To identify a proportional relationship, we need to check if the ratio of \( y \) to \( x \) is constant for each pair of values in the table.
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First table:
- \( y = 2 \) when \( x = 4 \) → \( \frac{y}{x} = \frac{2}{4} = 0.5 \)
- \( y = 3 \) when \( x = 9 \) → \( \frac{y}{x} = \frac{3}{9} = \frac{1}{3} \)
- \( y = 4 \) when \( x = 16 \) → \( \frac{y}{x} = \frac{4}{16} = \frac{1}{4} \)
The ratios are not constant.
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Second table:
- \( y = 2 \) when \( x = 10 \) → \( \frac{y}{x} = \frac{2}{10} = 0.2 \)
- \( y = 3 \) when \( x = 15 \) → \( \frac{y}{x} = \frac{3}{15} = 0.2 \)
- \( y = 8 \) when \( x = 40 \) → \( \frac{y}{x} = \frac{8}{40} = 0.2 \)
The ratios are constant at 0.2.
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Third table:
- \( y = 10 \) when \( x = 2 \) → \( \frac{y}{x} = \frac{10}{2} = 5 \)
- \( y = 20 \) when \( x = 5 \) → \( \frac{y}{x} = \frac{20}{5} = 4 \)
- \( y = 30 \) when \( x = 6 \) → \( \frac{y}{x} = \frac{30}{6} = 5 \)
The ratios are not constant.
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Fourth table:
- \( y = 2 \) when \( x = 8 \) → \( \frac{y}{x} = \frac{2}{8} = 0.25 \)
- \( y = 3 \) when \( x = 12 \) → \( \frac{y}{x} = \frac{3}{12} = 0.25 \)
- \( y = 4 \) when \( x = 40 \) → \( \frac{y}{x} = \frac{4}{40} = 0.1 \)
The ratios are not constant.
Based on this analysis, the second table demonstrates a proportional relationship, as the ratio of \( y \) to \( x \) is constant (0.2).