Let's determine if the table shows a proportional relationship by examining the ratios [tex]\(\frac{y}{x}\)[/tex] for each pair of values.
The table lists the following values:
| [tex]\(x\)[/tex] | 25.6 | 72.8 | 77.2 |
|------|------|------|------|
| [tex]\(y\)[/tex] | 6.4 | 18.2 | 19.3 |
To check if [tex]\(y\)[/tex] is proportional to [tex]\(x\)[/tex], we need to compute the ratios [tex]\(\frac{y}{x}\)[/tex] for each pair and see if they are all equivalent.
1. For the first pair ([tex]\(x = 25.6\)[/tex] and [tex]\(y = 6.4\)[/tex]):
[tex]\[
\frac{y}{x} = \frac{6.4}{25.6} = 0.25
\][/tex]
2. For the second pair ([tex]\(x = 72.8\)[/tex] and [tex]\(y = 18.2\)[/tex]):
[tex]\[
\frac{y}{x} = \frac{18.2}{72.8} = 0.25
\][/tex]
3. For the third pair ([tex]\(x = 77.2\)[/tex] and [tex]\(y = 19.3\)[/tex]):
[tex]\[
\frac{y}{x} = \frac{19.3}{77.2} = 0.25
\][/tex]
All the computed ratios [tex]\(\frac{y}{x}\)[/tex] are equal to [tex]\(0.25\)[/tex]. This means that the [tex]\(y\)[/tex]-values are consistently [tex]\(0.25\)[/tex] times their corresponding [tex]\(x\)[/tex]-values, indicating a consistent proportional relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex].
Therefore, we can conclude that:
Yes, it is proportional because all [tex]\(\frac{y}{x}\)[/tex] ratios are equivalent to [tex]\(0.25\)[/tex].
