To determine the equation that relates the number of purple seashells ([tex]\( p \)[/tex]) to the number of white seashells ([tex]\( w \)[/tex]), we need to analyze the given data:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|}
\hline
\text{White Seashells} & 3 & 6 & 9 & 12 & 15 & 18 \\
\hline
\text{Purple Seashells} & 5 & 10 & 15 & 20 & 25 & 30 \\
\hline
\end{array}
\][/tex]
First, we'll find the ratio of the number of purple seashells ([tex]\( p \)[/tex]) to the number of white seashells ([tex]\( w \)[/tex]) for each pair of data points. The formula to calculate the ratio is:
[tex]\[ \text{ratio} = \frac{p}{w} \][/tex]
Calculate the ratios for each corresponding pair:
1. For [tex]\( w = 3 \)[/tex] and [tex]\( p = 5 \)[/tex]:
[tex]\[
\frac{5}{3} = 1.6666666666666667
\][/tex]
2. For [tex]\( w = 6 \)[/tex] and [tex]\( p = 10 \)[/tex]:
[tex]\[
\frac{10}{6} = 1.6666666666666667
\][/tex]
3. For [tex]\( w = 9 \)[/tex] and [tex]\( p = 15 \)[/tex]:
[tex]\[
\frac{15}{9} = 1.6666666666666667
\][/tex]
4. For [tex]\( w = 12 \)[/tex] and [tex]\( p = 20 \)[/tex]:
[tex]\[
\frac{20}{12} = 1.6666666666666667
\][/tex]
5. For [tex]\( w = 15 \)[/tex] and [tex]\( p = 25 \)[/tex]:
[tex]\[
\frac{25}{15} = 1.6666666666666667
\][/tex]
6. For [tex]\( w = 18 \)[/tex] and [tex]\( p = 30 \)[/tex]:
[tex]\[
\frac{30}{18} = 1.6666666666666667
\][/tex]
As we can see, the ratio [tex]\( \frac{p}{w} \)[/tex] is consistent and equals approximately 1.6667 for all pairs. This indicates a linear relationship between [tex]\( p \)[/tex] and [tex]\( w \)[/tex]. The consistent ratio suggests that:
[tex]\[ p = k \cdot w \][/tex]
where [tex]\( k \)[/tex] is the constant ratio. In this case, [tex]\( k \approx 1.6667 \)[/tex].
Rewriting the constant ratio in fractional form, we get:
[tex]\[
k = \frac{5}{3}
\][/tex]
Thus, the relationship between [tex]\( p \)[/tex] and [tex]\( w \)[/tex] is:
[tex]\[ p = \frac{5}{3} w \][/tex]
Hence, the correct equation that relates the number of purple shells [tex]\( p \)[/tex] to the number of white shells [tex]\( w \)[/tex] is:
[tex]\[ \boxed{p = \frac{5}{3} w} \][/tex]
