To determine which table demonstrates a proportional relationship, we need to check if the ratio of \( y \) to \( x \) is constant across all pairs of values.
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For the first table: \[ \frac{y}{x} = \frac{2}{4} = 0.5, \quad \frac{3}{9} \approx 0.33, \quad \frac{4}{16} = 0.25 \] The ratios are not constant.
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For the second table: \[ \frac{y}{x} = \frac{10}{2} = 5, \quad \frac{20}{5} = 4, \quad \frac{30}{6} = 5 \] The ratios are not constant.
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For the third table: \[ \frac{y}{x} = \frac{2}{10} = 0.2, \quad \frac{3}{15} = 0.2, \quad \frac{8}{40} = 0.2 \] The ratios are constant and equal to 0.2.
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For the fourth table: \[ \frac{y}{x} = \frac{2}{8} = 0.25, \quad \frac{3}{12} = 0.25, \quad \frac{4}{40} = 0.1 \] The ratios are not constant.
Conclusion: The third table (\( x,y \): 10,2; 15,3; 40,8) demonstrates a proportional relationship with a constant ratio of \( 0.2 \).
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