To determine if the table shows a proportional relationship between \( x \) and \( y \), you need to check if the ratios \( \frac{y}{x} \) are the same for all pairs of \( (x, y) \).
Let's calculate the ratios for each pair:
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For \( x = 30 \) and \( y = 150 \): \[ \frac{y}{x} = \frac{150}{30} = 5 \]
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For \( x = \frac{1}{6} \) and \( y = \frac{5}{6} \): \[ \frac{y}{x} = \frac{\frac{5}{6}}{\frac{1}{6}} = 5 \]
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For \( x = 199 \) and \( y = 995 \): \[ \frac{y}{x} = \frac{995}{199} = 5 \]
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For \( x = \frac{2}{15} \) and \( y = \frac{2}{3} \): \[ \frac{y}{x} = \frac{\frac{2}{3}}{\frac{2}{15}} = \frac{2}{3} \times \frac{15}{2} = 5 \]
Since all the ratios \( \frac{y}{x} \) are equal (all equal to 5), the table does show a proportional relationship.
Therefore, the table does show a proportional relationship between \( x \) and \( y \).
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