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. eynn did not pay any money up front when she bought her car, and she pays \( \$ 275 \) 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. Click the arrows to choose an answer from each menu. A graph that shows the number of months and the total amount pald by Hank would Choose... A graph that shows the number of months and the total amount paid by Lynn would Choose... The relationship between the number of months and the total amount paid is proportional for Choose...

To determine whether the relationships between the number of months and the total amount paid are proportional for Hank and Lynn, let's analyze their payment structures: 1. **Hank's Payments:** - **Up-front Payment:** \$2,000 - **Monthly Payment:** \$200 The total amount Hank pays can be expressed as: \[ \text{Total Paid} = \$2,000 + \$200 \times (\text{Number of Months}) \] This equation represents a linear relationship but **not a proportional relationship** because there is an initial payment (\$2,000) that doesn't depend on the number of months. A proportional relationship requires that when the number of months is zero, the total amount paid is also zero. 2. **Lynn's Payments:** - **Up-front Payment:** \$0 - **Monthly Payment:** \$275 The total amount Lynn pays can be expressed as: \[ \text{Total Paid} = \$275 \times (\text{Number of Months}) \] This equation represents a **proportional relationship** because there is no initial payment. When the number of months is zero, the total amount paid is also zero, satisfying the condition for proportionality. **Dropdown Selections:** 1. **A graph that shows the number of months and the total amount paid by Hank would:** - **Not Pass Through the Origin** 2. **A graph that shows the number of months and the total amount paid by Lynn would:** - **Pass Through the Origin** 3. **The relationship between the number of months and the total amount paid is proportional for:** - **Lynn only**