To identify which table shows a constant of proportionality of 2 for the ratio of string instruments to percussion instruments, we need to examine the ratios of string to percussion instruments in each table.
A constant of proportionality [tex]\( k \)[/tex] for the ratio of string instruments to percussion instruments implies that:
[tex]\[ \frac{\text{String}}{\text{Percussion}} = k \][/tex]
In this case, [tex]\( k = 2 \)[/tex]. So we need to find which table consistently holds the ratio [tex]\( \frac{\text{String}}{\text{Percussion}} = 2 \)[/tex].
Let's analyze each table:
### Table 1:
[tex]\[
\begin{tabular}{|c|c|}
\hline Percussion & String \\
\hline 3 & 2 \\
\hline 6 & 4 \\
\hline 9 & 6 \\
\hline
\end{tabular}
\][/tex]
- For (3, 2): [tex]\( \frac{2}{3} \neq 2 \)[/tex]
- For (6, 4): [tex]\( \frac{4}{6} = \frac{2}{3} \neq 2 \)[/tex]
- For (9, 6): [tex]\( \frac{6}{9} = \frac{2}{3} \neq 2 \)[/tex]
### Table 2:
[tex]\[
\begin{tabular}{|c|c|}
\hline Percussion & String \\
\hline 5 & 2 \\
\hline 10 & 4 \\
\hline 15 & 6 \\
\hline
\end{tabular}
\][/tex]
- For (5, 2): [tex]\( \frac{2}{5} \neq 2 \)[/tex]
- For (10, 4): [tex]\( \frac{4}{10} = \frac{2}{5} \neq 2 \)[/tex]
- For (15, 6): [tex]\( \frac{6}{15} = \frac{2}{5} \neq 2 \)[/tex]
### Table 3:
[tex]\[
\begin{tabular}{|c|c|}
\hline Percussion & String \\
\hline 1 & 2 \\
\hline 2 & 4 \\
\hline 3 & 6 \\
\hline
\end{tabular}
\][/tex]
- For (1, 2): [tex]\( \frac{2}{1} = 2 \)[/tex]
- For (2, 4): [tex]\( \frac{4}{2} = 2 \)[/tex]
- For (3, 6): [tex]\( \frac{6}{3} = 2 \)[/tex]
### Table 4:
[tex]\[
\begin{tabular}{|c|c|}
\hline Percussion & String \\
\hline 4 & 2 \\
\hline 8 & 4 \\
\hline 12 & 6 \\
\hline
\end{tabular}
\][/tex]
- For (4, 2): [tex]\( \frac{2}{4} = \frac{1}{2} \neq 2 \)[/tex]
- For (8, 4): [tex]\( \frac{4}{8} = \frac{1}{2} \neq 2 \)[/tex]
- For (12, 6): [tex]\( \frac{6}{12} = \frac{1}{2} \neq 2 \)[/tex]
After evaluating all the tables, we find that Table 3 consistently shows a constant of proportionality of 2 for the ratio of string instruments to percussion instruments. Therefore, the correct table is:
[tex]\[
\begin{tabular}{|c|c|}
\hline Percussion & String \\
\hline 1 & 2 \\
\hline 2 & 4 \\
\hline 3 & 6 \\
\hline
\end{tabular}
\][/tex]
So, the correct answer is Table 3.
