Identify the table that demonstrates a proportional relationship.(1 point)

To identify a table that demonstrates a proportional relationship, we need to check if the ratio \( \frac{y}{x} \) is constant across all pairs in the table.

Let's analyze each table:

1. First table:

   x     y
   2    10
   5    20
   6    30
   

   Ratios:

   • \( \frac{10}{2} = 5 \)
   • \( \frac{20}{5} = 4 \)
   • \( \frac{30}{6} = 5 \)

   Since the ratios are not constant, this table does not show a proportional relationship.

2. Second table:

   x     y
   8    2
   12   3
   40   4
   

   Ratios:

   • \( \frac{2}{8} = 0.25 \)
   • \( \frac{3}{12} = 0.25 \)
   • \( \frac{4}{40} = 0.1 \)

   Since the ratios are not constant, this table does not show a proportional relationship.

3. Third table:

   x     y
   10    2
   15    3
   40    8
   

   Ratios:

   • \( \frac{2}{10} = 0.2 \)
   • \( \frac{3}{15} = 0.2 \)
   • \( \frac{8}{40} = 0.2 \)

   Since all ratios are equal to 0.2, this table shows a proportional relationship.

4. Fourth table:

   x     y
   4    2
   9    3
   16   4
   

   Ratios:

   • \( \frac{2}{4} = 0.5 \)
   • \( \frac{3}{9} \approx 0.333 \)
   • \( \frac{4}{16} = 0.25 \)

   Since the ratios are not constant, this table does not show a proportional relationship.

Conclusion

The third table:

x     y
10    2
15    3
40    8

demonstrates a proportional relationship.