Does the table show a proportional relationship? If so, what is the constant of proportionality (unit rate)?

Question

Hours: 2, 3, 4, 5
$ Earned: $15, $22.50, $30, $45

Answers:
A. No, there is no constant rate of change
B. Yes, and the constant of proportionality is 5
C. Yes, and the constant of proportionality is 7.50

Answer 1

To determine if the relationship is proportional, we can examine whether the ratio of the amount earned to the hours worked remains constant.

First, let's find the constant of proportionality (unit rate) for each pair of hours and earned amounts:

1. For 2 hours: \[ \text{Earned} = 15 \quad \Rightarrow \quad \frac{15}{2} = 7.50 \]

2. For 3 hours: \[ \text{Earned} = 22.50 \quad \Rightarrow \quad \frac{22.50}{3} = 7.50 \]

3. For 4 hours: \[ \text{Earned} = 30 \quad \Rightarrow \quad \frac{30}{4} = 7.50 \]

4. For 5 hours: \[ \text{Earned} = 45 \quad \Rightarrow \quad \frac{45}{5} = 9.00 \]

Now we compare the ratios we calculated:

• For 2 hours: 7.50
• For 3 hours: 7.50
• For 4 hours: 7.50
• For 5 hours: 9.00

Since the ratio for the 5 hours is different from the others, we see that there is no constant rate of change throughout all intervals.

Thus, the answer is: A. No, there is no constant rate of change.