Tony is offering two repayment plans to Phil for a long overdue loan.  Offer 1 is to receive a visit from an enforcer and the debt is due in full at once.  Offer 2 is to pay back ​$3 comma 700 at the end of the year at an interest rate of 21​% until Phil pays off the loan principal.  Phil owes Tony ​$13 comma 000.  How long will it take for Phil to pay off the loan if he takes offer​ 2?
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Part 1
How long will it take for Phil to pay off the loan if he takes offer​ 2? 

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.

Tony is offering two repayment plans to Phil for a long overdue loan. Offer 1 is to receive a visit from an enforcer and the debt is due in full at once. Offer 2 is to pay back $3,700 at the end of the year at an interest rate of 21% until Phil pays off the loan principal. Phil owes Tony $13,000. How long will it take for Phil to pay off the loan if he takes offer 2?

Discover how to calculate the time it will take to pay off a $13,000 loan with annual payments of $3,700 and a 21% interest rate. This step-by-step guide covers key concepts such as compound interest and loan repayment iterations. Suitable for Grades 10-12.

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