To determine which table represents a proportional relationship, we need to check if the ratio \( \frac{y}{x} \) is constant for each pair of \( (x, y) \).
Option A:
- For \( x = 1, y = 0 \): \( \frac{y}{x} = \frac{0}{1} = 0 \)
- For \( x = 4, y = 3 \): \( \frac{y}{x} = \frac{3}{4} = 0.75 \)
- For \( x = 7, y = 6 \): \( \frac{y}{x} = \frac{6}{7} \approx 0.857 \)
- For \( x = 10, y = 9 \): \( \frac{y}{x} = \frac{9}{10} = 0.9 \)
The ratios are not constant. So, A is not proportional.
Option B:
- For \( x = 2, y = 3 \): \( \frac{y}{x} = \frac{3}{2} = 1.5 \)
- For \( x = 4, y = 4 \): \( \frac{y}{x} = \frac{4}{4} = 1 \)
- For \( x = 6, y = 5 \): \( \frac{y}{x} = \frac{5}{6} \approx 0.833 \)
- For \( x = 8, y = 6 \): \( \frac{y}{x} = \frac{6}{8} = 0.75 \)
The ratios are not constant. So, B is not proportional.
Option C:
- For \( x = 2, y = 3 \): \( \frac{y}{x} = \frac{3}{2} = 1.5 \)
- For \( x = 3, y = 6 \): \( \frac{y}{x} = \frac{6}{3} = 2 \)
- For \( x = 4, y = 9 \): \( \frac{y}{x} = \frac{9}{4} = 2.25 \)
- For \( x = 5, y = 12 \): \( \frac{y}{x} = \frac{12}{5} = 2.4 \)
The ratios are not constant. So, C is not proportional.
Option D:
- For \( x = 4, y = 6 \): \( \frac{y}{x} = \frac{6}{4} = 1.5 \)
- For \( x = 6, y = 9 \): \( \frac{y}{x} = \frac{9}{6} = 1.5 \)
- For \( x = 8, y = 12 \): \( \frac{y}{x} = \frac{12}{8} = 1.5 \)
- For \( x = 10, y = 15 \): \( \frac{y}{x} = \frac{15}{10} = 1.5 \)
The ratio \( \frac{y}{x} \) is constant at 1.5. Thus, D is proportional.
The table that represents a proportional relationship is: Option D.
4 answers