Let's solve this step-by-step.
1. Determine the ratios of book price to number of pages for each ordered pair:
We need to calculate the ratio [tex]\( \frac{y}{x} \)[/tex] for each pair.
2. First ordered pair (100 pages, [tex]$5.00):
\[
\frac{5.00}{100} = 0.05
\]
So, the ratio is 0.05.
3. Second ordered pair (400 pages, $[/tex]20.00):
[tex]\[
\frac{20.00}{400} = 0.05
\][/tex]
Let [tex]\( a = 0.05 \)[/tex].
4. Third ordered pair (125 pages, [tex]$6.25):
\[
\frac{6.25}{125} = 0.05
\]
Let \( b = 0.05 \).
5. Fourth ordered pair (350 pages, $[/tex]15.00):
[tex]\[
\frac{15.00}{350} = 0.04
\][/tex]
Let [tex]\( c = 0.04 \)[/tex].
Now, we can fill in the table with the ratios:
\begin{tabular}{|c|c|c|}
\hline
[tex]$x$[/tex] Number of Pages & [tex]$y$[/tex] Price of Book (\[tex]$) & Ratio of $[/tex]y:x$ \\
\hline
100 & 5.00 & 0.05 \\
\hline
400 & 20.00 & 0.05 \\
\hline
125 & 6.25 & 0.05 \\
\hline
350 & 15.00 & 0.04 \\
\hline
\end{tabular}
So, we have:
[tex]\[ a = 0.05 \][/tex]
[tex]\[ b = 0.05 \][/tex]
[tex]\[ c = 0.04 \][/tex]
6. Check for direct variation:
A direct variation is present when all ratios [tex]\(\frac{y}{x}\)[/tex] are the same.
Since the ratios are 0.05, 0.05, 0.05, and 0.04, not all are equal. Therefore, the points in the table do not form a direct variation.
In conclusion:
[tex]\[ a = 0.05 \][/tex]
[tex]\[ b = 0.05 \][/tex]
[tex]\[ c = 0.04 \][/tex]
[tex]\[ \text{The ratios are not all the same, therefore, the points in the table do not form a direct variation.} \][/tex]
