Question

x - Number of Days y - Cost
6 2
9 3
24 8
3 1
Part A

Is this a proportional relationship?




Part B

If this is a proportional relationship, what is the constant? Write N/A if it is not proportional.

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Part C
​If this is a proportional relationship, write a direct proportion equation. Write N/A if it is not proportional.

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Answer 1

To determine whether the given relationship between the number of days (x) and cost (y) is proportional, we can analyze the values provided.

### Part A: Is this a proportional relationship?

A relationship is considered proportional if the ratio of \( y \) to \( x \) (i.e., \( \frac{y}{x} \)) is the same for all pairs of values.

Let's calculate \( \frac{y}{x} \) for each pair:

1. For (6, 2): \( \frac{2}{6} = \frac{1}{3} \)
2. For (9, 3): \( \frac{3}{9} = \frac{1}{3} \)
3. For (24, 8): \( \frac{8}{24} = \frac{1}{3} \)
4. For (3, 1): \( \frac{1}{3} = \frac{1}{3} \)

Since the ratio \( \frac{y}{x} \) is constant ( \( \frac{1}{3} \) ) across all pairs, this is a proportional relationship.

### Part B: If this is a proportional relationship, what is the constant?

The constant of proportionality can be determined from the consistent ratio we calculated.

**Constant:** \( \frac{1}{3} \)

### Part C: If this is a proportional relationship, write a direct proportion equation.

The direct proportion equation can be expressed in the form \( y = kx \), where \( k \) is the constant of proportionality.

Using the constant we found:

**Direct proportion equation:** \( y = \frac{1}{3}x \)

In summary:
- **Part A:** Yes, it is a proportional relationship.
- **Part B:** \( \frac{1}{3} \)
- **Part C:** \( y = \frac{1}{3}x \)