To determine if the table shows a proportional relationship, we need to verify if the ratios [tex]\(\frac{y}{x}\)[/tex] are consistent for all given pairs of [tex]\((x, y)\)[/tex] values.
Given the table:
[tex]\[
\begin{tabular}{|c|c|c|c|}
\hline
$x$ & 18.2 & 49.2 & 45.9 \\
\hline
$y$ & 9.1 & 24.6 & 15.3 \\
\hline
\end{tabular}
\][/tex]
We will calculate the ratios [tex]\(\frac{y}{x}\)[/tex] for each pair.
1. Calculate [tex]\(\frac{y}{x}\)[/tex] for the first pair [tex]\((18.2, 9.1)\)[/tex]:
[tex]\[
\frac{9.1}{18.2} = 0.5
\][/tex]
2. Calculate [tex]\(\frac{y}{x}\)[/tex] for the second pair [tex]\((49.2, 24.6)\)[/tex]:
[tex]\[
\frac{24.6}{49.2} = 0.5
\][/tex]
3. Calculate [tex]\(\frac{y}{x}\)[/tex] for the third pair [tex]\((45.9, 15.3)\)[/tex]:
[tex]\[
\frac{15.3}{45.9} = \frac{1}{3} \approx 0.333
\][/tex]
Now, we need to compare these ratios:
- The first ratio [tex]\(\frac{9.1}{18.2} = 0.5\)[/tex]
- The second ratio [tex]\(\frac{24.6}{49.2} = 0.5\)[/tex]
- The third ratio [tex]\(\frac{15.3}{45.9} \approx 0.333\)[/tex]
Clearly, [tex]\(\frac{9.1}{18.2}\)[/tex] and [tex]\(\frac{24.6}{49.2}\)[/tex] are equal to [tex]\(0.5\)[/tex], but [tex]\(\frac{15.3}{45.9}\)[/tex] is approximately [tex]\(0.333\)[/tex], which is not equal to [tex]\(0.5\)[/tex].
Since not all of the ratios [tex]\(\frac{y}{x}\)[/tex] are equal, the table does not show a proportional relationship.
Therefore, the correct answer is:
[tex]\[
\boxed{ \text{No, it is not proportional because } \frac{9.1}{18.2} \neq \frac{15.3}{45.9}. }
\][/tex]
