https://file.aiquickdraw.com/mathbot/file/1726578639124ojmaa4ls.jpg

Let's break down the situation based on the given information:

1. **Hank's Loan**: 
   - He paid $2,000 up front.
   - He pays $200 per month.

   The total amount Hank has paid after $ t $ months is given by:
   \[
   \text{Total payment by Hank} = 2000 + 200 \times t
   \]
   This is a **linear equation** but not proportional because of the initial $2,000 upfront payment. In proportional relationships, the graph passes through the origin (0, 0), but Hank’s graph would start at $2,000.

2. **Lynn's Loan**:
   - She pays $275 per month, without any upfront payment.

   The total amount Lynn has paid after $ t $ months is given by:
   \[
   \text{Total payment by Lynn} = 275 \times t
   \]
   This is also a **linear equation**, but it **is** proportional, as the graph passes through the origin, indicating no initial payment.

### Answers to the drop-down menu:
- A graph that shows the number of months and the total amount paid by Hank would: **be a straight line that does not start at (0, 0)**.
- A graph that shows the number of months and the total amount paid by Lynn would: **be a straight line that starts at (0, 0)**.
- The relationship between the number of months and the total amount paid is proportional for: **Lynn but not Hank**.

Would you like a more detailed breakdown or further questions on this? Here are 5 relative questions for further exploration:

1. How do you determine if a relationship is proportional from a graph?
2. What are other real-world examples of non-proportional relationships?
3. How would Hank’s equation change if he did not pay an upfront cost?
4. How can we calculate the total amount Hank will pay after a specific number of months?
5. How would the slope of Lynn's line compare to Hank’s if they both paid the same monthly amount?

**Tip**: In a proportional relationship, all points on the graph are aligned along a straight line that passes through the origin (0, 0).

Hank and Lynn are both paying off car loans. Hank paid $2,000 up front when he bought his car, and he pays $200 each month. Lynn did not pay any money up front when she bought her car, and she pays $275 each month. Is the relationship between the number of months and the total amount paid proportional for both Hank’s and Lynn’s loans? Use the drop-down menus to explain your answer.

Learn how to determine if the relationship between the number of months and the total amount paid for Hank and Lynn’s car loans is proportional. Explore linear relationships and proportionality with a step-by-step breakdown. Ideal for students in Grades 6-8.

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