Question

The following table represents the total cost, in dollars (y) to join a gym for x number of months. The cost includes a one-time joining fee of 10$. Does the data in the table represent a proportional relationship? How do you know?

Answer 1

The table does not represent a proportional relationship because the ratio of the y-values to the corresponding x-value at each data point is not constant

y/x ≠ Constant

Explanation:

The table of values for the total cost in dollars (y) to join a gym for "x" number of months is presented as follows;

x; 1, 2, 3, 4, 5

y; 25, 40, 55, 70, 85

The rate of change of the data in the table is given by the following formula;

`Slope, \, m =(y_(2)-y_(1))/(x_(2)-x_(1))`

Between the 1st and the 3rd point, we have;

m = (55 - 25)/(3 - 1) = 15

Between the 1st and the 5th point, we have;

m = (85 - 25)/(5 - 1) = 15

Therefore, the rate of change of the y values per unit change of the x-value is 15

Therefore, the "x" and y-values have a linear relationship

However, for a proportional relationship, we have;

y/x = Constant

At the 1st point, we have;

25/1 = 25

At the 3rd point, we have;

55/3 = 18.
`\overline 3`

∴ y/x is not constant and the the data in the table does not represent a proportional relationship.