Question

Which table shows a proportional relationship between x and y?

Responses

x 2 3 4 5
y 7 9 11 13
x 2 3 4 5 y 7 9 11 13

x 3 6 8 10
y 12 15 18 20
x 3 6 8 10 y 12 15 18 20

x 1 2 7 8
y 0.5 1 3.5 4
x 1 2 7 8 y 0.5 1 3.5 4

x 3 7.5 15 20
y 1 2.5 4 6
x 3 7.5 15 20 y 1 2.5 4 6

Answer 1

To determine which table shows a proportional relationship between \(x\) and \(y\), we can check if the ratio \( \frac{y}{x} \) is constant for each pair of values. That is, for a proportional relationship, each value of \(y\) divided by its corresponding value of \(x\) should yield the same result.

Let's analyze each of the options:

1. **First table:**
- \( \frac{7}{2} = 3.5 \)
- \( \frac{9}{3} = 3.0 \)
- \( \frac{11}{4} = 2.75 \)
- \( \frac{13}{5} = 2.6 \)

(Not a constant ratio; therefore, not proportional)

2. **Second table:**
- \( \frac{12}{3} = 4 \)
- \( \frac{15}{6} = 2.5 \)
- \( \frac{18}{8} = 2.25 \)
- \( \frac{20}{10} = 2.0 \)

(Not a constant ratio; therefore, not proportional)

3. **Third table:**
- \( \frac{0.5}{1} = 0.5 \)
- \( \frac{1}{2} = 0.5 \)
- \( \frac{3.5}{7} = 0.5 \)
- \( \frac{4}{8} = 0.5 \)

(Constant ratio of \(0.5\); therefore, this is proportional)

4. **Fourth table:**
- \( \frac{1}{3} \approx 0.333 \)
- \( \frac{2.5}{7.5} \approx 0.333 \)
- \( \frac{4}{15} \approx 0.267 \)
- \( \frac{6}{20} = 0.3 \)

(Not a constant ratio; therefore, not proportional)

From this analysis, the **third table** shows a proportional relationship between \(x\) and \(y\).