To determine if the table shows a proportional relationship, we need to check if the ratios [tex]\(\frac{y}{x}\)[/tex] are consistent for every pair [tex]\((x, y)\)[/tex] in the table.
The values in the table are:
\begin{tabular}{|l|l|l|l|}
\hline
[tex]$x$[/tex] & 36.75 & 16.8 & 77.1 \\
\hline
[tex]$y$[/tex] & 12.25 & 5.6 & 25.7 \\
\hline
\end{tabular}
Let's calculate the ratios [tex]\(\frac{y}{x}\)[/tex] for each pair:
1. For the pair [tex]\( (36.75, 12.25) \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{12.25}{36.75} \approx 0.3333333333333333
\][/tex]
2. For the pair [tex]\( (16.8, 5.6) \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{5.6}{16.8} \approx 0.3333333333333333
\][/tex]
3. For the pair [tex]\( (77.1, 25.7) \)[/tex]:
[tex]\[
\frac{y}{x} = \frac{25.7}{77.1} \approx 0.33333333333333337
\][/tex]
Now, we have the ratios [tex]\([0.3333333333333333, 0.3333333333333333, 0.33333333333333337]\)[/tex].
To determine if these ratios indicate a proportional relationship:
- We observe that while the ratios are very close to each other, the last ratio [tex]\( 0.33333333333333337 \)[/tex] is not exactly equal to the others.
Therefore, even though the ratios are quite similar, they are not perfectly identical.
The correct interpretation based on these calculated ratios would be:
- No, it is not proportional because the ratios are not exactly equal.
However, given the options provided, the most fitting answer is:
- No, it is not proportional because [tex]\(\frac{12.25}{36.75} \neq \frac{16.8}{5.6}\)[/tex].
