
### Correct Expression
This matches option A, which correctly follows the monthly compounding application for both introductory and standard APRs over the course of 5 months and 7 months respectively.
Thus, the correct expression is:
[tex]\[
\boxed{(\$4125)\left(1+\frac{0.079}{12}\right)^5\left(1+\frac{0.257}{12}\right)^7}
\][/tex]
