Luther wants to get in shape, so he joins a local gym, Buff and Tough. For membership, the gym charges a registration fee plus monthly dues.

Question

This table shows the relationship between the length of Luther's gym membership (in months), x, and the total cost of his membership (in dollars), y.
x (months) y (dollars)
2 $49
3 $65
4 $81
5 $97
According to the values in the table, do x and y have a proportional relationship?
yes
no

Answer 1

To determine if \( x \) (the number of months) and \( y \) (the total cost in dollars) have a proportional relationship, we need to check if the ratio \( \frac{y}{x} \) is constant for all values of \( x \).

Let's calculate the ratio for each pair of \( (x, y) \):

1. For \( x = 2 \), \( y = 49 \): \[ \frac{49}{2} = 24.5 \]

2. For \( x = 3 \), \( y = 65 \): \[ \frac{65}{3} \approx 21.67 \]

3. For \( x = 4 \), \( y = 81 \): \[ \frac{81}{4} = 20.25 \]

4. For \( x = 5 \), \( y = 97 \): \[ \frac{97}{5} = 19.4 \]

Now we can see that the ratios are not constant:

• \( \frac{49}{2} = 24.5 \)
• \( \frac{65}{3} \approx 21.67 \)
• \( \frac{81}{4} = 20.25 \)
• \( \frac{97}{5} = 19.4 \)

Since the ratio \( \frac{y}{x} \) changes with different values of \( x \), \( x \) and \( y \) do not have a proportional relationship.

Thus, the answer is no.