Answer :
To determine if there is a proportional relationship between the variables \( x \) and \( y \) given in the table, we need to check if the ratio \(\frac{y}{x}\) is constant for all pairs \((x, y)\).
We'll evaluate the ratio for each pair of values:
1. For \( x = \frac{1}{4} \) and \( y = 3 \):
[tex]\[ \frac{y}{x} = \frac{3}{\frac{1}{4}} = 3 \times 4 = 12 \][/tex]
2. For \( x = \frac{2}{4} \) (which simplifies to \(\frac{1}{2}\)) and \( y = 6 \):
[tex]\[ \frac{y}{x} = \frac{6}{\frac{1}{2}} = 6 \times 2 = 12 \][/tex]
3. For \( x = \frac{3}{4} \) and \( y = 9 \):
[tex]\[ \frac{y}{x} = \frac{9}{\frac{3}{4}} = 9 \times \frac{4}{3} = 12 \][/tex]
Since the ratio \(\frac{y}{x}\) is constant and is equal to 12 for all given pairs of \((x, y)\), we can conclude that there is a proportional relationship between \( x \) and \( y \).
Therefore, the answer is:
(A) Yes
We'll evaluate the ratio for each pair of values:
1. For \( x = \frac{1}{4} \) and \( y = 3 \):
[tex]\[ \frac{y}{x} = \frac{3}{\frac{1}{4}} = 3 \times 4 = 12 \][/tex]
2. For \( x = \frac{2}{4} \) (which simplifies to \(\frac{1}{2}\)) and \( y = 6 \):
[tex]\[ \frac{y}{x} = \frac{6}{\frac{1}{2}} = 6 \times 2 = 12 \][/tex]
3. For \( x = \frac{3}{4} \) and \( y = 9 \):
[tex]\[ \frac{y}{x} = \frac{9}{\frac{3}{4}} = 9 \times \frac{4}{3} = 12 \][/tex]
Since the ratio \(\frac{y}{x}\) is constant and is equal to 12 for all given pairs of \((x, y)\), we can conclude that there is a proportional relationship between \( x \) and \( y \).
Therefore, the answer is:
(A) Yes
