Which of the following tables represents a proportional relationship?

A.
x 0 1 2 3
y 0 4 5 6

B.
x 0 2 4 6
y 2 4 8 12

C.
x 0 1 2 3
y 0 3 6 9

D.
x 0 5 10 15
y 0 10 15 20

1 answer

A proportional relationship can be identified by the consistency of the ratio \( \frac{y}{x} \) across all pairs of values (excluding \( x = 0 \), where \( y \) should also be \( 0 \)).

Let's analyze each table:

A.

  • \( \frac{0}{0} \) (undefined)
  • \( \frac{4}{1} = 4 \)
  • \( \frac{5}{2} = 2.5 \)
  • \( \frac{6}{3} = 2 \)
    The ratios are not consistent.

B.

  • \( \frac{2}{2} = 1 \)
  • \( \frac{4}{4} = 1 \)
  • \( \frac{8}{6} = 1.33 \)
  • \( \frac{12}{6} = 2 \)
    The ratios are not consistent.

C.

  • \( \frac{0}{0} \) (undefined)
  • \( \frac{3}{1} = 3 \)
  • \( \frac{6}{2} = 3 \)
  • \( \frac{9}{3} = 3 \)
    The ratios are consistent and equal to \( 3 \).

D.

  • \( \frac{0}{0} \) (undefined)
  • \( \frac{10}{5} = 2 \)
  • \( \frac{15}{10} = 1.5 \)
  • \( \frac{20}{15} = 1.33 \)
    The ratios are not consistent.

Based on this analysis, the only option that represents a proportional relationship is:

C.
\( y = 3x \) (which is a constant multiple of \( x \)).