Question

1. frank and allie purchased cell phone plans through different providers. their costs for several minutes are shown. graph each plan to determine whose plan is proportional to the number of minutes the phone is used. explain your reasoning.

cell phone plans

time (min)	frank’s cost ($)	allie’s cos
3	1.50	4.50
6	3.00	5.00

(graph with x-axis and y-axis labeled, grid lines)

Explanation:

Step1: Identify proportionality condition

A proportional relationship has a constant ratio $\frac{\text{cost}}{\text{time}}$ and passes through $(0,0)$.

Step2: Calculate Frank’s ratio

For Frank: $\frac{1.50}{3}=0.5$, $\frac{3.00}{6}=0.5$. Ratio is constant, and $(0,0)$ is included.

Step3: Calculate Allie’s ratio

For Allie: $\frac{4.50 - 4.00}{3 - 0} \approx 0.167$, $\frac{5.00 - 4.50}{6 - 3} \approx 0.167$, but starts at $(0,4.00)$ (non-zero intercept).

Answer:

Frank's plan is proportional. His cost-to-time ratio is constant ($0.5$ per minute) and the graph passes through the origin, satisfying proportionality. Allie's plan has a fixed cost ($4.00$ at 0 minutes) so it is not proportional.