Sure, let's analyze each table step-by-step to determine whether the ratios of ingredient [tex]\( Y \)[/tex] to ingredient [tex]\( X \)[/tex] remain constant.
### Table 1:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Servings} & X & Y \\
\hline
1 & 1 & 2 \\
\hline
2 & 2 & 3 \\
\hline
3 & 3 & 4 \\
\hline
\end{array}
\][/tex]
First, we'll calculate the ratios [tex]\( \frac{Y}{X} \)[/tex]:
1. [tex]\( \frac{2}{1} = 2 \)[/tex]
2. [tex]\( \frac{3}{2} = 1.5 \)[/tex]
3. [tex]\( \frac{4}{3} \approx 1.33 \)[/tex]
The ratios are 2, 1.5, and approximately 1.33; these are not constant.
### Table 2:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Servings} & X & Y \\
\hline
1 & 1 & 2 \\
\hline
2 & 2 & 4 \\
\hline
3 & 3 & 8 \\
\hline
\end{array}
\][/tex]
Next, we'll calculate the ratios [tex]\( \frac{Y}{X} \)[/tex]:
1. [tex]\( \frac{2}{1} = 2 \)[/tex]
2. [tex]\( \frac{4}{2} = 2 \)[/tex]
3. [tex]\( \frac{8}{3} \approx 2.67 \)[/tex]
The ratios are 2, 2, and approximately 2.67; these are not constant.
### Table 3:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Servings} & X & Y \\
\hline
1 & 1 & 2 \\
\hline
2 & 2 & 3 \\
\hline
3 & 3 & 5 \\
\hline
\end{array}
\][/tex]
Next, we'll calculate the ratios [tex]\( \frac{Y}{X} \)[/tex]:
1. [tex]\( \frac{2}{1} = 2 \)[/tex]
2. [tex]\( \frac{3}{2} = 1.5 \)[/tex]
3. [tex]\( \frac{5}{3} \approx 1.67 \)[/tex]
The ratios are 2, 1.5, and approximately 1.67; these are not constant.
### Table 4:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Servings} & X & Y \\
\hline
1 & 1 & 2 \\
\hline
2 & 2 & 4 \\
\hline
\end{array}
\][/tex]
Finally, we'll calculate the ratios [tex]\( \frac{Y}{X} \)[/tex]:
1. [tex]\( \frac{2}{1} = 2 \)[/tex]
2. [tex]\( \frac{4}{2} = 2 \)[/tex]
The ratios are 2 and 2; these are constant.
Thus, the table where the ingredient ratios [tex]\( Y \)[/tex] to [tex]\( X \)[/tex] remain constant is Table 4. Therefore, the correct table index is 4.
