To solve this problem, we need to calculate the time it will take for Phil to pay off a loan of $13,000 if he chooses Offer 2, which involves paying $3,700 at the end of each year with an interest rate of 21%.

Step-by-step Solution:

1. Understanding the Loan Terms:

   • Principal (P): $13,000
   • Annual Payment (A): $3,700
   • Annual Interest Rate (r): 21% (or 0.21 as a decimal)

2. Interest Calculation:

   • At the end of each year, interest is applied to the remaining balance.
   • After adding the interest, Phil pays $3,700, which reduces the remaining loan balance.

3. Calculate the Remaining Balance Each Year:

   • The remaining balance after each year can be calculated using the formula: $\text{Remaining Balance after Year } n = (\text{Previous Year's Balance} \times (1 + r)) - A$

4. Iterate Year by Year Until the Loan is Paid Off:

Let's perform the calculations iteratively.It will take 8 years for Phil to pay off the loan if he takes Offer 2 and makes annual payments of $3,700 with an interest rate of 21%.

Would you like more details on the yearly calculations or have any questions?

Here are some related questions you might consider:

1. How much will Phil have paid in total over the 8 years, including interest?
2. What would the remaining balance be at the end of each year?
3. What would happen if the interest rate were different, say 15% instead of 21%?
4. How much would Phil need to pay annually to pay off the loan in 5 years instead?
5. How does compound interest affect loan repayment duration?

Tip: When dealing with loans, understanding how interest affects the balance over time is crucial to manage debt effectively.