Answer :
To determine which option represents a proportional relationship between variables \( x \) and \( y \), we need to recall the definition of a proportional relationship. Two variables are proportional if one variable is always a constant multiple of the other. Mathematically, this can be expressed as \( y = k \times x \), where \( k \) is the constant of proportionality.
Let's evaluate each given option:
- Option (A): \( x = \frac{1}{8} \times y \)
This option suggests that \( x \) is a constant multiple of \( y \), with the constant of proportionality being \( \frac{1}{8} \). This fits the definition of a proportional relationship where one variable is directly proportional to the other.
- Option (B): \( x = \frac{1}{8} \times \frac{1}{y} \)
Here, \( x \) is proportional to the reciprocal of \( y \), not to \( y \) itself. This is not a direct proportional relationship because multiplying by the reciprocal does not fit the form \( y = k \times x \).
- Option (C): \( x = \frac{1}{8} - y \)
This equation represents a linear relationship, but not a proportional one. In a proportional relationship, the equation should involve multiplication by a constant, not subtraction.
- Option (D): \( \frac{1}{8} \times \frac{1}{x} = y \)
This suggests that \( y \) is proportional to the reciprocal of \( x \). This does not represent a direct proportional relationship between \( x \) and \( y \) since it involves reciprocals.
- Option (E): \( x + y = 8 \)
This represents a linear relationship where the sum of \( x \) and \( y \) is constant. However, it does not fit the form \( y = k \times x \), as required for a proportional relationship because it involves addition instead of multiplication by a constant.
Considering the definitions and examining each option, the correct answer is:
Option (A): \( x = \frac{1}{8} \times y \)
This equation demonstrates a proportional relationship where [tex]\( x \)[/tex] is consistently [tex]\( \frac{1}{8} \)[/tex] of [tex]\( y \)[/tex].
Let's evaluate each given option:
- Option (A): \( x = \frac{1}{8} \times y \)
This option suggests that \( x \) is a constant multiple of \( y \), with the constant of proportionality being \( \frac{1}{8} \). This fits the definition of a proportional relationship where one variable is directly proportional to the other.
- Option (B): \( x = \frac{1}{8} \times \frac{1}{y} \)
Here, \( x \) is proportional to the reciprocal of \( y \), not to \( y \) itself. This is not a direct proportional relationship because multiplying by the reciprocal does not fit the form \( y = k \times x \).
- Option (C): \( x = \frac{1}{8} - y \)
This equation represents a linear relationship, but not a proportional one. In a proportional relationship, the equation should involve multiplication by a constant, not subtraction.
- Option (D): \( \frac{1}{8} \times \frac{1}{x} = y \)
This suggests that \( y \) is proportional to the reciprocal of \( x \). This does not represent a direct proportional relationship between \( x \) and \( y \) since it involves reciprocals.
- Option (E): \( x + y = 8 \)
This represents a linear relationship where the sum of \( x \) and \( y \) is constant. However, it does not fit the form \( y = k \times x \), as required for a proportional relationship because it involves addition instead of multiplication by a constant.
Considering the definitions and examining each option, the correct answer is:
Option (A): \( x = \frac{1}{8} \times y \)
This equation demonstrates a proportional relationship where [tex]\( x \)[/tex] is consistently [tex]\( \frac{1}{8} \)[/tex] of [tex]\( y \)[/tex].
