Question

Which table shows a proportional relationship between x and y?

y, 9, 11, 16
x, 3, 5, 10

y. 8, 16, 32
x, 3, 6, 12

y, 5, 15, 30
x, 4, 12, 16

y, 4, 8, 16
x, 1, 3, 9

Answer 1

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:

1. **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.

2. **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).

3. **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.

4. **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**.