To determine which table shows a proportional relationship between \( x \) and \( y \), we need to check if the ratio \( \frac{y}{x} \) is constant for all pairs of values in each table.
Let's analyze each table:
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Table 1: \[ \begin{align*} y: & 9, 11, 16 \ x: & 3, 5, 10 \ \end{align*} \] Ratios:
- \( \frac{9}{3} = 3 \)
- \( \frac{11}{5} = 2.2 \)
- \( \frac{16}{10} = 1.6 \)
The ratios are not constant.
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Table 2: \[ \begin{align*} y: & 8, 16, 32 \ x: & 3, 6, 12 \ \end{align*} \] Ratios:
- \( \frac{8}{3} \approx 2.67 \)
- \( \frac{16}{6} \approx 2.67 \)
- \( \frac{32}{12} \approx 2.67 \)
The ratios are constant (approximately 2.67).
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Table 3: \[ \begin{align*} y: & 5, 15, 30 \ x: & 4, 12, 16 \ \end{align*} \] Ratios:
- \( \frac{5}{4} = 1.25 \)
- \( \frac{15}{12} = 1.25 \)
- \( \frac{30}{16} = 1.875 \)
The ratios are not constant.
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Table 4: \[ \begin{align*} y: & 4, 8, 16 \ x: & 1, 3, 9 \ \end{align*} \] Ratios:
- \( \frac{4}{1} = 4 \)
- \( \frac{8}{3} \approx 2.67 \)
- \( \frac{16}{9} \approx 1.78 \)
The ratios are not constant.
The only table that shows a proportional relationship is Table 2.