To determine which table represents a direct proportion, we need to check if there is a consistent ratio between the values of $x$ and $y$ in each table.

A relationship is a direct proportion if $y = kx$, where $k$ (the constant of proportionality) remains the same for each pair of $x$ and $y$.

Analyzing Each Table:

1. Table A:

   • Ratios: $\frac{y}{x} = \frac{2}{5}, \frac{4}{10}, \frac{6}{15}, \frac{8}{20}$.
   • All ratios are $\frac{2}{5}$, which is constant. Therefore, Table A shows direct proportion.

2. Table B:

   • Ratios: $\frac{y}{x} = \frac{2}{1}, \frac{5}{2}, \frac{7}{3}, \frac{8}{4}$.
   • Ratios are not consistent. Thus, Table B does not represent direct proportion.

3. Table C:

   • Ratios: $\frac{y}{x} = \frac{1}{2}, \frac{3}{4}, \frac{5}{6}, \frac{7}{8}$.
   • Ratios are not consistent. Thus, Table C does not represent direct proportion.

4. Table D:

   • Ratios: $\frac{y}{x} = \frac{4}{2}, \frac{8}{4}, \frac{12}{6}, \frac{16}{8}$.
   • All ratios are $2$, which is constant. Therefore, Table D shows direct proportion.

Conclusion:

Both Table A and Table D represent direct proportion, but since the question is asking for a single answer, if only one choice can be made, you would generally select Table A (as it is likely meant to be the primary answer in multiple-choice settings).

Would you like more details on this analysis or have any questions?

Further Questions:

1. What is the constant of proportionality for each table?
2. How can you identify inverse proportion in similar tables?
3. Why does a constant ratio confirm direct proportion?
4. Can we graph these tables to visualize direct proportion?
5. How would the values change if these tables represented inverse proportions?

Tip:

When checking for direct proportion, calculating the ratio $\frac{y}{x}$ for each pair helps confirm if it remains constant.