Question

Which table shows a proportional relationship between x and y?

A.
x y
2 9
4 18
6 27
8 36
10 45
B.
x y
2 2
4 8
6 18
8 32
10 50
C.
x y
1 4
2 7
3 10
4 13
5 16
D.
x y
1 5
2 6
3 7
4 8
5 9

Answer 1

A proportional relationship means that the ratio \( \frac{y}{x} \) remains constant for all pairs of \( (x, y) \).

Let’s analyze each table:

**Table A:**
```
x | y
----------
2 | 9 → \(\frac{9}{2} = 4.5\)
4 | 18 → \(\frac{18}{4} = 4.5\)
6 | 27 → \(\frac{27}{6} = 4.5\)
8 | 36 → \(\frac{36}{8} = 4.5\)
10 | 45 → \(\frac{45}{10} = 4.5\)
```
The ratios are all equal (\(4.5\)), so Table A shows a proportional relationship.

**Table B:**
```
x | y
----------
2 | 2 → \(\frac{2}{2} = 1\)
4 | 8 → \(\frac{8}{4} = 2\)
6 | 18 → \(\frac{18}{6} = 3\)
8 | 32 → \(\frac{32}{8} = 4\)
10 | 50 → \(\frac{50}{10} = 5\)
```
The ratios are not equal, so Table B does not show a proportional relationship.

**Table C:**
```
x | y
----------
1 | 4 → \(\frac{4}{1} = 4\)
2 | 7 → \(\frac{7}{2} = 3.5\)
3 | 10 → \(\frac{10}{3} \approx 3.33\)
4 | 13 → \(\frac{13}{4} = 3.25\)
5 | 16 → \(\frac{16}{5} = 3.2\)
```
The ratios are not equal, so Table C does not show a proportional relationship.

**Table D:**
```
x | y
----------
1 | 5 → \(\frac{5}{1} = 5\)
2 | 6 → \(\frac{6}{2} = 3\)
3 | 7 → \(\frac{7}{3} \approx 2.33\)
4 | 8 → \(\frac{8}{4} = 2\)
5 | 9 → \(\frac{9}{5} = 1.8\)
```
The ratios are not equal, so Table D does not show a proportional relationship.

**Conclusion:** Only Table A shows a proportional relationship between \( x \) and \( y \).