Question

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.
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**.