Answer :
To determine which table has a constant of proportionality between [tex]\(y\)[/tex] and [tex]\(x\)[/tex] of [tex]\(\frac{1}{6}\)[/tex], we need to check if the relationship between [tex]\(x\)[/tex] and [tex]\(y\)[/tex] in each table is described by the equation [tex]\( y = \frac{1}{6}x \)[/tex].
### Table A:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 15 & 5 \\ 19 \frac{1}{2} & 6 \frac{1}{2} \\ 36 & 12 \\ \hline \end{array} \][/tex]
Checking each pair:
1. For [tex]\( x = 15 \)[/tex], [tex]\( y = 5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 15 = \frac{15}{6} = 2.5 \quad \text{(not 5)} \][/tex]
2. For [tex]\( x = 19 \frac{1}{2} = 19.5 \)[/tex], [tex]\( y = 6 \frac{1}{2} = 6.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 19.5 = \frac{19.5}{6} \approx 3.25 \quad \text{(not 6.5)} \][/tex]
3. For [tex]\( x = 36 \)[/tex], [tex]\( y = 12 \)[/tex]:
[tex]\[ \frac{1}{6} \times 36 = \frac{36}{6} = 6 \quad \text{(not 12)} \][/tex]
### Table B:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 12 & 2 \\ 13 \frac{1}{2} & 3 \frac{1}{2} \\ 24 & 14 \\ \hline \end{array} \][/tex]
Checking each pair:
1. For [tex]\( x = 12 \)[/tex], [tex]\( y = 2 \)[/tex]:
[tex]\[ \frac{1}{6} \times 12 = \frac{12}{6} = 2 \quad \text{(Correct)} \][/tex]
2. For [tex]\( x = 13 \frac{1}{2} = 13.5 \)[/tex], [tex]\( y = 3 \frac{1}{2} = 3.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 13.5 = \frac{13.5}{6} = 2.25 \quad \text{(not 3.5)} \][/tex]
3. For [tex]\( x = 24 \)[/tex], [tex]\( y = 14 \)[/tex]:
[tex]\[ \frac{1}{6} \times 24 = \frac{24}{6} = 4 \quad \text{(not 14)} \][/tex]
### Table C:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 18 & 3 \\ 27 & 4 \frac{1}{2} \\ 33 & 5 \frac{1}{2} \\ \hline \end{array} \][/tex]
Checking each pair:
1. For [tex]\( x = 18 \)[/tex], [tex]\( y = 3 \)[/tex]:
[tex]\[ \frac{1}{6} \times 18 = \frac{18}{6} = 3 \quad \text{(Correct)} \][/tex]
2. For [tex]\( x = 27 \)[/tex], [tex]\( y = 4 \frac{1}{2} = 4.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 27 = \frac{27}{6} = 4.5 \quad \text{(Correct)} \][/tex]
3. For [tex]\( x = 33 \)[/tex], [tex]\( y = 5 \frac{1}{2} = 5.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 33 = \frac{33}{6} = 5.5 \quad \text{(Correct)} \][/tex]
From this, we can conclude which table follows the given constant of proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
- Table A does not satisfy the condition.
- Table B does not satisfy the condition.
- Table C satisfies the condition.
Therefore, the table that has a constant of proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] of [tex]\( \frac{1}{6} \)[/tex] is:
[tex]\[ \boxed{\text{Table C}} \][/tex]
### Table A:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 15 & 5 \\ 19 \frac{1}{2} & 6 \frac{1}{2} \\ 36 & 12 \\ \hline \end{array} \][/tex]
Checking each pair:
1. For [tex]\( x = 15 \)[/tex], [tex]\( y = 5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 15 = \frac{15}{6} = 2.5 \quad \text{(not 5)} \][/tex]
2. For [tex]\( x = 19 \frac{1}{2} = 19.5 \)[/tex], [tex]\( y = 6 \frac{1}{2} = 6.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 19.5 = \frac{19.5}{6} \approx 3.25 \quad \text{(not 6.5)} \][/tex]
3. For [tex]\( x = 36 \)[/tex], [tex]\( y = 12 \)[/tex]:
[tex]\[ \frac{1}{6} \times 36 = \frac{36}{6} = 6 \quad \text{(not 12)} \][/tex]
### Table B:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 12 & 2 \\ 13 \frac{1}{2} & 3 \frac{1}{2} \\ 24 & 14 \\ \hline \end{array} \][/tex]
Checking each pair:
1. For [tex]\( x = 12 \)[/tex], [tex]\( y = 2 \)[/tex]:
[tex]\[ \frac{1}{6} \times 12 = \frac{12}{6} = 2 \quad \text{(Correct)} \][/tex]
2. For [tex]\( x = 13 \frac{1}{2} = 13.5 \)[/tex], [tex]\( y = 3 \frac{1}{2} = 3.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 13.5 = \frac{13.5}{6} = 2.25 \quad \text{(not 3.5)} \][/tex]
3. For [tex]\( x = 24 \)[/tex], [tex]\( y = 14 \)[/tex]:
[tex]\[ \frac{1}{6} \times 24 = \frac{24}{6} = 4 \quad \text{(not 14)} \][/tex]
### Table C:
[tex]\[ \begin{array}{|c|c|} \hline x & y \\ \hline 18 & 3 \\ 27 & 4 \frac{1}{2} \\ 33 & 5 \frac{1}{2} \\ \hline \end{array} \][/tex]
Checking each pair:
1. For [tex]\( x = 18 \)[/tex], [tex]\( y = 3 \)[/tex]:
[tex]\[ \frac{1}{6} \times 18 = \frac{18}{6} = 3 \quad \text{(Correct)} \][/tex]
2. For [tex]\( x = 27 \)[/tex], [tex]\( y = 4 \frac{1}{2} = 4.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 27 = \frac{27}{6} = 4.5 \quad \text{(Correct)} \][/tex]
3. For [tex]\( x = 33 \)[/tex], [tex]\( y = 5 \frac{1}{2} = 5.5 \)[/tex]:
[tex]\[ \frac{1}{6} \times 33 = \frac{33}{6} = 5.5 \quad \text{(Correct)} \][/tex]
From this, we can conclude which table follows the given constant of proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex]:
- Table A does not satisfy the condition.
- Table B does not satisfy the condition.
- Table C satisfies the condition.
Therefore, the table that has a constant of proportionality between [tex]\( y \)[/tex] and [tex]\( x \)[/tex] of [tex]\( \frac{1}{6} \)[/tex] is:
[tex]\[ \boxed{\text{Table C}} \][/tex]
