Certainly! Let's determine the constant of proportionality for the given pairs of values [tex]\((x, y)\)[/tex] in the table. The pairs are:
- [tex]\( (x_1, y_1) = (16, 10) \)[/tex]
- [tex]\( (x_2, y_2) = (24, 15) \)[/tex]
- [tex]\( (x_3, y_3) = (48, 30) \)[/tex]
To find the constant of proportionality, we need to divide [tex]\( y \)[/tex] by [tex]\( x \)[/tex] for each pair and ensure that the ratio [tex]\(\frac{y}{x}\)[/tex] is consistent across all pairs. Let’s perform the following calculations:
1. For [tex]\( (16, 10) \)[/tex]:
[tex]\[
\frac{y_1}{x_1} = \frac{10}{16} = 0.625
\][/tex]
2. For [tex]\( (24, 15) \)[/tex]:
[tex]\[
\frac{y_2}{x_2} = \frac{15}{24} = 0.625
\][/tex]
3. For [tex]\( (48, 30) \)[/tex]:
[tex]\[
\frac{y_3}{x_3} = \frac{30}{48} = 0.625
\][/tex]
We observe that the ratio [tex]\(\frac{y}{x}\)[/tex] is the same for all pairs, which means that the values are directly proportional.
So, the constant of proportionality for the given table is:
[tex]\[
\boxed{0.625}
\][/tex]
Additionally, the list of constants found for each pair is:
[tex]\[
[0.625, 0.625, 0.625]
\][/tex]
This consistency across all pairs affirms that the values are directly proportional with a constant of proportionality of [tex]\(0.625\)[/tex].
